Formulas and Chapter Summaries

Physics 123

Ross L Spencer Harold T Stokes

With additions from the textbook by Allred

1

Serway Chapter 14 Fluid Mechanics

Density and Pressure

The density of a material is defined to be the ratio between the mass of a small piece of it divided

by the volume of the small piece

m

V

Water has a density of 1000 kgm3 while air has a density of 13 kgm3 at sea level and 11

kgm3 in Provo Sea water has a density of 103 of water Ethanol (ethyl alcohol) is 0806

The pressure exerted on a small surface is defined to be the force applied to it divided by its

area

FP

A

At sea level atmospheric pressure is 1013 times 105 pascals where 1 Pa = 1 Nm2 Here in Provo

atmospheric pressure is about 86 times 104 Pa Sometimes pressure will be expressed in pounds-per-

square-inch (psi) or sometimes in torr 1 psi= 6895 times 103 Pa and 1 torr= 1333 Pa

Hydrostatics

If h measures distance below the surface of a liquid then the pressure as a function of the depth h

is given by

oP P gh

where Po is the pressure at the surface of the liquid usually atmospheric pressure In 123 liquids

are imcompressible ρ does not vary with depth and g = 980 m2s

Pascalrsquos Principle

A change in the pressure applied to an enclosed fluid is transmitted undiminished to every

portion of the fluid and to the walls of the containing vessel (Blaise Pascal 1652)

Archimedes Principle

A body wholly or partially immersed in a fluid will be buoyed up by a force equal to the weight

of the fluid that it displaces (Archimedes ca 220 BC) This principle is a handy computational

tool but it does not explain the real reason that objects in a fluid experience an upward force

The reason is that the pressure on the bottom of the object is greater than the pressure at the top

so there is a net upward pressure force The buoyant force is given by the formula

B Vg

2

where ρ is the density of the fluid V is the submerged volume of the object and g = 980 ms2 is

the acceleration of gravity If the object is floating on the liquid then the submerged volume V is

only that portion which is below the liquid surface

Equation of Continuity

For a fluid whose density is constant the rate of fluid flow or volume flux Φ measured in units

of m3s for a cross-section of fluid streamlines having area A and flow speed v is given by

Av

Bernoullirsquos Equation

The principle of conservation of energy for steady fluid flow states Along each streamline of the

flow the following quantity has a constant value

21

2P v gy

where y is measured upward against gravity Note that y in this equation and h in the hydrostatic

pressure equation have opposite signs This law works for gases and liquids The demos are

mostly for gases and are set up so that ρ does not vary

3

Serway Chapter 16

Longitudinal and Transverse Waves

A longitudinal wave is one in which the direction of vibration of the medium and the direction of

propagation of the wave are in the same direction Sound waves in the air in water and in solids

are longitudinal waves

A transverse wave is one in which the direction of vibration of the medium is perpendicular to

the direction of propagation of the wave Ocean waves waves on ropes and electromagnetic

waves are transverse waves

Wavelength and Wavenumber

The wavelength λ is the period in space of the wave ie it is the distance from one wave crest to

the next The wavenumber k (units reciprocal length) is related to the wavelength by the formula

2k

Frequency and Angular Frequency

The period T is the period in time of the wave ie it is the time between the arrival of one wave

crest and the next at some point in space The frequency f is the number of crests that arrive per

second which is just 1T and the angular frequency ω is 2π times the frequency Both have units

of per second (reciprocal time)

1 22f f

T T

Wave Speed

The wave speed of a wave is defined to be the the speed at which the wave crests or troughs

move through the medium This speed is related to the frequency and the wavelength by the

formulas

v fk

where k is the wavenumber of the wave v is also called the phase velocity

Traveling Waves

Traveling waves are oscillations in which the wave crests and troughs glide smoothly through the

medium Waves on oceans and lakes are traveling waves The mathematical form of a harmonic

(sinusoidal) traveling wave is

4

siny x t A kx t

where A is the amplitude of the wave As usual ω and k are related by the relation ω = kv where

v is the wave speed the speed at which the crests travel the minus sign means that the wave is

traveling to the right towards higher values of x φ is the phase and constrols where the maxima

and minima occur

Speed of Transverse Waves on a Rope

The velocity of transverse waves on a perfectly flexible rope is given by

Tv

where T is the tension in the rope and where μ is its linear mass density (mass per unit length)

Common Test Questions

A common question is to be given an equation for a wave and to be asked what is amplitude

velocity etc Another common type of question is to be given facts about a wave like its

direction velocity and magnitude at a given place and time and to be asked which of several

equations correctly describe the wave

5

Serway Chapter 17

Sound Speed in Solids Liquids and Gases

The speed of sound in a liquid or solid of bulk modulus B and volume mass density ρ

Bv

The speed of sound in air is

331 1 273v m s T C

The speed of sound in air at room temperature is about 343 ms Notice it doesnrsquot depend on

pressure or frequency

Power Intensity and Loudness

The intensity of a sound wave is defined to be the power per unit area ie wattsm2 in the wave

For spherical waves traveling away from a small source emitting waves with average power Рav

the intensity falls off with distance from the source r according to the inverse-square law

av24

Ir

Hence the intensity I2 at distance r2 is related to the intensity I1 at a different distance r1 by

21 2

22 1

I r

I r

Experiments on human hearing have shown that we hear intensity differences logarithmically so

the decibel loudness scale for sound intensity was invented The loudness β of a sound in

decibels is related to its intensity I in Wm2 by the formula

1010logo

I

I

Where Io is a sound intensity near the threshold of hearing defined to be Io = 10-12 Wm2 Note

that on this scale a sound is made 10 times more intense by adding 10 decibels Remember that

intensity is proportional to velocity squared Amplitude is the A is the previous chapterrsquos

equation for a traveling wave If you make the A 10 times as much the intensity will increase by

100 times (20 dB)

Doppler Effect

If sound waves are traveling through a medium and if either the receiver of the waves or the

source of waves is moving then the frequency received is related to the frequency emitted by

6

0

s

v vf f

v v

where fprime is the frequency detected by the observer f is the frequency emitted by the source v is

the speed of the waves vo is the speed of the observer and vs is the speed of the source This

formula assumes that the source and receiver are either moving directly toward each other or

directly away from each other To know which signs to use remember that when observer and

source approach each other the observed frequency is higher while if they move away from each

other it is lower Just examine the signs in the formula and make the answer come out right

For electromagnetic waves (light radio waves X-rays) traveling in vacuum Einsteinrsquos theory of

relativity (and careful experiments) show that the Doppler shift is given by

r

r

c vf f

c v

where vr is the relative speed between the source and the observer

For all kinds of waves (sound light etc) if the relative speed of the source and the observer is

small compared to the speed of the waves then there is a simple approximation to the Doppler

effect For example if the relative speed is 1 of the wave speed then the frequency shifts by

1 Remember that this is only an approximation

Shock Waves

When an object moves through a medium at a speed greater than the speed of waves a V-shaped

shock wave is produced The V-shaped wake behind a speeding boat is a good example of this

effect and the cone of sonic-boom behind a supersonic aircraft is another The angle that V-line

makes with the direction of travel of the source is given by

sins

v

v

where v is the wave speed and vs is the source speed

7

Serway Chapter 18

Principle of Linear Superposition

We say that a system obeys the principle of linear superposition if two or more different motions

of the system can simply be added together to find the net motion of the system Light waves

obey this principle as they propagate through the air as can be seen by shining two flashlights so

that their beams cross The beams propagate along without affecting each other (Light sabers are

a spectacular but unfortunately fictional example of systems that do not obey the principle of

superposition) Wave pulses on an ideal rope also obey this principle two different pulses pass

through each other without change Standing waves are an example of this effect being simply

the linear superposition of two traveling waves of the same frequency but moving in opposite

directions Light waves in matter do not always obey this principle For instance two powerful

laser beams could be made to cross in a piece of glass in such a way that their combined heating

effect in the crossing region could melt the glass and scatter the beams in complicated ways This

is an example of a nonlinear effect

Interference

When two or more waves are present in the same medium at the same time their net effect may

often be obtained simply by adding them at each point in the medium according to the principle

of linear superposition (Note this wonrsquot work if the medium is nonlinear) When this addition

makes the total amplitude be greater than the individual amplitudes of the various waves we say

that the interference is constructive When the addition produces cancellation and an amplitude

less than the amplitudes of the separate waves we have destructive interference

Standing Waves

A standing wave is the superposition of two identical traveling waves moving in opposite

directions Nodes are places where the two waves perfectly destructively interfere to produce

zero amplitude at all times Anti-nodes are places where the two waves perfectly constructively

interfere to produce an amplitude maximum The distance between nodes is λ2 Standing waves

on a string fixed at both ends have nodes at each end of the string Standing waves in an air

column enclosed in a tube have displacement anti-nodes at open ends of the tube and

displacement nodes at closed ends The frequency of the standing wave with the lowest possible

frequency is called the fundamental frequency Standing waves on strings or in air columns all

have frequencies which are integer multiples of the fundamental frequency and are called

8

harmonics (The fundamental is called the ldquofirst harmonicrdquo)

Beats

Beats are heard when two waves with slightly different frequencies f1 and f2 are combined The

waves constructively interfere for a number of cycles then destructively interfere for a number

of cycles We hear a periodic ldquowah-wahrdquo frequency equal to the difference of the two wave

frequencies

1 2bf f f

Musical Instruments

Musical instruments produce tones by exciting standing waves on strings (violins piano) and in

tubes (trumpet organ) The fundamental frequency of the standing wave is called the pitch of the

tone The pitch of concert A is 440 Hz by definition Two tones are an octave apart if one pitch

has twice the frequency of the other In written music there are 12 intervals in each octave with

the ratio between successive intervals equal to 2112 = 105946 The ratios for each tone in an

octave starting at A and ending at the next higher A are

A A B C C D D E F F G G A

1 10595 11225 11892 12599 13348 14142 14983 15874 16818 17818 18877 2

A musical tone is actually a superposition of the fundamental frequency and the higher

harmonics The tone quality of a musical instrument is determined by the amplitudes of the

various harmonics that it produces A violin and a trumpet can play the same pitch but they

donrsquot sound at all alike to our ears The difference between them is in the various amplitudes of

their harmonics

9

Serway Chapter 19

Temperature

Formally temperature is what is measured by a thermometer Roughly high temperature is what

we call hot and low temperature is what we call cold On the atomic level temperature refers to

the kinetic energy of the molecules A collection of molecules is called ldquohotrdquo if the molecules

have rapid random motion while a collection of molecules is called ldquocoldrdquo if the random motion

is slow When two bodies are placed in close contact with each other they exchange molecular

kinetic energy until they come to the same temperature This is the microscopic picture of the

Zeroth Law of Thermodynamics

Absolute Zero

Absolute zero is the lowest possible temperature that any object can have This is the temperature

at which all of the energy than can be removed an object has been removed (This removable

energy we call thermal energy) There is still motion at absolute zero Electrons continue to orbit

around atomic nuclei and even atoms continue to move about with a small amount of kinetic

energy but this small energy cannot be removed from the object For example at absolute zero

helium is a liquid whose atoms still move and slide past each other

Temperature Scales

Kelvin Scale Absolute zero is at T = 0 K water freezes at T = 27315 K room temperature is

around T = 295 K and water boils at T = 373 K Note we donrsquot use a deg symbol Kelvin is

prefered SI Unit

Celsius Scale Absolute zero is at T = -273degC water freezes at T = 0degC room temperature is

around T = 22degC and water boils at T = 100degC

Fahrenheit Scale Absolute zero is at T = -459degF water freezes at TF = 32degF room temperature

is around T = 72degF and water boils at TF = 212degF TF =18TC + 32 Notice that temperature

differences are the same for the Kelvin and Celsius scales

Thermal Expansion

When materials are heated they usually expand and when they are cooled they usually contract

(Water near freezing is a spectacular counterexample it works the other way around) The

coefficient of linear expansion is defined by the relation

1

i

L

L T

10

where Li is the initial length of a rod of the material and ΔL is the change in its length due to a

small temperature change ΔT The coefficient of volume expansion is defined similarly

1

i

V

V T

where Vi is the initial volume of a piece of material and where ΔV is the change in its volume due

to a small temperature change ΔT

Avogadrorsquos Number (N or NA)

One mole of any substance corresponds to 6022 times 1023 molecules The atomic mass given on the

periodic table (p A32-33 in the text) is the mass in grams of one mole of atoms For example

the atomic mass of oxygen is 15999 gmol The mass of a single atom is given by the same

number in units of atomic mass units (u) For example the mass of an oxygen atom is 15999 u

(on the average)

Ideal Gas Law (an example of an equation of state)

When the molecules of a gas are sufficiently inert and widely separated that interactions between

them are negligible we say that it is an ideal gas The pressure P volume V and temperature T

(in kelvins) of such a gas are State Variables and are related by the ideal gas law

Bor PV=NkPV nRT T

where n is the number of moles of the gas where R is the gas constant

8314 Jmol KR

where N is the number of molecules and where kB is Boltzmannrsquos constant

231380 10 J KBk

It works well for air at atmosphere pressure and even better for partial vaccuums The relative

ease of measuring pressure and the linear relationship between pressure and temperature (if V

and n are held fixed) makes an ideal gas an ideal thermometer Thermometers can be based on

properties of solids or liquids but the behavior of these materials with temperature is more

complicated

11

Serway Chapter 20

Heat

Heat is energy that flows between a system and its environment because of a tempera-

ture difference between them The units of heat are Joules as expected for an energy

Unfortunately there are several competing units of energy They are related by

1 cal 4186 J 1 Cal 4186 J 1 Btu 1054 J

Heat Capacity

There is often a simple linear relation between the heat that flows in or out of part of a

system and the temperature change that results from this energy transfer When this

linear relation holds it is convenient to define the heat capacity C and the specific

heat c as follows

For the entire object the heat Q it absorbs is given by Q = C(Tf - Ti)

For a particular material the specific heat is defined by c = Cm which is the heat

capacity per unit mass so that

Q = mc(Tf ndash Ti)

Note C has units of energy (J or Cal)(Kelvin kg)

Heats of Transformation or Latent Heat Q = plusmn mL

When a substance changes phase from solid to liquid or from liquid to gas it absorbs

heat without a change in temperature The latent heat or heat of transformation is

usually given per unit mass of the substance For example for water the heat of fusion

(melting) is L = 333 kJkg while the heat of vaporization (boiling) is L = 2260 kJkg

Note that heat for boiling is considerably bigger than melting for water You have to

be careful with signs heat is given off (negative) if you go down in temperature and

condense steam

Work

In general the small amount of work done on a system as a force Fon is exerted on it

through a vector displacement dx is given by

on xdW d F

12

But if the displacement is done very slowly (as we always assume in thermodynamics)

then the force exerted on the system and the force exerted by the system are in

balance so the force exerted by the system is ndash Fon In thermodynamics it is more

convenient to talk about the force exerted by the system so we change the above

formula for the work done on the system to

xdW d F

where F is the force exerted by the system This has confused students for more than a

century now but this is the way your book and many other books do it so you are

stuck You will need to memorize the minus sign in this definition of the work to be

able to use your textbook

There are many chances to get signs wrong in this and the next two chapters (Mosiah

2 )

When an external agent changes the volume of a gas at pressure P by a small amount

dV the (small amount) of work done on the system is given by

dW PdV

Notice that this minus sign is just what we need to make dW be positive if the external

agent compresses the gas for then dV is negative If on the other hand the external

agent gives way allowing the gas to expand against it then dV is positive and we say

that the work done on the gas is negative

The work done on the system (eg by the gas in a cylinder) in a thermodynamic

process is the area under the curve in a PV diagram It is positive for compressions

and negative for expansions If the volume of gas remains constant in a process then

no work is done by the gas

Cyclic processes are important For cyclic processes represented by PV diagrams the

magnitude of the net work during one cycle is simply the area enclosed by the cycle

on the diagram Be careful to keep track of signs when you are calculating that

enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes

1

Path A-B B-C C-D D-A A to A net

Q

W

ΔU

ΔS

Internal Energy

The energy stored in a substance is called its internal energy Eint This energy may be

stored as random kinetic energy or as potential energy in each molecule (stretched

chemical bonds electrons in excited states etc) For ideal gases all states with the

same temperature will have the same Eint

First Law of Thermodynamics

The change ΔEint in the internal energy of a system is given by

intE Q W

where Q is the heat absorbed by the system and where W is the work done on the

system Hence if a system absorbs heat (and if Wge0) the internal energy increases

Likewise if the system does work (W on the system is negative) and if Qge0 the

internal energy decreases Potential Pitfall Many times people talk about work done

by the system It is the minus of W on the system Donrsquot get tripped up

Processes

Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated

from the environment A process may be approximately adiabatic if it happens so

rapidly that heat does not have time to enter or leave the system Work + or ndash is done

and ΔEint = W

Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing

on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal

energy and temperature does not change (Note the difference between an adiabatic

process and a free expansion is that NO work is done in the adiabatic free expansion)

Isobaric process The pressure is held fixed ΔP = 0 For example usually the

pressure increases when a gas is heated but if it were allowed to expand during the

2

heating process in just the right way its pressure could remain fixed In isobaric

processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)

Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is

then zero and so we have ΔEint = Q

Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint

so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in

an isothermal process The work done on the gas is then given by

lnf

i

Vi

Vf

VW PdV nRT

V

Heat Conduction

The quantity P is defined to be the rate at which heat flows through an object and is a

power having units of watts It is analogous to electric current which is the rate at

which charge flows through an object If the flow of heat through a slab of length L

and cross-sectional area A is steady in time then P is given by the equation

h cT TdQkA

dt L

P =

where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The

heat flows of course because of this temperature difference The quantity k is called

the thermal conductivity and is a constant that is characteristic of the material It is

analogous to the electrical conductivity h cT TL is sometimes called the temperature

gradient and is written dTd dTdx

R-Values

It is common to have the heat-conducting properties of materials described by their R-

values especially for insulating materials like fiberglass batting The connection

between k and R is R = Lk where L is the material thickness In this country R-values

always have units of 2ft F hourBtu

Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11

(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)

The heat flow rate through a slab of area A is given by

3

h cT TA

R

P

in units of Btuhour Note that A must be in square feet and the temperatures must be

in degrees Fahrenheit

Convection

Convection is the transfer of thermal energy by flow of material For instance a home

furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct

throughout the house instead it quickly pumps warm air to all of the rooms

Generally convection is a much faster way to transfer heat than conduction

Radiation

Electromagnetic radiation can also transfer heat When you warm yourself near a

campfire which has burned itself down into a bed of glowing embers you are

receiving radiant heat from the infrared portion of the electromagnetic spectrum The

rate at which an object emits radiant heat is given by Stefanrsquos law

4AeTP

where P is the radiated power in watts σ is a constant

8 2 45696 10 W m K

A is the surface area of the object in m2 and T is the temperature in kelvins The

constant e is called the emissivity and it varies from substance to substance A perfect

absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =

0 Hence black objects radiate very well while shiny ones do not Also an object that

is hotter than its surroundings radiates more energy than it absorbs whereas an object

that is cooler than its surroundings absorbs more energy than it radiates

Terminology

Transfer variables vs state variables

Energy transfer by heat as well as work done depends on the initial final and

intermediate states of the system They are transfer variables But their sum (Q + W =

4

Eint) is a state variable

Figure 205

5

Serway Chapter 21

Kinetic Theory

The ideal gas law works for all atoms and molecules at low pressure It is rather

amazing that it does Kinetic theory explains why The properties of an ideal gas can

be understood by thinking of it as N rapidly moving particles of mass m As these

particles collide with the container walls momentum is imparted to the walls which

we call the force of gas pressure In this picture the pressure is related to the average

of the square of the particle velocity 2v by

22 1( )

3 2

NP mv

V

Using the ideal gas law we obtain the average translational kinetic energy per

molecule

21 3

2 2 Bmv k T

The rms speed is then given by

2rms

3 3Bk T RTv v

m M

where M is the molecular mass in kgmol

Degrees of Freedom

Roughly speaking a degree of freedom is a way in which a molecule can store energy

For instance since there are three different directions in space along which a molecule

can move there are three degrees of freedom for the translational kinetic energy

There are also three different axes of rotation about which a polyatomic molecule can

spin so we say there are three degrees of freedom for the rotational kinetic energy

There are even degrees of freedom associated with the various ways in which a

molecule can vibrate and with the different energy levels in which the electrons of

the molecule can exist

Internal Energy and Degrees of Freedom The internal energy of an ideal gas made

up of molecules with J degrees of freedom is given by

int 2 2 B

J JE nRT Nk T

6

Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of

molar heat capacities CV and CP These are the heat capacities per mole and the

subscript V on CV means that the volume is being held constant while for CP the

pressure is held constant For example to raise the temperature of n moles of a gas

whose pressure is held constant by 10 K we would have to supply an amount of heat Q

= nCP (10) K

Molar Specific Heat of an Ideal Gas at Constant Volume

VQ nC T

3monatomic

2VC R

5diatomic

2VC R

5polyatomic

2VC

Real gases deviate from these formulas because in addition to the translational and ro-

tational degrees of freedom they also have vibrational and electronic degrees of

freedom These are unimportant at low temperatures due to quantum mechanical

effects but become increasingly important at higher temperatures The rough rule is

No of degrees of freedom

2VC R

Molar Specific Heat of an Ideal Gas at Constant Pressure

PQ nC T

P VC C R

The internal energy of an ideal gas depends only on the temperature

int VE nC T

Adiabatic Processes in an Ideal Gas

7

An adiabatic process is one in which no heat is exchanged between the system and the

environment When an ideal gas expands or contracts adiabatically not only does its

pressure change as expected from the ideal gas law but its temperature changes as

well Under these conditions the final pressure Pf can be computed from the initial

pressure Pi and from the final and initial volumes Vf and Vi by

or constantf f i iP V PV PV

where γ = CPCV The quantity γ is called the adiabatic exponent Note that this

doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be

combined with the adiabatic law for pressure given above to obtain the adiabatic law

for temperatures

1 constantTV

Compressions in sound waves are adiabatic because they happen too rapidly for any

appreciable amount of heat to flow This is why the adiabatic exponent appears in the

formula for the speed of sound in an ideal gas

RTv

M

Note that v depends only on T and not on P Because it depends only on the

temperature the speed of sound is the same in Provo as at sea level in spite of the

lower pressure here due to the difference in elevation

Equipartition of Energy

Every kind of molecule has a certain number of degrees of freedom which are

independent ways in which it can store energy Each such degree of freedom has

associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1

2 RT per mole)

(Note since a molecule has so many possible degrees of freedom it would seem that

there should be a lot of 12 sBk T to spread around But because energy is quantized

some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high

enough that 12 Bk T is as big as the lowest quantum of energy

8

Serway Chapter 22

Second Law of Thermodynamics

There are several equivalent forms of this important law

Kelvin It is not possible to change heat completely into work with no other change

taking place Or in other words there are no perfect heat engines

Clausius It is not possible for heat to flow from one body to another body at a higher

temperature with no other change taking place Or in other words there are no

perfect refrigerators

Entropy In any thermodynamic process that proceeds from one equilibrium state to

another the entropy of the system + environment either remains unchanged or

increases The total entropy never decreases This law is a bit of an oddity among the

laws of physics because it is not absolute Things are forbidden by the second law not

because it is impossible for them to happen but because it is extremely unlikely for

them to happen (See below for more information about entropy)

Reversible and Irreversible Processes

A reversible process is one which occurs so slowly that it is in thermal equilibrium (or

very nearly so) at all times A hallmark of such processes is that a motion picture of

them looks perfectly normal whether run forward or backward Imagine for instance

the slow expansion of a gas at constant temperature in a cylinder whose volume is being

increased by a slowly moving piston Run the movie backwards and what do you see

You see the slow compression of a gas at constant temperature which looks perfectly

normal

An irreversible process is one which occurs in such a way that thermal equilibrium is

not maintained throughout the process The mark of this kind of process is that a motion

picture of it looks very odd when run backward Imagine the sudden expansion of a gas

into a previously evacuated chamber because a hole was punched in the wall between a

pressurized chamber and the evacuated one Run the movie backward and what do you

see You see the gas in the soon-to-be-evacuated chamber gather itself together and

stream through a tiny hole into a chamber in which there is already plenty of gas If you

have ever seen this happen get in touch with the support group for those who have

witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW

9

Heat Engines

Because of the vexing difference in sign between work done on and system and work done

by a system we will invent a new work variable Weng Heat engines do work and so the net

W for these engines is negative But in engineering applications hidden minus signs are

regarded as evil so for heat engines we donrsquot talk about W instead we talk about its

magnitude engW W So for heat engines the first law is

int engE Q W

But for heat pumps and refrigerators work is done on the system so we use the usual work

W when we talk about these systems

A heat engine is a machine that absorbs heat converts part of it to work and exhausts the

rest The heat must be absorbed at high temperature and exhausted at low temperature If the

absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then

eng h cW Q Q

and the efficiency of the engine is defined to be

eng

h

We

Q

A perfect engine would convert the heat hQ completely into work Weng giving an effi-

ciency of e = 1 Energy conservation alone allows a perfect engine but the second law

requires e lt 1

Refrigerators and Heat Pumps

A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high

temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine

by some source of power A heat pump is a machine that either works like a refrigerator

keeping a place cold by transferring heat from this cold place to a higher temperature

environment (cooling mode like an air conditioner) or it functions as a heater

transferring heat into a warm place from a cooler one (heating mode like a window unit

that heats a house by extracting thermal energy from the cold outdoors) The coefficient

of performance of a refrigerator or of a heat pump in cooling mode is defined to be

COP cooling modecQ

W

10

For a heat pump in heating mode the coefficient of performance is

COP heating modehQ

W

Note that we donrsquot have to use the engineering work here because in these systems

work is done on the system and W is naturally positive

A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without

doing any work giving an infinite coefficient of performance Energy conservation alone

allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump

in heating mode would transfer Qh into the house without doing any work and so would

also have an infinite coefficient of performance The second law forbids this too A good

coefficient of performance for a real device would be around 5 or 6

Carnot Cycle

The most efficient of all possible engines is one that uses the Carnot cycle This cycle

employs an ideal gas has no friction and operates very slowly so that the gas can be in

thermal equilibrium at all parts of the cycle This means of course that it canrsquot

possibly be built and even if it could be built it would not run fast enough to be useful

Nevertheless this cycle is very important because it gives an upper bound on the

efficiency of real engines There cannot possibly be an engine that is more efficient

than one based on the Carnot cycle This cycle consists of the following four steps

1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its

volume The reason that heat is absorbed is that expansion tends to cool the gas but

thermal contact with the environment at Th keeps the temperature high by heat

conduction into the ideal gas

2 The ideal gas further increases its volume by an adiabatic expansion This expansion

causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc

3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in

volume The reason that heat is exhausted is that compression tends to heat the gas

but thermal contact with the environment at Tc keeps the temperature low by heat

conduction out of the ideal gas

4 The gas is adiabatically compressed back to its original volume (the volume it started

with in step 1) This compression heats the gas from Tc up to Th

11

The efficiency of a Carnot engine is given by the very simple formula

1 cC

h

Te

T

where the temperatures must in be Kelvin No real engine can be more efficient than

this

The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode

is given by

COP cooling mode cC

h c

T

T T

and no real refrigerator can have a coefficient of performance greater than this

The coefficient of performance of a Carnot heat pump in heating mode is

COP heating mode hC

h c

T

T T

So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency

To make the reversible steps in the cycle really reversible they would have to occur

infinitely slowly So the price you pay for making a perfect engine is that it takes

forever to get it to do any work

Entropy

The entropy of a system is defined in terms of its molecular makeup and measures

roughly the disorder of the system If the system is packed into a very small volume

then it is quite ordered and the entropy will be low If it occupies a large volume the

entropy is high (To see what this has to do with disorder note that socks in a drawer

occupy a small volume while socks on the bed in the corner by the door and

hanging from the chandelier occupy a large volume) If the system is very cold then

the molecules hardly move and may even reach out to each other and form a crystal

This is a highly ordered state and therefore has low entropy If the system is very hot

with rapidly speeding molecules crashing into the container walls and bouncing off

each other things are disordered and the entropy is high

It is possible to calculate the entropy of a system in terms of its macroscopic thermody-

namic properties ie pressure volume temperature number of moles etc The key to this

calculation is the concept of a reversible process A reversible process is one that is

12

carried out without friction and so slowly that the process can be reversed at any stage by

making an infinitesimal change in the environment of the system The slow expansion of the

gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment

is reversible If we turned around and began slowly to compress the gas would just slowly

exhaust heat to the environment in the exact reverse way that it absorbed it during expansion

Most processes however are irreversible For example if a gas-filled box were suddenly

increased in size so that the particles were free to wander into the void created by the sudden

expansion then the gas would eventually fill the new volume uniformly at the same

temperature as before the expansion (The temperature is unchanged in this imaginary

process because the kinetic energy of the molecules would be unaffected by such an

instantaneous expansion of the container walls) This imaginary but highly thought-

stimulating process is called a free expansion and it is impossible to reverse it During the

expansion we didnrsquot push on any of the molecules so reversing this process would mean

making them go back into their original volume without pushing on them they simply will

not cooperate to this extent Another way to see that just pushing them back where they came

from does not reverse the free expansion is to think about what would happen if we just

compressed either adiabatically or isothermally An adiabatic compression back to the

original volume would heat the gas above its original temperature and an isothermal

compression would require that heat be exhausted to the environment But the free expansion

involved neither temperature changes nor heat exchanges so neither of these two processes

is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in

a way that takes us back to the initial state

It is possible to calculate the change of entropy for both reversible and irreversible processes

Letrsquos consider a reversible process first In a reversible process the entropy change is given

by the formula

dQS

T

where dQ is the amount of heat added to the system during a small step of the process

The total energy change during the process may then simply be calculated by integration

f f

f i i i

dQS S S dS

T

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11

1

Serway Chapter 14 Fluid Mechanics

Density and Pressure

The density of a material is defined to be the ratio between the mass of a small piece of it divided

by the volume of the small piece

m

V

Water has a density of 1000 kgm3 while air has a density of 13 kgm3 at sea level and 11

kgm3 in Provo Sea water has a density of 103 of water Ethanol (ethyl alcohol) is 0806

The pressure exerted on a small surface is defined to be the force applied to it divided by its

area

FP

A

At sea level atmospheric pressure is 1013 times 105 pascals where 1 Pa = 1 Nm2 Here in Provo

atmospheric pressure is about 86 times 104 Pa Sometimes pressure will be expressed in pounds-per-

square-inch (psi) or sometimes in torr 1 psi= 6895 times 103 Pa and 1 torr= 1333 Pa

Hydrostatics

If h measures distance below the surface of a liquid then the pressure as a function of the depth h

is given by

oP P gh

where Po is the pressure at the surface of the liquid usually atmospheric pressure In 123 liquids

are imcompressible ρ does not vary with depth and g = 980 m2s

Pascalrsquos Principle

A change in the pressure applied to an enclosed fluid is transmitted undiminished to every

portion of the fluid and to the walls of the containing vessel (Blaise Pascal 1652)

Archimedes Principle

A body wholly or partially immersed in a fluid will be buoyed up by a force equal to the weight

of the fluid that it displaces (Archimedes ca 220 BC) This principle is a handy computational

tool but it does not explain the real reason that objects in a fluid experience an upward force

The reason is that the pressure on the bottom of the object is greater than the pressure at the top

so there is a net upward pressure force The buoyant force is given by the formula

B Vg

2

where ρ is the density of the fluid V is the submerged volume of the object and g = 980 ms2 is

the acceleration of gravity If the object is floating on the liquid then the submerged volume V is

only that portion which is below the liquid surface

Equation of Continuity

For a fluid whose density is constant the rate of fluid flow or volume flux Φ measured in units

of m3s for a cross-section of fluid streamlines having area A and flow speed v is given by

Av

Bernoullirsquos Equation

The principle of conservation of energy for steady fluid flow states Along each streamline of the

flow the following quantity has a constant value

21

2P v gy

where y is measured upward against gravity Note that y in this equation and h in the hydrostatic

pressure equation have opposite signs This law works for gases and liquids The demos are

mostly for gases and are set up so that ρ does not vary

3

Serway Chapter 16

Longitudinal and Transverse Waves

A longitudinal wave is one in which the direction of vibration of the medium and the direction of

propagation of the wave are in the same direction Sound waves in the air in water and in solids

are longitudinal waves

A transverse wave is one in which the direction of vibration of the medium is perpendicular to

the direction of propagation of the wave Ocean waves waves on ropes and electromagnetic

waves are transverse waves

Wavelength and Wavenumber

The wavelength λ is the period in space of the wave ie it is the distance from one wave crest to

the next The wavenumber k (units reciprocal length) is related to the wavelength by the formula

2k

Frequency and Angular Frequency

The period T is the period in time of the wave ie it is the time between the arrival of one wave

crest and the next at some point in space The frequency f is the number of crests that arrive per

second which is just 1T and the angular frequency ω is 2π times the frequency Both have units

of per second (reciprocal time)

1 22f f

T T

Wave Speed

The wave speed of a wave is defined to be the the speed at which the wave crests or troughs

move through the medium This speed is related to the frequency and the wavelength by the

formulas

v fk

where k is the wavenumber of the wave v is also called the phase velocity

Traveling Waves

Traveling waves are oscillations in which the wave crests and troughs glide smoothly through the

medium Waves on oceans and lakes are traveling waves The mathematical form of a harmonic

(sinusoidal) traveling wave is

4

siny x t A kx t

where A is the amplitude of the wave As usual ω and k are related by the relation ω = kv where

v is the wave speed the speed at which the crests travel the minus sign means that the wave is

traveling to the right towards higher values of x φ is the phase and constrols where the maxima

and minima occur

Speed of Transverse Waves on a Rope

The velocity of transverse waves on a perfectly flexible rope is given by

Tv

where T is the tension in the rope and where μ is its linear mass density (mass per unit length)

Common Test Questions

A common question is to be given an equation for a wave and to be asked what is amplitude

velocity etc Another common type of question is to be given facts about a wave like its

direction velocity and magnitude at a given place and time and to be asked which of several

equations correctly describe the wave

5

Serway Chapter 17

Sound Speed in Solids Liquids and Gases

The speed of sound in a liquid or solid of bulk modulus B and volume mass density ρ

Bv

The speed of sound in air is

331 1 273v m s T C

The speed of sound in air at room temperature is about 343 ms Notice it doesnrsquot depend on

pressure or frequency

Power Intensity and Loudness

The intensity of a sound wave is defined to be the power per unit area ie wattsm2 in the wave

For spherical waves traveling away from a small source emitting waves with average power Рav

the intensity falls off with distance from the source r according to the inverse-square law

av24

Ir

Hence the intensity I2 at distance r2 is related to the intensity I1 at a different distance r1 by

21 2

22 1

I r

I r

Experiments on human hearing have shown that we hear intensity differences logarithmically so

the decibel loudness scale for sound intensity was invented The loudness β of a sound in

decibels is related to its intensity I in Wm2 by the formula

1010logo

I

I

Where Io is a sound intensity near the threshold of hearing defined to be Io = 10-12 Wm2 Note

that on this scale a sound is made 10 times more intense by adding 10 decibels Remember that

intensity is proportional to velocity squared Amplitude is the A is the previous chapterrsquos

equation for a traveling wave If you make the A 10 times as much the intensity will increase by

100 times (20 dB)

Doppler Effect

If sound waves are traveling through a medium and if either the receiver of the waves or the

source of waves is moving then the frequency received is related to the frequency emitted by

6

0

s

v vf f

v v

where fprime is the frequency detected by the observer f is the frequency emitted by the source v is

the speed of the waves vo is the speed of the observer and vs is the speed of the source This

formula assumes that the source and receiver are either moving directly toward each other or

directly away from each other To know which signs to use remember that when observer and

source approach each other the observed frequency is higher while if they move away from each

other it is lower Just examine the signs in the formula and make the answer come out right

For electromagnetic waves (light radio waves X-rays) traveling in vacuum Einsteinrsquos theory of

relativity (and careful experiments) show that the Doppler shift is given by

r

r

c vf f

c v

where vr is the relative speed between the source and the observer

For all kinds of waves (sound light etc) if the relative speed of the source and the observer is

small compared to the speed of the waves then there is a simple approximation to the Doppler

effect For example if the relative speed is 1 of the wave speed then the frequency shifts by

1 Remember that this is only an approximation

Shock Waves

When an object moves through a medium at a speed greater than the speed of waves a V-shaped

shock wave is produced The V-shaped wake behind a speeding boat is a good example of this

effect and the cone of sonic-boom behind a supersonic aircraft is another The angle that V-line

makes with the direction of travel of the source is given by

sins

v

v

where v is the wave speed and vs is the source speed

7

Serway Chapter 18

Principle of Linear Superposition

We say that a system obeys the principle of linear superposition if two or more different motions

of the system can simply be added together to find the net motion of the system Light waves

obey this principle as they propagate through the air as can be seen by shining two flashlights so

that their beams cross The beams propagate along without affecting each other (Light sabers are

a spectacular but unfortunately fictional example of systems that do not obey the principle of

superposition) Wave pulses on an ideal rope also obey this principle two different pulses pass

through each other without change Standing waves are an example of this effect being simply

the linear superposition of two traveling waves of the same frequency but moving in opposite

directions Light waves in matter do not always obey this principle For instance two powerful

laser beams could be made to cross in a piece of glass in such a way that their combined heating

effect in the crossing region could melt the glass and scatter the beams in complicated ways This

is an example of a nonlinear effect

Interference

When two or more waves are present in the same medium at the same time their net effect may

often be obtained simply by adding them at each point in the medium according to the principle

of linear superposition (Note this wonrsquot work if the medium is nonlinear) When this addition

makes the total amplitude be greater than the individual amplitudes of the various waves we say

that the interference is constructive When the addition produces cancellation and an amplitude

less than the amplitudes of the separate waves we have destructive interference

Standing Waves

A standing wave is the superposition of two identical traveling waves moving in opposite

directions Nodes are places where the two waves perfectly destructively interfere to produce

zero amplitude at all times Anti-nodes are places where the two waves perfectly constructively

interfere to produce an amplitude maximum The distance between nodes is λ2 Standing waves

on a string fixed at both ends have nodes at each end of the string Standing waves in an air

column enclosed in a tube have displacement anti-nodes at open ends of the tube and

displacement nodes at closed ends The frequency of the standing wave with the lowest possible

frequency is called the fundamental frequency Standing waves on strings or in air columns all

have frequencies which are integer multiples of the fundamental frequency and are called

8

harmonics (The fundamental is called the ldquofirst harmonicrdquo)

Beats

Beats are heard when two waves with slightly different frequencies f1 and f2 are combined The

waves constructively interfere for a number of cycles then destructively interfere for a number

of cycles We hear a periodic ldquowah-wahrdquo frequency equal to the difference of the two wave

frequencies

1 2bf f f

Musical Instruments

Musical instruments produce tones by exciting standing waves on strings (violins piano) and in

tubes (trumpet organ) The fundamental frequency of the standing wave is called the pitch of the

tone The pitch of concert A is 440 Hz by definition Two tones are an octave apart if one pitch

has twice the frequency of the other In written music there are 12 intervals in each octave with

the ratio between successive intervals equal to 2112 = 105946 The ratios for each tone in an

octave starting at A and ending at the next higher A are

A A B C C D D E F F G G A

1 10595 11225 11892 12599 13348 14142 14983 15874 16818 17818 18877 2

A musical tone is actually a superposition of the fundamental frequency and the higher

harmonics The tone quality of a musical instrument is determined by the amplitudes of the

various harmonics that it produces A violin and a trumpet can play the same pitch but they

donrsquot sound at all alike to our ears The difference between them is in the various amplitudes of

their harmonics

9

Serway Chapter 19

Temperature

Formally temperature is what is measured by a thermometer Roughly high temperature is what

we call hot and low temperature is what we call cold On the atomic level temperature refers to

the kinetic energy of the molecules A collection of molecules is called ldquohotrdquo if the molecules

have rapid random motion while a collection of molecules is called ldquocoldrdquo if the random motion

is slow When two bodies are placed in close contact with each other they exchange molecular

kinetic energy until they come to the same temperature This is the microscopic picture of the

Zeroth Law of Thermodynamics

Absolute Zero

Absolute zero is the lowest possible temperature that any object can have This is the temperature

at which all of the energy than can be removed an object has been removed (This removable

energy we call thermal energy) There is still motion at absolute zero Electrons continue to orbit

around atomic nuclei and even atoms continue to move about with a small amount of kinetic

energy but this small energy cannot be removed from the object For example at absolute zero

helium is a liquid whose atoms still move and slide past each other

Temperature Scales

Kelvin Scale Absolute zero is at T = 0 K water freezes at T = 27315 K room temperature is

around T = 295 K and water boils at T = 373 K Note we donrsquot use a deg symbol Kelvin is

prefered SI Unit

Celsius Scale Absolute zero is at T = -273degC water freezes at T = 0degC room temperature is

around T = 22degC and water boils at T = 100degC

Fahrenheit Scale Absolute zero is at T = -459degF water freezes at TF = 32degF room temperature

is around T = 72degF and water boils at TF = 212degF TF =18TC + 32 Notice that temperature

differences are the same for the Kelvin and Celsius scales

Thermal Expansion

When materials are heated they usually expand and when they are cooled they usually contract

(Water near freezing is a spectacular counterexample it works the other way around) The

coefficient of linear expansion is defined by the relation

1

i

L

L T

10

where Li is the initial length of a rod of the material and ΔL is the change in its length due to a

small temperature change ΔT The coefficient of volume expansion is defined similarly

1

i

V

V T

where Vi is the initial volume of a piece of material and where ΔV is the change in its volume due

to a small temperature change ΔT

Avogadrorsquos Number (N or NA)

One mole of any substance corresponds to 6022 times 1023 molecules The atomic mass given on the

periodic table (p A32-33 in the text) is the mass in grams of one mole of atoms For example

the atomic mass of oxygen is 15999 gmol The mass of a single atom is given by the same

number in units of atomic mass units (u) For example the mass of an oxygen atom is 15999 u

(on the average)

Ideal Gas Law (an example of an equation of state)

When the molecules of a gas are sufficiently inert and widely separated that interactions between

them are negligible we say that it is an ideal gas The pressure P volume V and temperature T

(in kelvins) of such a gas are State Variables and are related by the ideal gas law

Bor PV=NkPV nRT T

where n is the number of moles of the gas where R is the gas constant

8314 Jmol KR

where N is the number of molecules and where kB is Boltzmannrsquos constant

231380 10 J KBk

It works well for air at atmosphere pressure and even better for partial vaccuums The relative

ease of measuring pressure and the linear relationship between pressure and temperature (if V

and n are held fixed) makes an ideal gas an ideal thermometer Thermometers can be based on

properties of solids or liquids but the behavior of these materials with temperature is more

complicated

11

Serway Chapter 20

Heat

Heat is energy that flows between a system and its environment because of a tempera-

ture difference between them The units of heat are Joules as expected for an energy

Unfortunately there are several competing units of energy They are related by

1 cal 4186 J 1 Cal 4186 J 1 Btu 1054 J

Heat Capacity

There is often a simple linear relation between the heat that flows in or out of part of a

system and the temperature change that results from this energy transfer When this

linear relation holds it is convenient to define the heat capacity C and the specific

heat c as follows

For the entire object the heat Q it absorbs is given by Q = C(Tf - Ti)

For a particular material the specific heat is defined by c = Cm which is the heat

capacity per unit mass so that

Q = mc(Tf ndash Ti)

Note C has units of energy (J or Cal)(Kelvin kg)

Heats of Transformation or Latent Heat Q = plusmn mL

When a substance changes phase from solid to liquid or from liquid to gas it absorbs

heat without a change in temperature The latent heat or heat of transformation is

usually given per unit mass of the substance For example for water the heat of fusion

(melting) is L = 333 kJkg while the heat of vaporization (boiling) is L = 2260 kJkg

Note that heat for boiling is considerably bigger than melting for water You have to

be careful with signs heat is given off (negative) if you go down in temperature and

condense steam

Work

In general the small amount of work done on a system as a force Fon is exerted on it

through a vector displacement dx is given by

on xdW d F

12

But if the displacement is done very slowly (as we always assume in thermodynamics)

then the force exerted on the system and the force exerted by the system are in

balance so the force exerted by the system is ndash Fon In thermodynamics it is more

convenient to talk about the force exerted by the system so we change the above

formula for the work done on the system to

xdW d F

where F is the force exerted by the system This has confused students for more than a

century now but this is the way your book and many other books do it so you are

stuck You will need to memorize the minus sign in this definition of the work to be

able to use your textbook

There are many chances to get signs wrong in this and the next two chapters (Mosiah

2 )

When an external agent changes the volume of a gas at pressure P by a small amount

dV the (small amount) of work done on the system is given by

dW PdV

Notice that this minus sign is just what we need to make dW be positive if the external

agent compresses the gas for then dV is negative If on the other hand the external

agent gives way allowing the gas to expand against it then dV is positive and we say

that the work done on the gas is negative

The work done on the system (eg by the gas in a cylinder) in a thermodynamic

process is the area under the curve in a PV diagram It is positive for compressions

and negative for expansions If the volume of gas remains constant in a process then

no work is done by the gas

Cyclic processes are important For cyclic processes represented by PV diagrams the

magnitude of the net work during one cycle is simply the area enclosed by the cycle

on the diagram Be careful to keep track of signs when you are calculating that

enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes

1

Path A-B B-C C-D D-A A to A net

Q

W

ΔU

ΔS

Internal Energy

The energy stored in a substance is called its internal energy Eint This energy may be

stored as random kinetic energy or as potential energy in each molecule (stretched

chemical bonds electrons in excited states etc) For ideal gases all states with the

same temperature will have the same Eint

First Law of Thermodynamics

The change ΔEint in the internal energy of a system is given by

intE Q W

where Q is the heat absorbed by the system and where W is the work done on the

system Hence if a system absorbs heat (and if Wge0) the internal energy increases

Likewise if the system does work (W on the system is negative) and if Qge0 the

internal energy decreases Potential Pitfall Many times people talk about work done

by the system It is the minus of W on the system Donrsquot get tripped up

Processes

Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated

from the environment A process may be approximately adiabatic if it happens so

rapidly that heat does not have time to enter or leave the system Work + or ndash is done

and ΔEint = W

Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing

on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal

energy and temperature does not change (Note the difference between an adiabatic

process and a free expansion is that NO work is done in the adiabatic free expansion)

Isobaric process The pressure is held fixed ΔP = 0 For example usually the

pressure increases when a gas is heated but if it were allowed to expand during the

2

heating process in just the right way its pressure could remain fixed In isobaric

processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)

Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is

then zero and so we have ΔEint = Q

Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint

so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in

an isothermal process The work done on the gas is then given by

lnf

i

Vi

Vf

VW PdV nRT

V

Heat Conduction

The quantity P is defined to be the rate at which heat flows through an object and is a

power having units of watts It is analogous to electric current which is the rate at

which charge flows through an object If the flow of heat through a slab of length L

and cross-sectional area A is steady in time then P is given by the equation

h cT TdQkA

dt L

P =

where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The

heat flows of course because of this temperature difference The quantity k is called

the thermal conductivity and is a constant that is characteristic of the material It is

analogous to the electrical conductivity h cT TL is sometimes called the temperature

gradient and is written dTd dTdx

R-Values

It is common to have the heat-conducting properties of materials described by their R-

values especially for insulating materials like fiberglass batting The connection

between k and R is R = Lk where L is the material thickness In this country R-values

always have units of 2ft F hourBtu

Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11

(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)

The heat flow rate through a slab of area A is given by

3

h cT TA

R

P

in units of Btuhour Note that A must be in square feet and the temperatures must be

in degrees Fahrenheit

Convection

Convection is the transfer of thermal energy by flow of material For instance a home

furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct

throughout the house instead it quickly pumps warm air to all of the rooms

Generally convection is a much faster way to transfer heat than conduction

Radiation

Electromagnetic radiation can also transfer heat When you warm yourself near a

campfire which has burned itself down into a bed of glowing embers you are

receiving radiant heat from the infrared portion of the electromagnetic spectrum The

rate at which an object emits radiant heat is given by Stefanrsquos law

4AeTP

where P is the radiated power in watts σ is a constant

8 2 45696 10 W m K

A is the surface area of the object in m2 and T is the temperature in kelvins The

constant e is called the emissivity and it varies from substance to substance A perfect

absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =

0 Hence black objects radiate very well while shiny ones do not Also an object that

is hotter than its surroundings radiates more energy than it absorbs whereas an object

that is cooler than its surroundings absorbs more energy than it radiates

Terminology

Transfer variables vs state variables

Energy transfer by heat as well as work done depends on the initial final and

intermediate states of the system They are transfer variables But their sum (Q + W =

4

Eint) is a state variable

Figure 205

5

Serway Chapter 21

Kinetic Theory

The ideal gas law works for all atoms and molecules at low pressure It is rather

amazing that it does Kinetic theory explains why The properties of an ideal gas can

be understood by thinking of it as N rapidly moving particles of mass m As these

particles collide with the container walls momentum is imparted to the walls which

we call the force of gas pressure In this picture the pressure is related to the average

of the square of the particle velocity 2v by

22 1( )

3 2

NP mv

V

Using the ideal gas law we obtain the average translational kinetic energy per

molecule

21 3

2 2 Bmv k T

The rms speed is then given by

2rms

3 3Bk T RTv v

m M

where M is the molecular mass in kgmol

Degrees of Freedom

Roughly speaking a degree of freedom is a way in which a molecule can store energy

For instance since there are three different directions in space along which a molecule

can move there are three degrees of freedom for the translational kinetic energy

There are also three different axes of rotation about which a polyatomic molecule can

spin so we say there are three degrees of freedom for the rotational kinetic energy

There are even degrees of freedom associated with the various ways in which a

molecule can vibrate and with the different energy levels in which the electrons of

the molecule can exist

Internal Energy and Degrees of Freedom The internal energy of an ideal gas made

up of molecules with J degrees of freedom is given by

int 2 2 B

J JE nRT Nk T

6

Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of

molar heat capacities CV and CP These are the heat capacities per mole and the

subscript V on CV means that the volume is being held constant while for CP the

pressure is held constant For example to raise the temperature of n moles of a gas

whose pressure is held constant by 10 K we would have to supply an amount of heat Q

= nCP (10) K

Molar Specific Heat of an Ideal Gas at Constant Volume

VQ nC T

3monatomic

2VC R

5diatomic

2VC R

5polyatomic

2VC

Real gases deviate from these formulas because in addition to the translational and ro-

tational degrees of freedom they also have vibrational and electronic degrees of

freedom These are unimportant at low temperatures due to quantum mechanical

effects but become increasingly important at higher temperatures The rough rule is

No of degrees of freedom

2VC R

Molar Specific Heat of an Ideal Gas at Constant Pressure

PQ nC T

P VC C R

The internal energy of an ideal gas depends only on the temperature

int VE nC T

Adiabatic Processes in an Ideal Gas

7

An adiabatic process is one in which no heat is exchanged between the system and the

environment When an ideal gas expands or contracts adiabatically not only does its

pressure change as expected from the ideal gas law but its temperature changes as

well Under these conditions the final pressure Pf can be computed from the initial

pressure Pi and from the final and initial volumes Vf and Vi by

or constantf f i iP V PV PV

where γ = CPCV The quantity γ is called the adiabatic exponent Note that this

doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be

combined with the adiabatic law for pressure given above to obtain the adiabatic law

for temperatures

1 constantTV

Compressions in sound waves are adiabatic because they happen too rapidly for any

appreciable amount of heat to flow This is why the adiabatic exponent appears in the

formula for the speed of sound in an ideal gas

RTv

M

Note that v depends only on T and not on P Because it depends only on the

temperature the speed of sound is the same in Provo as at sea level in spite of the

lower pressure here due to the difference in elevation

Equipartition of Energy

Every kind of molecule has a certain number of degrees of freedom which are

independent ways in which it can store energy Each such degree of freedom has

associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1

2 RT per mole)

(Note since a molecule has so many possible degrees of freedom it would seem that

there should be a lot of 12 sBk T to spread around But because energy is quantized

some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high

enough that 12 Bk T is as big as the lowest quantum of energy

8

Serway Chapter 22

Second Law of Thermodynamics

There are several equivalent forms of this important law

Kelvin It is not possible to change heat completely into work with no other change

taking place Or in other words there are no perfect heat engines

Clausius It is not possible for heat to flow from one body to another body at a higher

temperature with no other change taking place Or in other words there are no

perfect refrigerators

Entropy In any thermodynamic process that proceeds from one equilibrium state to

another the entropy of the system + environment either remains unchanged or

increases The total entropy never decreases This law is a bit of an oddity among the

laws of physics because it is not absolute Things are forbidden by the second law not

because it is impossible for them to happen but because it is extremely unlikely for

them to happen (See below for more information about entropy)

Reversible and Irreversible Processes

A reversible process is one which occurs so slowly that it is in thermal equilibrium (or

very nearly so) at all times A hallmark of such processes is that a motion picture of

them looks perfectly normal whether run forward or backward Imagine for instance

the slow expansion of a gas at constant temperature in a cylinder whose volume is being

increased by a slowly moving piston Run the movie backwards and what do you see

You see the slow compression of a gas at constant temperature which looks perfectly

normal

An irreversible process is one which occurs in such a way that thermal equilibrium is

not maintained throughout the process The mark of this kind of process is that a motion

picture of it looks very odd when run backward Imagine the sudden expansion of a gas

into a previously evacuated chamber because a hole was punched in the wall between a

pressurized chamber and the evacuated one Run the movie backward and what do you

see You see the gas in the soon-to-be-evacuated chamber gather itself together and

stream through a tiny hole into a chamber in which there is already plenty of gas If you

have ever seen this happen get in touch with the support group for those who have

witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW

9

Heat Engines

Because of the vexing difference in sign between work done on and system and work done

by a system we will invent a new work variable Weng Heat engines do work and so the net

W for these engines is negative But in engineering applications hidden minus signs are

regarded as evil so for heat engines we donrsquot talk about W instead we talk about its

magnitude engW W So for heat engines the first law is

int engE Q W

But for heat pumps and refrigerators work is done on the system so we use the usual work

W when we talk about these systems

A heat engine is a machine that absorbs heat converts part of it to work and exhausts the

rest The heat must be absorbed at high temperature and exhausted at low temperature If the

absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then

eng h cW Q Q

and the efficiency of the engine is defined to be

eng

h

We

Q

A perfect engine would convert the heat hQ completely into work Weng giving an effi-

ciency of e = 1 Energy conservation alone allows a perfect engine but the second law

requires e lt 1

Refrigerators and Heat Pumps

A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high

temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine

by some source of power A heat pump is a machine that either works like a refrigerator

keeping a place cold by transferring heat from this cold place to a higher temperature

environment (cooling mode like an air conditioner) or it functions as a heater

transferring heat into a warm place from a cooler one (heating mode like a window unit

that heats a house by extracting thermal energy from the cold outdoors) The coefficient

of performance of a refrigerator or of a heat pump in cooling mode is defined to be

COP cooling modecQ

W

10

For a heat pump in heating mode the coefficient of performance is

COP heating modehQ

W

Note that we donrsquot have to use the engineering work here because in these systems

work is done on the system and W is naturally positive

A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without

doing any work giving an infinite coefficient of performance Energy conservation alone

allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump

in heating mode would transfer Qh into the house without doing any work and so would

also have an infinite coefficient of performance The second law forbids this too A good

coefficient of performance for a real device would be around 5 or 6

Carnot Cycle

The most efficient of all possible engines is one that uses the Carnot cycle This cycle

employs an ideal gas has no friction and operates very slowly so that the gas can be in

thermal equilibrium at all parts of the cycle This means of course that it canrsquot

possibly be built and even if it could be built it would not run fast enough to be useful

Nevertheless this cycle is very important because it gives an upper bound on the

efficiency of real engines There cannot possibly be an engine that is more efficient

than one based on the Carnot cycle This cycle consists of the following four steps

1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its

volume The reason that heat is absorbed is that expansion tends to cool the gas but

thermal contact with the environment at Th keeps the temperature high by heat

conduction into the ideal gas

2 The ideal gas further increases its volume by an adiabatic expansion This expansion

causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc

3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in

volume The reason that heat is exhausted is that compression tends to heat the gas

but thermal contact with the environment at Tc keeps the temperature low by heat

conduction out of the ideal gas

4 The gas is adiabatically compressed back to its original volume (the volume it started

with in step 1) This compression heats the gas from Tc up to Th

11

The efficiency of a Carnot engine is given by the very simple formula

1 cC

h

Te

T

where the temperatures must in be Kelvin No real engine can be more efficient than

this

The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode

is given by

COP cooling mode cC

h c

T

T T

and no real refrigerator can have a coefficient of performance greater than this

The coefficient of performance of a Carnot heat pump in heating mode is

COP heating mode hC

h c

T

T T

So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency

To make the reversible steps in the cycle really reversible they would have to occur

infinitely slowly So the price you pay for making a perfect engine is that it takes

forever to get it to do any work

Entropy

The entropy of a system is defined in terms of its molecular makeup and measures

roughly the disorder of the system If the system is packed into a very small volume

then it is quite ordered and the entropy will be low If it occupies a large volume the

entropy is high (To see what this has to do with disorder note that socks in a drawer

occupy a small volume while socks on the bed in the corner by the door and

hanging from the chandelier occupy a large volume) If the system is very cold then

the molecules hardly move and may even reach out to each other and form a crystal

This is a highly ordered state and therefore has low entropy If the system is very hot

with rapidly speeding molecules crashing into the container walls and bouncing off

each other things are disordered and the entropy is high

It is possible to calculate the entropy of a system in terms of its macroscopic thermody-

namic properties ie pressure volume temperature number of moles etc The key to this

calculation is the concept of a reversible process A reversible process is one that is

12

carried out without friction and so slowly that the process can be reversed at any stage by

making an infinitesimal change in the environment of the system The slow expansion of the

gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment

is reversible If we turned around and began slowly to compress the gas would just slowly

exhaust heat to the environment in the exact reverse way that it absorbed it during expansion

Most processes however are irreversible For example if a gas-filled box were suddenly

increased in size so that the particles were free to wander into the void created by the sudden

expansion then the gas would eventually fill the new volume uniformly at the same

temperature as before the expansion (The temperature is unchanged in this imaginary

process because the kinetic energy of the molecules would be unaffected by such an

instantaneous expansion of the container walls) This imaginary but highly thought-

stimulating process is called a free expansion and it is impossible to reverse it During the

expansion we didnrsquot push on any of the molecules so reversing this process would mean

making them go back into their original volume without pushing on them they simply will

not cooperate to this extent Another way to see that just pushing them back where they came

from does not reverse the free expansion is to think about what would happen if we just

compressed either adiabatically or isothermally An adiabatic compression back to the

original volume would heat the gas above its original temperature and an isothermal

compression would require that heat be exhausted to the environment But the free expansion

involved neither temperature changes nor heat exchanges so neither of these two processes

is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in

a way that takes us back to the initial state

It is possible to calculate the change of entropy for both reversible and irreversible processes

Letrsquos consider a reversible process first In a reversible process the entropy change is given

by the formula

dQS

T

where dQ is the amount of heat added to the system during a small step of the process

The total energy change during the process may then simply be calculated by integration

f f

f i i i

dQS S S dS

T

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11

2

where ρ is the density of the fluid V is the submerged volume of the object and g = 980 ms2 is

the acceleration of gravity If the object is floating on the liquid then the submerged volume V is

only that portion which is below the liquid surface

Equation of Continuity

For a fluid whose density is constant the rate of fluid flow or volume flux Φ measured in units

of m3s for a cross-section of fluid streamlines having area A and flow speed v is given by

Av

Bernoullirsquos Equation

The principle of conservation of energy for steady fluid flow states Along each streamline of the

flow the following quantity has a constant value

21

2P v gy

where y is measured upward against gravity Note that y in this equation and h in the hydrostatic

pressure equation have opposite signs This law works for gases and liquids The demos are

mostly for gases and are set up so that ρ does not vary

3

Serway Chapter 16

Longitudinal and Transverse Waves

A longitudinal wave is one in which the direction of vibration of the medium and the direction of

propagation of the wave are in the same direction Sound waves in the air in water and in solids

are longitudinal waves

A transverse wave is one in which the direction of vibration of the medium is perpendicular to

the direction of propagation of the wave Ocean waves waves on ropes and electromagnetic

waves are transverse waves

Wavelength and Wavenumber

The wavelength λ is the period in space of the wave ie it is the distance from one wave crest to

the next The wavenumber k (units reciprocal length) is related to the wavelength by the formula

2k

Frequency and Angular Frequency

The period T is the period in time of the wave ie it is the time between the arrival of one wave

crest and the next at some point in space The frequency f is the number of crests that arrive per

second which is just 1T and the angular frequency ω is 2π times the frequency Both have units

of per second (reciprocal time)

1 22f f

T T

Wave Speed

The wave speed of a wave is defined to be the the speed at which the wave crests or troughs

move through the medium This speed is related to the frequency and the wavelength by the

formulas

v fk

where k is the wavenumber of the wave v is also called the phase velocity

Traveling Waves

Traveling waves are oscillations in which the wave crests and troughs glide smoothly through the

medium Waves on oceans and lakes are traveling waves The mathematical form of a harmonic

(sinusoidal) traveling wave is

4

siny x t A kx t

where A is the amplitude of the wave As usual ω and k are related by the relation ω = kv where

v is the wave speed the speed at which the crests travel the minus sign means that the wave is

traveling to the right towards higher values of x φ is the phase and constrols where the maxima

and minima occur

Speed of Transverse Waves on a Rope

The velocity of transverse waves on a perfectly flexible rope is given by

Tv

where T is the tension in the rope and where μ is its linear mass density (mass per unit length)

Common Test Questions

A common question is to be given an equation for a wave and to be asked what is amplitude

velocity etc Another common type of question is to be given facts about a wave like its

direction velocity and magnitude at a given place and time and to be asked which of several

equations correctly describe the wave

5

Serway Chapter 17

Sound Speed in Solids Liquids and Gases

The speed of sound in a liquid or solid of bulk modulus B and volume mass density ρ

Bv

The speed of sound in air is

331 1 273v m s T C

The speed of sound in air at room temperature is about 343 ms Notice it doesnrsquot depend on

pressure or frequency

Power Intensity and Loudness

The intensity of a sound wave is defined to be the power per unit area ie wattsm2 in the wave

For spherical waves traveling away from a small source emitting waves with average power Рav

the intensity falls off with distance from the source r according to the inverse-square law

av24

Ir

Hence the intensity I2 at distance r2 is related to the intensity I1 at a different distance r1 by

21 2

22 1

I r

I r

Experiments on human hearing have shown that we hear intensity differences logarithmically so

the decibel loudness scale for sound intensity was invented The loudness β of a sound in

decibels is related to its intensity I in Wm2 by the formula

1010logo

I

I

Where Io is a sound intensity near the threshold of hearing defined to be Io = 10-12 Wm2 Note

that on this scale a sound is made 10 times more intense by adding 10 decibels Remember that

intensity is proportional to velocity squared Amplitude is the A is the previous chapterrsquos

equation for a traveling wave If you make the A 10 times as much the intensity will increase by

100 times (20 dB)

Doppler Effect

If sound waves are traveling through a medium and if either the receiver of the waves or the

source of waves is moving then the frequency received is related to the frequency emitted by

6

0

s

v vf f

v v

where fprime is the frequency detected by the observer f is the frequency emitted by the source v is

the speed of the waves vo is the speed of the observer and vs is the speed of the source This

formula assumes that the source and receiver are either moving directly toward each other or

directly away from each other To know which signs to use remember that when observer and

source approach each other the observed frequency is higher while if they move away from each

other it is lower Just examine the signs in the formula and make the answer come out right

For electromagnetic waves (light radio waves X-rays) traveling in vacuum Einsteinrsquos theory of

relativity (and careful experiments) show that the Doppler shift is given by

r

r

c vf f

c v

where vr is the relative speed between the source and the observer

For all kinds of waves (sound light etc) if the relative speed of the source and the observer is

small compared to the speed of the waves then there is a simple approximation to the Doppler

effect For example if the relative speed is 1 of the wave speed then the frequency shifts by

1 Remember that this is only an approximation

Shock Waves

When an object moves through a medium at a speed greater than the speed of waves a V-shaped

shock wave is produced The V-shaped wake behind a speeding boat is a good example of this

effect and the cone of sonic-boom behind a supersonic aircraft is another The angle that V-line

makes with the direction of travel of the source is given by

sins

v

v

where v is the wave speed and vs is the source speed

7

Serway Chapter 18

Principle of Linear Superposition

We say that a system obeys the principle of linear superposition if two or more different motions

of the system can simply be added together to find the net motion of the system Light waves

obey this principle as they propagate through the air as can be seen by shining two flashlights so

that their beams cross The beams propagate along without affecting each other (Light sabers are

a spectacular but unfortunately fictional example of systems that do not obey the principle of

superposition) Wave pulses on an ideal rope also obey this principle two different pulses pass

through each other without change Standing waves are an example of this effect being simply

the linear superposition of two traveling waves of the same frequency but moving in opposite

directions Light waves in matter do not always obey this principle For instance two powerful

laser beams could be made to cross in a piece of glass in such a way that their combined heating

effect in the crossing region could melt the glass and scatter the beams in complicated ways This

is an example of a nonlinear effect

Interference

When two or more waves are present in the same medium at the same time their net effect may

often be obtained simply by adding them at each point in the medium according to the principle

of linear superposition (Note this wonrsquot work if the medium is nonlinear) When this addition

makes the total amplitude be greater than the individual amplitudes of the various waves we say

that the interference is constructive When the addition produces cancellation and an amplitude

less than the amplitudes of the separate waves we have destructive interference

Standing Waves

A standing wave is the superposition of two identical traveling waves moving in opposite

directions Nodes are places where the two waves perfectly destructively interfere to produce

zero amplitude at all times Anti-nodes are places where the two waves perfectly constructively

interfere to produce an amplitude maximum The distance between nodes is λ2 Standing waves

on a string fixed at both ends have nodes at each end of the string Standing waves in an air

column enclosed in a tube have displacement anti-nodes at open ends of the tube and

displacement nodes at closed ends The frequency of the standing wave with the lowest possible

frequency is called the fundamental frequency Standing waves on strings or in air columns all

have frequencies which are integer multiples of the fundamental frequency and are called

8

harmonics (The fundamental is called the ldquofirst harmonicrdquo)

Beats

Beats are heard when two waves with slightly different frequencies f1 and f2 are combined The

waves constructively interfere for a number of cycles then destructively interfere for a number

of cycles We hear a periodic ldquowah-wahrdquo frequency equal to the difference of the two wave

frequencies

1 2bf f f

Musical Instruments

Musical instruments produce tones by exciting standing waves on strings (violins piano) and in

tubes (trumpet organ) The fundamental frequency of the standing wave is called the pitch of the

tone The pitch of concert A is 440 Hz by definition Two tones are an octave apart if one pitch

has twice the frequency of the other In written music there are 12 intervals in each octave with

the ratio between successive intervals equal to 2112 = 105946 The ratios for each tone in an

octave starting at A and ending at the next higher A are

A A B C C D D E F F G G A

1 10595 11225 11892 12599 13348 14142 14983 15874 16818 17818 18877 2

A musical tone is actually a superposition of the fundamental frequency and the higher

harmonics The tone quality of a musical instrument is determined by the amplitudes of the

various harmonics that it produces A violin and a trumpet can play the same pitch but they

donrsquot sound at all alike to our ears The difference between them is in the various amplitudes of

their harmonics

9

Serway Chapter 19

Temperature

Formally temperature is what is measured by a thermometer Roughly high temperature is what

we call hot and low temperature is what we call cold On the atomic level temperature refers to

the kinetic energy of the molecules A collection of molecules is called ldquohotrdquo if the molecules

have rapid random motion while a collection of molecules is called ldquocoldrdquo if the random motion

is slow When two bodies are placed in close contact with each other they exchange molecular

kinetic energy until they come to the same temperature This is the microscopic picture of the

Zeroth Law of Thermodynamics

Absolute Zero

Absolute zero is the lowest possible temperature that any object can have This is the temperature

at which all of the energy than can be removed an object has been removed (This removable

energy we call thermal energy) There is still motion at absolute zero Electrons continue to orbit

around atomic nuclei and even atoms continue to move about with a small amount of kinetic

energy but this small energy cannot be removed from the object For example at absolute zero

helium is a liquid whose atoms still move and slide past each other

Temperature Scales

Kelvin Scale Absolute zero is at T = 0 K water freezes at T = 27315 K room temperature is

around T = 295 K and water boils at T = 373 K Note we donrsquot use a deg symbol Kelvin is

prefered SI Unit

Celsius Scale Absolute zero is at T = -273degC water freezes at T = 0degC room temperature is

around T = 22degC and water boils at T = 100degC

Fahrenheit Scale Absolute zero is at T = -459degF water freezes at TF = 32degF room temperature

is around T = 72degF and water boils at TF = 212degF TF =18TC + 32 Notice that temperature

differences are the same for the Kelvin and Celsius scales

Thermal Expansion

When materials are heated they usually expand and when they are cooled they usually contract

(Water near freezing is a spectacular counterexample it works the other way around) The

coefficient of linear expansion is defined by the relation

1

i

L

L T

10

where Li is the initial length of a rod of the material and ΔL is the change in its length due to a

small temperature change ΔT The coefficient of volume expansion is defined similarly

1

i

V

V T

where Vi is the initial volume of a piece of material and where ΔV is the change in its volume due

to a small temperature change ΔT

Avogadrorsquos Number (N or NA)

One mole of any substance corresponds to 6022 times 1023 molecules The atomic mass given on the

periodic table (p A32-33 in the text) is the mass in grams of one mole of atoms For example

the atomic mass of oxygen is 15999 gmol The mass of a single atom is given by the same

number in units of atomic mass units (u) For example the mass of an oxygen atom is 15999 u

(on the average)

Ideal Gas Law (an example of an equation of state)

When the molecules of a gas are sufficiently inert and widely separated that interactions between

them are negligible we say that it is an ideal gas The pressure P volume V and temperature T

(in kelvins) of such a gas are State Variables and are related by the ideal gas law

Bor PV=NkPV nRT T

where n is the number of moles of the gas where R is the gas constant

8314 Jmol KR

where N is the number of molecules and where kB is Boltzmannrsquos constant

231380 10 J KBk

It works well for air at atmosphere pressure and even better for partial vaccuums The relative

ease of measuring pressure and the linear relationship between pressure and temperature (if V

and n are held fixed) makes an ideal gas an ideal thermometer Thermometers can be based on

properties of solids or liquids but the behavior of these materials with temperature is more

complicated

11

Serway Chapter 20

Heat

Heat is energy that flows between a system and its environment because of a tempera-

ture difference between them The units of heat are Joules as expected for an energy

Unfortunately there are several competing units of energy They are related by

1 cal 4186 J 1 Cal 4186 J 1 Btu 1054 J

Heat Capacity

There is often a simple linear relation between the heat that flows in or out of part of a

system and the temperature change that results from this energy transfer When this

linear relation holds it is convenient to define the heat capacity C and the specific

heat c as follows

For the entire object the heat Q it absorbs is given by Q = C(Tf - Ti)

For a particular material the specific heat is defined by c = Cm which is the heat

capacity per unit mass so that

Q = mc(Tf ndash Ti)

Note C has units of energy (J or Cal)(Kelvin kg)

Heats of Transformation or Latent Heat Q = plusmn mL

When a substance changes phase from solid to liquid or from liquid to gas it absorbs

heat without a change in temperature The latent heat or heat of transformation is

usually given per unit mass of the substance For example for water the heat of fusion

(melting) is L = 333 kJkg while the heat of vaporization (boiling) is L = 2260 kJkg

Note that heat for boiling is considerably bigger than melting for water You have to

be careful with signs heat is given off (negative) if you go down in temperature and

condense steam

Work

In general the small amount of work done on a system as a force Fon is exerted on it

through a vector displacement dx is given by

on xdW d F

12

But if the displacement is done very slowly (as we always assume in thermodynamics)

then the force exerted on the system and the force exerted by the system are in

balance so the force exerted by the system is ndash Fon In thermodynamics it is more

convenient to talk about the force exerted by the system so we change the above

formula for the work done on the system to

xdW d F

where F is the force exerted by the system This has confused students for more than a

century now but this is the way your book and many other books do it so you are

stuck You will need to memorize the minus sign in this definition of the work to be

able to use your textbook

There are many chances to get signs wrong in this and the next two chapters (Mosiah

2 )

When an external agent changes the volume of a gas at pressure P by a small amount

dV the (small amount) of work done on the system is given by

dW PdV

Notice that this minus sign is just what we need to make dW be positive if the external

agent compresses the gas for then dV is negative If on the other hand the external

agent gives way allowing the gas to expand against it then dV is positive and we say

that the work done on the gas is negative

The work done on the system (eg by the gas in a cylinder) in a thermodynamic

process is the area under the curve in a PV diagram It is positive for compressions

and negative for expansions If the volume of gas remains constant in a process then

no work is done by the gas

Cyclic processes are important For cyclic processes represented by PV diagrams the

magnitude of the net work during one cycle is simply the area enclosed by the cycle

on the diagram Be careful to keep track of signs when you are calculating that

enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes

1

Path A-B B-C C-D D-A A to A net

Q

W

ΔU

ΔS

Internal Energy

The energy stored in a substance is called its internal energy Eint This energy may be

stored as random kinetic energy or as potential energy in each molecule (stretched

chemical bonds electrons in excited states etc) For ideal gases all states with the

same temperature will have the same Eint

First Law of Thermodynamics

The change ΔEint in the internal energy of a system is given by

intE Q W

where Q is the heat absorbed by the system and where W is the work done on the

system Hence if a system absorbs heat (and if Wge0) the internal energy increases

Likewise if the system does work (W on the system is negative) and if Qge0 the

internal energy decreases Potential Pitfall Many times people talk about work done

by the system It is the minus of W on the system Donrsquot get tripped up

Processes

Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated

from the environment A process may be approximately adiabatic if it happens so

rapidly that heat does not have time to enter or leave the system Work + or ndash is done

and ΔEint = W

Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing

on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal

energy and temperature does not change (Note the difference between an adiabatic

process and a free expansion is that NO work is done in the adiabatic free expansion)

Isobaric process The pressure is held fixed ΔP = 0 For example usually the

pressure increases when a gas is heated but if it were allowed to expand during the

2

heating process in just the right way its pressure could remain fixed In isobaric

processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)

Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is

then zero and so we have ΔEint = Q

Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint

so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in

an isothermal process The work done on the gas is then given by

lnf

i

Vi

Vf

VW PdV nRT

V

Heat Conduction

The quantity P is defined to be the rate at which heat flows through an object and is a

power having units of watts It is analogous to electric current which is the rate at

which charge flows through an object If the flow of heat through a slab of length L

and cross-sectional area A is steady in time then P is given by the equation

h cT TdQkA

dt L

P =

where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The

heat flows of course because of this temperature difference The quantity k is called

the thermal conductivity and is a constant that is characteristic of the material It is

analogous to the electrical conductivity h cT TL is sometimes called the temperature

gradient and is written dTd dTdx

R-Values

It is common to have the heat-conducting properties of materials described by their R-

values especially for insulating materials like fiberglass batting The connection

between k and R is R = Lk where L is the material thickness In this country R-values

always have units of 2ft F hourBtu

Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11

(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)

The heat flow rate through a slab of area A is given by

3

h cT TA

R

P

in units of Btuhour Note that A must be in square feet and the temperatures must be

in degrees Fahrenheit

Convection

Convection is the transfer of thermal energy by flow of material For instance a home

furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct

throughout the house instead it quickly pumps warm air to all of the rooms

Generally convection is a much faster way to transfer heat than conduction

Radiation

Electromagnetic radiation can also transfer heat When you warm yourself near a

campfire which has burned itself down into a bed of glowing embers you are

receiving radiant heat from the infrared portion of the electromagnetic spectrum The

rate at which an object emits radiant heat is given by Stefanrsquos law

4AeTP

where P is the radiated power in watts σ is a constant

8 2 45696 10 W m K

A is the surface area of the object in m2 and T is the temperature in kelvins The

constant e is called the emissivity and it varies from substance to substance A perfect

absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =

0 Hence black objects radiate very well while shiny ones do not Also an object that

is hotter than its surroundings radiates more energy than it absorbs whereas an object

that is cooler than its surroundings absorbs more energy than it radiates

Terminology

Transfer variables vs state variables

Energy transfer by heat as well as work done depends on the initial final and

intermediate states of the system They are transfer variables But their sum (Q + W =

4

Eint) is a state variable

Figure 205

5

Serway Chapter 21

Kinetic Theory

The ideal gas law works for all atoms and molecules at low pressure It is rather

amazing that it does Kinetic theory explains why The properties of an ideal gas can

be understood by thinking of it as N rapidly moving particles of mass m As these

particles collide with the container walls momentum is imparted to the walls which

we call the force of gas pressure In this picture the pressure is related to the average

of the square of the particle velocity 2v by

22 1( )

3 2

NP mv

V

Using the ideal gas law we obtain the average translational kinetic energy per

molecule

21 3

2 2 Bmv k T

The rms speed is then given by

2rms

3 3Bk T RTv v

m M

where M is the molecular mass in kgmol

Degrees of Freedom

Roughly speaking a degree of freedom is a way in which a molecule can store energy

For instance since there are three different directions in space along which a molecule

can move there are three degrees of freedom for the translational kinetic energy

There are also three different axes of rotation about which a polyatomic molecule can

spin so we say there are three degrees of freedom for the rotational kinetic energy

There are even degrees of freedom associated with the various ways in which a

molecule can vibrate and with the different energy levels in which the electrons of

the molecule can exist

Internal Energy and Degrees of Freedom The internal energy of an ideal gas made

up of molecules with J degrees of freedom is given by

int 2 2 B

J JE nRT Nk T

6

Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of

molar heat capacities CV and CP These are the heat capacities per mole and the

subscript V on CV means that the volume is being held constant while for CP the

pressure is held constant For example to raise the temperature of n moles of a gas

whose pressure is held constant by 10 K we would have to supply an amount of heat Q

= nCP (10) K

Molar Specific Heat of an Ideal Gas at Constant Volume

VQ nC T

3monatomic

2VC R

5diatomic

2VC R

5polyatomic

2VC

Real gases deviate from these formulas because in addition to the translational and ro-

tational degrees of freedom they also have vibrational and electronic degrees of

freedom These are unimportant at low temperatures due to quantum mechanical

effects but become increasingly important at higher temperatures The rough rule is

No of degrees of freedom

2VC R

Molar Specific Heat of an Ideal Gas at Constant Pressure

PQ nC T

P VC C R

The internal energy of an ideal gas depends only on the temperature

int VE nC T

Adiabatic Processes in an Ideal Gas

7

An adiabatic process is one in which no heat is exchanged between the system and the

environment When an ideal gas expands or contracts adiabatically not only does its

pressure change as expected from the ideal gas law but its temperature changes as

well Under these conditions the final pressure Pf can be computed from the initial

pressure Pi and from the final and initial volumes Vf and Vi by

or constantf f i iP V PV PV

where γ = CPCV The quantity γ is called the adiabatic exponent Note that this

doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be

combined with the adiabatic law for pressure given above to obtain the adiabatic law

for temperatures

1 constantTV

Compressions in sound waves are adiabatic because they happen too rapidly for any

appreciable amount of heat to flow This is why the adiabatic exponent appears in the

formula for the speed of sound in an ideal gas

RTv

M

Note that v depends only on T and not on P Because it depends only on the

temperature the speed of sound is the same in Provo as at sea level in spite of the

lower pressure here due to the difference in elevation

Equipartition of Energy

Every kind of molecule has a certain number of degrees of freedom which are

independent ways in which it can store energy Each such degree of freedom has

associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1

2 RT per mole)

(Note since a molecule has so many possible degrees of freedom it would seem that

there should be a lot of 12 sBk T to spread around But because energy is quantized

some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high

enough that 12 Bk T is as big as the lowest quantum of energy

8

Serway Chapter 22

Second Law of Thermodynamics

There are several equivalent forms of this important law

Kelvin It is not possible to change heat completely into work with no other change

taking place Or in other words there are no perfect heat engines

Clausius It is not possible for heat to flow from one body to another body at a higher

temperature with no other change taking place Or in other words there are no

perfect refrigerators

Entropy In any thermodynamic process that proceeds from one equilibrium state to

another the entropy of the system + environment either remains unchanged or

increases The total entropy never decreases This law is a bit of an oddity among the

laws of physics because it is not absolute Things are forbidden by the second law not

because it is impossible for them to happen but because it is extremely unlikely for

them to happen (See below for more information about entropy)

Reversible and Irreversible Processes

A reversible process is one which occurs so slowly that it is in thermal equilibrium (or

very nearly so) at all times A hallmark of such processes is that a motion picture of

them looks perfectly normal whether run forward or backward Imagine for instance

the slow expansion of a gas at constant temperature in a cylinder whose volume is being

increased by a slowly moving piston Run the movie backwards and what do you see

You see the slow compression of a gas at constant temperature which looks perfectly

normal

An irreversible process is one which occurs in such a way that thermal equilibrium is

not maintained throughout the process The mark of this kind of process is that a motion

picture of it looks very odd when run backward Imagine the sudden expansion of a gas

into a previously evacuated chamber because a hole was punched in the wall between a

pressurized chamber and the evacuated one Run the movie backward and what do you

see You see the gas in the soon-to-be-evacuated chamber gather itself together and

stream through a tiny hole into a chamber in which there is already plenty of gas If you

have ever seen this happen get in touch with the support group for those who have

witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW

9

Heat Engines

Because of the vexing difference in sign between work done on and system and work done

by a system we will invent a new work variable Weng Heat engines do work and so the net

W for these engines is negative But in engineering applications hidden minus signs are

regarded as evil so for heat engines we donrsquot talk about W instead we talk about its

magnitude engW W So for heat engines the first law is

int engE Q W

But for heat pumps and refrigerators work is done on the system so we use the usual work

W when we talk about these systems

A heat engine is a machine that absorbs heat converts part of it to work and exhausts the

rest The heat must be absorbed at high temperature and exhausted at low temperature If the

absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then

eng h cW Q Q

and the efficiency of the engine is defined to be

eng

h

We

Q

A perfect engine would convert the heat hQ completely into work Weng giving an effi-

ciency of e = 1 Energy conservation alone allows a perfect engine but the second law

requires e lt 1

Refrigerators and Heat Pumps

A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high

temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine

by some source of power A heat pump is a machine that either works like a refrigerator

keeping a place cold by transferring heat from this cold place to a higher temperature

environment (cooling mode like an air conditioner) or it functions as a heater

transferring heat into a warm place from a cooler one (heating mode like a window unit

that heats a house by extracting thermal energy from the cold outdoors) The coefficient

of performance of a refrigerator or of a heat pump in cooling mode is defined to be

COP cooling modecQ

W

10

For a heat pump in heating mode the coefficient of performance is

COP heating modehQ

W

Note that we donrsquot have to use the engineering work here because in these systems

work is done on the system and W is naturally positive

A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without

doing any work giving an infinite coefficient of performance Energy conservation alone

allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump

in heating mode would transfer Qh into the house without doing any work and so would

also have an infinite coefficient of performance The second law forbids this too A good

coefficient of performance for a real device would be around 5 or 6

Carnot Cycle

The most efficient of all possible engines is one that uses the Carnot cycle This cycle

employs an ideal gas has no friction and operates very slowly so that the gas can be in

thermal equilibrium at all parts of the cycle This means of course that it canrsquot

possibly be built and even if it could be built it would not run fast enough to be useful

Nevertheless this cycle is very important because it gives an upper bound on the

efficiency of real engines There cannot possibly be an engine that is more efficient

than one based on the Carnot cycle This cycle consists of the following four steps

1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its

volume The reason that heat is absorbed is that expansion tends to cool the gas but

thermal contact with the environment at Th keeps the temperature high by heat

conduction into the ideal gas

2 The ideal gas further increases its volume by an adiabatic expansion This expansion

causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc

3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in

volume The reason that heat is exhausted is that compression tends to heat the gas

but thermal contact with the environment at Tc keeps the temperature low by heat

conduction out of the ideal gas

4 The gas is adiabatically compressed back to its original volume (the volume it started

with in step 1) This compression heats the gas from Tc up to Th

11

The efficiency of a Carnot engine is given by the very simple formula

1 cC

h

Te

T

where the temperatures must in be Kelvin No real engine can be more efficient than

this

The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode

is given by

COP cooling mode cC

h c

T

T T

and no real refrigerator can have a coefficient of performance greater than this

The coefficient of performance of a Carnot heat pump in heating mode is

COP heating mode hC

h c

T

T T

So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency

To make the reversible steps in the cycle really reversible they would have to occur

infinitely slowly So the price you pay for making a perfect engine is that it takes

forever to get it to do any work

Entropy

The entropy of a system is defined in terms of its molecular makeup and measures

roughly the disorder of the system If the system is packed into a very small volume

then it is quite ordered and the entropy will be low If it occupies a large volume the

entropy is high (To see what this has to do with disorder note that socks in a drawer

occupy a small volume while socks on the bed in the corner by the door and

hanging from the chandelier occupy a large volume) If the system is very cold then

the molecules hardly move and may even reach out to each other and form a crystal

This is a highly ordered state and therefore has low entropy If the system is very hot

with rapidly speeding molecules crashing into the container walls and bouncing off

each other things are disordered and the entropy is high

It is possible to calculate the entropy of a system in terms of its macroscopic thermody-

namic properties ie pressure volume temperature number of moles etc The key to this

calculation is the concept of a reversible process A reversible process is one that is

12

carried out without friction and so slowly that the process can be reversed at any stage by

making an infinitesimal change in the environment of the system The slow expansion of the

gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment

is reversible If we turned around and began slowly to compress the gas would just slowly

exhaust heat to the environment in the exact reverse way that it absorbed it during expansion

Most processes however are irreversible For example if a gas-filled box were suddenly

increased in size so that the particles were free to wander into the void created by the sudden

expansion then the gas would eventually fill the new volume uniformly at the same

temperature as before the expansion (The temperature is unchanged in this imaginary

process because the kinetic energy of the molecules would be unaffected by such an

instantaneous expansion of the container walls) This imaginary but highly thought-

stimulating process is called a free expansion and it is impossible to reverse it During the

expansion we didnrsquot push on any of the molecules so reversing this process would mean

making them go back into their original volume without pushing on them they simply will

not cooperate to this extent Another way to see that just pushing them back where they came

from does not reverse the free expansion is to think about what would happen if we just

compressed either adiabatically or isothermally An adiabatic compression back to the

original volume would heat the gas above its original temperature and an isothermal

compression would require that heat be exhausted to the environment But the free expansion

involved neither temperature changes nor heat exchanges so neither of these two processes

is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in

a way that takes us back to the initial state

It is possible to calculate the change of entropy for both reversible and irreversible processes

Letrsquos consider a reversible process first In a reversible process the entropy change is given

by the formula

dQS

T

where dQ is the amount of heat added to the system during a small step of the process

The total energy change during the process may then simply be calculated by integration

f f

f i i i

dQS S S dS

T

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11

3

Serway Chapter 16

Longitudinal and Transverse Waves

A longitudinal wave is one in which the direction of vibration of the medium and the direction of

propagation of the wave are in the same direction Sound waves in the air in water and in solids

are longitudinal waves

A transverse wave is one in which the direction of vibration of the medium is perpendicular to

the direction of propagation of the wave Ocean waves waves on ropes and electromagnetic

waves are transverse waves

Wavelength and Wavenumber

The wavelength λ is the period in space of the wave ie it is the distance from one wave crest to

the next The wavenumber k (units reciprocal length) is related to the wavelength by the formula

2k

Frequency and Angular Frequency

The period T is the period in time of the wave ie it is the time between the arrival of one wave

crest and the next at some point in space The frequency f is the number of crests that arrive per

second which is just 1T and the angular frequency ω is 2π times the frequency Both have units

of per second (reciprocal time)

1 22f f

T T

Wave Speed

The wave speed of a wave is defined to be the the speed at which the wave crests or troughs

move through the medium This speed is related to the frequency and the wavelength by the

formulas

v fk

where k is the wavenumber of the wave v is also called the phase velocity

Traveling Waves

Traveling waves are oscillations in which the wave crests and troughs glide smoothly through the

medium Waves on oceans and lakes are traveling waves The mathematical form of a harmonic

(sinusoidal) traveling wave is

4

siny x t A kx t

where A is the amplitude of the wave As usual ω and k are related by the relation ω = kv where

v is the wave speed the speed at which the crests travel the minus sign means that the wave is

traveling to the right towards higher values of x φ is the phase and constrols where the maxima

and minima occur

Speed of Transverse Waves on a Rope

The velocity of transverse waves on a perfectly flexible rope is given by

Tv

where T is the tension in the rope and where μ is its linear mass density (mass per unit length)

Common Test Questions

A common question is to be given an equation for a wave and to be asked what is amplitude

velocity etc Another common type of question is to be given facts about a wave like its

direction velocity and magnitude at a given place and time and to be asked which of several

equations correctly describe the wave

5

Serway Chapter 17

Sound Speed in Solids Liquids and Gases

The speed of sound in a liquid or solid of bulk modulus B and volume mass density ρ

Bv

The speed of sound in air is

331 1 273v m s T C

The speed of sound in air at room temperature is about 343 ms Notice it doesnrsquot depend on

pressure or frequency

Power Intensity and Loudness

The intensity of a sound wave is defined to be the power per unit area ie wattsm2 in the wave

For spherical waves traveling away from a small source emitting waves with average power Рav

the intensity falls off with distance from the source r according to the inverse-square law

av24

Ir

Hence the intensity I2 at distance r2 is related to the intensity I1 at a different distance r1 by

21 2

22 1

I r

I r

Experiments on human hearing have shown that we hear intensity differences logarithmically so

the decibel loudness scale for sound intensity was invented The loudness β of a sound in

decibels is related to its intensity I in Wm2 by the formula

1010logo

I

I

Where Io is a sound intensity near the threshold of hearing defined to be Io = 10-12 Wm2 Note

that on this scale a sound is made 10 times more intense by adding 10 decibels Remember that

intensity is proportional to velocity squared Amplitude is the A is the previous chapterrsquos

equation for a traveling wave If you make the A 10 times as much the intensity will increase by

100 times (20 dB)

Doppler Effect

If sound waves are traveling through a medium and if either the receiver of the waves or the

source of waves is moving then the frequency received is related to the frequency emitted by

6

0

s

v vf f

v v

where fprime is the frequency detected by the observer f is the frequency emitted by the source v is

the speed of the waves vo is the speed of the observer and vs is the speed of the source This

formula assumes that the source and receiver are either moving directly toward each other or

directly away from each other To know which signs to use remember that when observer and

source approach each other the observed frequency is higher while if they move away from each

other it is lower Just examine the signs in the formula and make the answer come out right

For electromagnetic waves (light radio waves X-rays) traveling in vacuum Einsteinrsquos theory of

relativity (and careful experiments) show that the Doppler shift is given by

r

r

c vf f

c v

where vr is the relative speed between the source and the observer

For all kinds of waves (sound light etc) if the relative speed of the source and the observer is

small compared to the speed of the waves then there is a simple approximation to the Doppler

effect For example if the relative speed is 1 of the wave speed then the frequency shifts by

1 Remember that this is only an approximation

Shock Waves

When an object moves through a medium at a speed greater than the speed of waves a V-shaped

shock wave is produced The V-shaped wake behind a speeding boat is a good example of this

effect and the cone of sonic-boom behind a supersonic aircraft is another The angle that V-line

makes with the direction of travel of the source is given by

sins

v

v

where v is the wave speed and vs is the source speed

7

Serway Chapter 18

Principle of Linear Superposition

We say that a system obeys the principle of linear superposition if two or more different motions

of the system can simply be added together to find the net motion of the system Light waves

obey this principle as they propagate through the air as can be seen by shining two flashlights so

that their beams cross The beams propagate along without affecting each other (Light sabers are

a spectacular but unfortunately fictional example of systems that do not obey the principle of

superposition) Wave pulses on an ideal rope also obey this principle two different pulses pass

through each other without change Standing waves are an example of this effect being simply

the linear superposition of two traveling waves of the same frequency but moving in opposite

directions Light waves in matter do not always obey this principle For instance two powerful

laser beams could be made to cross in a piece of glass in such a way that their combined heating

effect in the crossing region could melt the glass and scatter the beams in complicated ways This

is an example of a nonlinear effect

Interference

When two or more waves are present in the same medium at the same time their net effect may

often be obtained simply by adding them at each point in the medium according to the principle

of linear superposition (Note this wonrsquot work if the medium is nonlinear) When this addition

makes the total amplitude be greater than the individual amplitudes of the various waves we say

that the interference is constructive When the addition produces cancellation and an amplitude

less than the amplitudes of the separate waves we have destructive interference

Standing Waves

A standing wave is the superposition of two identical traveling waves moving in opposite

directions Nodes are places where the two waves perfectly destructively interfere to produce

zero amplitude at all times Anti-nodes are places where the two waves perfectly constructively

interfere to produce an amplitude maximum The distance between nodes is λ2 Standing waves

on a string fixed at both ends have nodes at each end of the string Standing waves in an air

column enclosed in a tube have displacement anti-nodes at open ends of the tube and

displacement nodes at closed ends The frequency of the standing wave with the lowest possible

frequency is called the fundamental frequency Standing waves on strings or in air columns all

have frequencies which are integer multiples of the fundamental frequency and are called

8

harmonics (The fundamental is called the ldquofirst harmonicrdquo)

Beats

Beats are heard when two waves with slightly different frequencies f1 and f2 are combined The

waves constructively interfere for a number of cycles then destructively interfere for a number

of cycles We hear a periodic ldquowah-wahrdquo frequency equal to the difference of the two wave

frequencies

1 2bf f f

Musical Instruments

Musical instruments produce tones by exciting standing waves on strings (violins piano) and in

tubes (trumpet organ) The fundamental frequency of the standing wave is called the pitch of the

tone The pitch of concert A is 440 Hz by definition Two tones are an octave apart if one pitch

has twice the frequency of the other In written music there are 12 intervals in each octave with

the ratio between successive intervals equal to 2112 = 105946 The ratios for each tone in an

octave starting at A and ending at the next higher A are

A A B C C D D E F F G G A

1 10595 11225 11892 12599 13348 14142 14983 15874 16818 17818 18877 2

A musical tone is actually a superposition of the fundamental frequency and the higher

harmonics The tone quality of a musical instrument is determined by the amplitudes of the

various harmonics that it produces A violin and a trumpet can play the same pitch but they

donrsquot sound at all alike to our ears The difference between them is in the various amplitudes of

their harmonics

9

Serway Chapter 19

Temperature

Formally temperature is what is measured by a thermometer Roughly high temperature is what

we call hot and low temperature is what we call cold On the atomic level temperature refers to

the kinetic energy of the molecules A collection of molecules is called ldquohotrdquo if the molecules

have rapid random motion while a collection of molecules is called ldquocoldrdquo if the random motion

is slow When two bodies are placed in close contact with each other they exchange molecular

kinetic energy until they come to the same temperature This is the microscopic picture of the

Zeroth Law of Thermodynamics

Absolute Zero

Absolute zero is the lowest possible temperature that any object can have This is the temperature

at which all of the energy than can be removed an object has been removed (This removable

energy we call thermal energy) There is still motion at absolute zero Electrons continue to orbit

around atomic nuclei and even atoms continue to move about with a small amount of kinetic

energy but this small energy cannot be removed from the object For example at absolute zero

helium is a liquid whose atoms still move and slide past each other

Temperature Scales

Kelvin Scale Absolute zero is at T = 0 K water freezes at T = 27315 K room temperature is

around T = 295 K and water boils at T = 373 K Note we donrsquot use a deg symbol Kelvin is

prefered SI Unit

Celsius Scale Absolute zero is at T = -273degC water freezes at T = 0degC room temperature is

around T = 22degC and water boils at T = 100degC

Fahrenheit Scale Absolute zero is at T = -459degF water freezes at TF = 32degF room temperature

is around T = 72degF and water boils at TF = 212degF TF =18TC + 32 Notice that temperature

differences are the same for the Kelvin and Celsius scales

Thermal Expansion

When materials are heated they usually expand and when they are cooled they usually contract

(Water near freezing is a spectacular counterexample it works the other way around) The

coefficient of linear expansion is defined by the relation

1

i

L

L T

10

where Li is the initial length of a rod of the material and ΔL is the change in its length due to a

small temperature change ΔT The coefficient of volume expansion is defined similarly

1

i

V

V T

where Vi is the initial volume of a piece of material and where ΔV is the change in its volume due

to a small temperature change ΔT

Avogadrorsquos Number (N or NA)

One mole of any substance corresponds to 6022 times 1023 molecules The atomic mass given on the

periodic table (p A32-33 in the text) is the mass in grams of one mole of atoms For example

the atomic mass of oxygen is 15999 gmol The mass of a single atom is given by the same

number in units of atomic mass units (u) For example the mass of an oxygen atom is 15999 u

(on the average)

Ideal Gas Law (an example of an equation of state)

When the molecules of a gas are sufficiently inert and widely separated that interactions between

them are negligible we say that it is an ideal gas The pressure P volume V and temperature T

(in kelvins) of such a gas are State Variables and are related by the ideal gas law

Bor PV=NkPV nRT T

where n is the number of moles of the gas where R is the gas constant

8314 Jmol KR

where N is the number of molecules and where kB is Boltzmannrsquos constant

231380 10 J KBk

It works well for air at atmosphere pressure and even better for partial vaccuums The relative

ease of measuring pressure and the linear relationship between pressure and temperature (if V

and n are held fixed) makes an ideal gas an ideal thermometer Thermometers can be based on

properties of solids or liquids but the behavior of these materials with temperature is more

complicated

11

Serway Chapter 20

Heat

Heat is energy that flows between a system and its environment because of a tempera-

ture difference between them The units of heat are Joules as expected for an energy

Unfortunately there are several competing units of energy They are related by

1 cal 4186 J 1 Cal 4186 J 1 Btu 1054 J

Heat Capacity

There is often a simple linear relation between the heat that flows in or out of part of a

system and the temperature change that results from this energy transfer When this

linear relation holds it is convenient to define the heat capacity C and the specific

heat c as follows

For the entire object the heat Q it absorbs is given by Q = C(Tf - Ti)

For a particular material the specific heat is defined by c = Cm which is the heat

capacity per unit mass so that

Q = mc(Tf ndash Ti)

Note C has units of energy (J or Cal)(Kelvin kg)

Heats of Transformation or Latent Heat Q = plusmn mL

When a substance changes phase from solid to liquid or from liquid to gas it absorbs

heat without a change in temperature The latent heat or heat of transformation is

usually given per unit mass of the substance For example for water the heat of fusion

(melting) is L = 333 kJkg while the heat of vaporization (boiling) is L = 2260 kJkg

Note that heat for boiling is considerably bigger than melting for water You have to

be careful with signs heat is given off (negative) if you go down in temperature and

condense steam

Work

In general the small amount of work done on a system as a force Fon is exerted on it

through a vector displacement dx is given by

on xdW d F

12

But if the displacement is done very slowly (as we always assume in thermodynamics)

then the force exerted on the system and the force exerted by the system are in

balance so the force exerted by the system is ndash Fon In thermodynamics it is more

convenient to talk about the force exerted by the system so we change the above

formula for the work done on the system to

xdW d F

where F is the force exerted by the system This has confused students for more than a

century now but this is the way your book and many other books do it so you are

stuck You will need to memorize the minus sign in this definition of the work to be

able to use your textbook

There are many chances to get signs wrong in this and the next two chapters (Mosiah

2 )

When an external agent changes the volume of a gas at pressure P by a small amount

dV the (small amount) of work done on the system is given by

dW PdV

Notice that this minus sign is just what we need to make dW be positive if the external

agent compresses the gas for then dV is negative If on the other hand the external

agent gives way allowing the gas to expand against it then dV is positive and we say

that the work done on the gas is negative

The work done on the system (eg by the gas in a cylinder) in a thermodynamic

process is the area under the curve in a PV diagram It is positive for compressions

and negative for expansions If the volume of gas remains constant in a process then

no work is done by the gas

Cyclic processes are important For cyclic processes represented by PV diagrams the

magnitude of the net work during one cycle is simply the area enclosed by the cycle

on the diagram Be careful to keep track of signs when you are calculating that

enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes

1

Path A-B B-C C-D D-A A to A net

Q

W

ΔU

ΔS

Internal Energy

The energy stored in a substance is called its internal energy Eint This energy may be

stored as random kinetic energy or as potential energy in each molecule (stretched

chemical bonds electrons in excited states etc) For ideal gases all states with the

same temperature will have the same Eint

First Law of Thermodynamics

The change ΔEint in the internal energy of a system is given by

intE Q W

where Q is the heat absorbed by the system and where W is the work done on the

system Hence if a system absorbs heat (and if Wge0) the internal energy increases

Likewise if the system does work (W on the system is negative) and if Qge0 the

internal energy decreases Potential Pitfall Many times people talk about work done

by the system It is the minus of W on the system Donrsquot get tripped up

Processes

Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated

from the environment A process may be approximately adiabatic if it happens so

rapidly that heat does not have time to enter or leave the system Work + or ndash is done

and ΔEint = W

Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing

on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal

energy and temperature does not change (Note the difference between an adiabatic

process and a free expansion is that NO work is done in the adiabatic free expansion)

Isobaric process The pressure is held fixed ΔP = 0 For example usually the

pressure increases when a gas is heated but if it were allowed to expand during the

2

heating process in just the right way its pressure could remain fixed In isobaric

processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)

Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is

then zero and so we have ΔEint = Q

Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint

so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in

an isothermal process The work done on the gas is then given by

lnf

i

Vi

Vf

VW PdV nRT

V

Heat Conduction

The quantity P is defined to be the rate at which heat flows through an object and is a

power having units of watts It is analogous to electric current which is the rate at

which charge flows through an object If the flow of heat through a slab of length L

and cross-sectional area A is steady in time then P is given by the equation

h cT TdQkA

dt L

P =

where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The

heat flows of course because of this temperature difference The quantity k is called

the thermal conductivity and is a constant that is characteristic of the material It is

analogous to the electrical conductivity h cT TL is sometimes called the temperature

gradient and is written dTd dTdx

R-Values

It is common to have the heat-conducting properties of materials described by their R-

values especially for insulating materials like fiberglass batting The connection

between k and R is R = Lk where L is the material thickness In this country R-values

always have units of 2ft F hourBtu

Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11

(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)

The heat flow rate through a slab of area A is given by

3

h cT TA

R

P

in units of Btuhour Note that A must be in square feet and the temperatures must be

in degrees Fahrenheit

Convection

Convection is the transfer of thermal energy by flow of material For instance a home

furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct

throughout the house instead it quickly pumps warm air to all of the rooms

Generally convection is a much faster way to transfer heat than conduction

Radiation

Electromagnetic radiation can also transfer heat When you warm yourself near a

campfire which has burned itself down into a bed of glowing embers you are

receiving radiant heat from the infrared portion of the electromagnetic spectrum The

rate at which an object emits radiant heat is given by Stefanrsquos law

4AeTP

where P is the radiated power in watts σ is a constant

8 2 45696 10 W m K

A is the surface area of the object in m2 and T is the temperature in kelvins The

constant e is called the emissivity and it varies from substance to substance A perfect

absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =

0 Hence black objects radiate very well while shiny ones do not Also an object that

is hotter than its surroundings radiates more energy than it absorbs whereas an object

that is cooler than its surroundings absorbs more energy than it radiates

Terminology

Transfer variables vs state variables

Energy transfer by heat as well as work done depends on the initial final and

intermediate states of the system They are transfer variables But their sum (Q + W =

4

Eint) is a state variable

Figure 205

5

Serway Chapter 21

Kinetic Theory

The ideal gas law works for all atoms and molecules at low pressure It is rather

amazing that it does Kinetic theory explains why The properties of an ideal gas can

be understood by thinking of it as N rapidly moving particles of mass m As these

particles collide with the container walls momentum is imparted to the walls which

we call the force of gas pressure In this picture the pressure is related to the average

of the square of the particle velocity 2v by

22 1( )

3 2

NP mv

V

Using the ideal gas law we obtain the average translational kinetic energy per

molecule

21 3

2 2 Bmv k T

The rms speed is then given by

2rms

3 3Bk T RTv v

m M

where M is the molecular mass in kgmol

Degrees of Freedom

Roughly speaking a degree of freedom is a way in which a molecule can store energy

For instance since there are three different directions in space along which a molecule

can move there are three degrees of freedom for the translational kinetic energy

There are also three different axes of rotation about which a polyatomic molecule can

spin so we say there are three degrees of freedom for the rotational kinetic energy

There are even degrees of freedom associated with the various ways in which a

molecule can vibrate and with the different energy levels in which the electrons of

the molecule can exist

Internal Energy and Degrees of Freedom The internal energy of an ideal gas made

up of molecules with J degrees of freedom is given by

int 2 2 B

J JE nRT Nk T

6

Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of

molar heat capacities CV and CP These are the heat capacities per mole and the

subscript V on CV means that the volume is being held constant while for CP the

pressure is held constant For example to raise the temperature of n moles of a gas

whose pressure is held constant by 10 K we would have to supply an amount of heat Q

= nCP (10) K

Molar Specific Heat of an Ideal Gas at Constant Volume

VQ nC T

3monatomic

2VC R

5diatomic

2VC R

5polyatomic

2VC

Real gases deviate from these formulas because in addition to the translational and ro-

tational degrees of freedom they also have vibrational and electronic degrees of

freedom These are unimportant at low temperatures due to quantum mechanical

effects but become increasingly important at higher temperatures The rough rule is

No of degrees of freedom

2VC R

Molar Specific Heat of an Ideal Gas at Constant Pressure

PQ nC T

P VC C R

The internal energy of an ideal gas depends only on the temperature

int VE nC T

Adiabatic Processes in an Ideal Gas

7

An adiabatic process is one in which no heat is exchanged between the system and the

environment When an ideal gas expands or contracts adiabatically not only does its

pressure change as expected from the ideal gas law but its temperature changes as

well Under these conditions the final pressure Pf can be computed from the initial

pressure Pi and from the final and initial volumes Vf and Vi by

or constantf f i iP V PV PV

where γ = CPCV The quantity γ is called the adiabatic exponent Note that this

doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be

combined with the adiabatic law for pressure given above to obtain the adiabatic law

for temperatures

1 constantTV

Compressions in sound waves are adiabatic because they happen too rapidly for any

appreciable amount of heat to flow This is why the adiabatic exponent appears in the

formula for the speed of sound in an ideal gas

RTv

M

Note that v depends only on T and not on P Because it depends only on the

temperature the speed of sound is the same in Provo as at sea level in spite of the

lower pressure here due to the difference in elevation

Equipartition of Energy

Every kind of molecule has a certain number of degrees of freedom which are

independent ways in which it can store energy Each such degree of freedom has

associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1

2 RT per mole)

(Note since a molecule has so many possible degrees of freedom it would seem that

there should be a lot of 12 sBk T to spread around But because energy is quantized

some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high

enough that 12 Bk T is as big as the lowest quantum of energy

8

Serway Chapter 22

Second Law of Thermodynamics

There are several equivalent forms of this important law

Kelvin It is not possible to change heat completely into work with no other change

taking place Or in other words there are no perfect heat engines

Clausius It is not possible for heat to flow from one body to another body at a higher

temperature with no other change taking place Or in other words there are no

perfect refrigerators

Entropy In any thermodynamic process that proceeds from one equilibrium state to

another the entropy of the system + environment either remains unchanged or

increases The total entropy never decreases This law is a bit of an oddity among the

laws of physics because it is not absolute Things are forbidden by the second law not

because it is impossible for them to happen but because it is extremely unlikely for

them to happen (See below for more information about entropy)

Reversible and Irreversible Processes

A reversible process is one which occurs so slowly that it is in thermal equilibrium (or

very nearly so) at all times A hallmark of such processes is that a motion picture of

them looks perfectly normal whether run forward or backward Imagine for instance

the slow expansion of a gas at constant temperature in a cylinder whose volume is being

increased by a slowly moving piston Run the movie backwards and what do you see

You see the slow compression of a gas at constant temperature which looks perfectly

normal

An irreversible process is one which occurs in such a way that thermal equilibrium is

not maintained throughout the process The mark of this kind of process is that a motion

picture of it looks very odd when run backward Imagine the sudden expansion of a gas

into a previously evacuated chamber because a hole was punched in the wall between a

pressurized chamber and the evacuated one Run the movie backward and what do you

see You see the gas in the soon-to-be-evacuated chamber gather itself together and

stream through a tiny hole into a chamber in which there is already plenty of gas If you

have ever seen this happen get in touch with the support group for those who have

witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW

9

Heat Engines

Because of the vexing difference in sign between work done on and system and work done

by a system we will invent a new work variable Weng Heat engines do work and so the net

W for these engines is negative But in engineering applications hidden minus signs are

regarded as evil so for heat engines we donrsquot talk about W instead we talk about its

magnitude engW W So for heat engines the first law is

int engE Q W

But for heat pumps and refrigerators work is done on the system so we use the usual work

W when we talk about these systems

A heat engine is a machine that absorbs heat converts part of it to work and exhausts the

rest The heat must be absorbed at high temperature and exhausted at low temperature If the

absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then

eng h cW Q Q

and the efficiency of the engine is defined to be

eng

h

We

Q

A perfect engine would convert the heat hQ completely into work Weng giving an effi-

ciency of e = 1 Energy conservation alone allows a perfect engine but the second law

requires e lt 1

Refrigerators and Heat Pumps

A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high

temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine

by some source of power A heat pump is a machine that either works like a refrigerator

keeping a place cold by transferring heat from this cold place to a higher temperature

environment (cooling mode like an air conditioner) or it functions as a heater

transferring heat into a warm place from a cooler one (heating mode like a window unit

that heats a house by extracting thermal energy from the cold outdoors) The coefficient

of performance of a refrigerator or of a heat pump in cooling mode is defined to be

COP cooling modecQ

W

10

For a heat pump in heating mode the coefficient of performance is

COP heating modehQ

W

Note that we donrsquot have to use the engineering work here because in these systems

work is done on the system and W is naturally positive

A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without

doing any work giving an infinite coefficient of performance Energy conservation alone

allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump

in heating mode would transfer Qh into the house without doing any work and so would

also have an infinite coefficient of performance The second law forbids this too A good

coefficient of performance for a real device would be around 5 or 6

Carnot Cycle

The most efficient of all possible engines is one that uses the Carnot cycle This cycle

employs an ideal gas has no friction and operates very slowly so that the gas can be in

thermal equilibrium at all parts of the cycle This means of course that it canrsquot

possibly be built and even if it could be built it would not run fast enough to be useful

Nevertheless this cycle is very important because it gives an upper bound on the

efficiency of real engines There cannot possibly be an engine that is more efficient

than one based on the Carnot cycle This cycle consists of the following four steps

1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its

volume The reason that heat is absorbed is that expansion tends to cool the gas but

thermal contact with the environment at Th keeps the temperature high by heat

conduction into the ideal gas

2 The ideal gas further increases its volume by an adiabatic expansion This expansion

causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc

3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in

volume The reason that heat is exhausted is that compression tends to heat the gas

but thermal contact with the environment at Tc keeps the temperature low by heat

conduction out of the ideal gas

4 The gas is adiabatically compressed back to its original volume (the volume it started

with in step 1) This compression heats the gas from Tc up to Th

11

The efficiency of a Carnot engine is given by the very simple formula

1 cC

h

Te

T

where the temperatures must in be Kelvin No real engine can be more efficient than

this

The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode

is given by

COP cooling mode cC

h c

T

T T

and no real refrigerator can have a coefficient of performance greater than this

The coefficient of performance of a Carnot heat pump in heating mode is

COP heating mode hC

h c

T

T T

So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency

To make the reversible steps in the cycle really reversible they would have to occur

infinitely slowly So the price you pay for making a perfect engine is that it takes

forever to get it to do any work

Entropy

The entropy of a system is defined in terms of its molecular makeup and measures

roughly the disorder of the system If the system is packed into a very small volume

then it is quite ordered and the entropy will be low If it occupies a large volume the

entropy is high (To see what this has to do with disorder note that socks in a drawer

occupy a small volume while socks on the bed in the corner by the door and

hanging from the chandelier occupy a large volume) If the system is very cold then

the molecules hardly move and may even reach out to each other and form a crystal

This is a highly ordered state and therefore has low entropy If the system is very hot

with rapidly speeding molecules crashing into the container walls and bouncing off

each other things are disordered and the entropy is high

It is possible to calculate the entropy of a system in terms of its macroscopic thermody-

namic properties ie pressure volume temperature number of moles etc The key to this

calculation is the concept of a reversible process A reversible process is one that is

12

carried out without friction and so slowly that the process can be reversed at any stage by

making an infinitesimal change in the environment of the system The slow expansion of the

gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment

is reversible If we turned around and began slowly to compress the gas would just slowly

exhaust heat to the environment in the exact reverse way that it absorbed it during expansion

Most processes however are irreversible For example if a gas-filled box were suddenly

increased in size so that the particles were free to wander into the void created by the sudden

expansion then the gas would eventually fill the new volume uniformly at the same

temperature as before the expansion (The temperature is unchanged in this imaginary

process because the kinetic energy of the molecules would be unaffected by such an

instantaneous expansion of the container walls) This imaginary but highly thought-

stimulating process is called a free expansion and it is impossible to reverse it During the

expansion we didnrsquot push on any of the molecules so reversing this process would mean

making them go back into their original volume without pushing on them they simply will

not cooperate to this extent Another way to see that just pushing them back where they came

from does not reverse the free expansion is to think about what would happen if we just

compressed either adiabatically or isothermally An adiabatic compression back to the

original volume would heat the gas above its original temperature and an isothermal

compression would require that heat be exhausted to the environment But the free expansion

involved neither temperature changes nor heat exchanges so neither of these two processes

is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in

a way that takes us back to the initial state

It is possible to calculate the change of entropy for both reversible and irreversible processes

Letrsquos consider a reversible process first In a reversible process the entropy change is given

by the formula

dQS

T

where dQ is the amount of heat added to the system during a small step of the process

The total energy change during the process may then simply be calculated by integration

f f

f i i i

dQS S S dS

T

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11

4

siny x t A kx t

where A is the amplitude of the wave As usual ω and k are related by the relation ω = kv where

v is the wave speed the speed at which the crests travel the minus sign means that the wave is

traveling to the right towards higher values of x φ is the phase and constrols where the maxima

and minima occur

Speed of Transverse Waves on a Rope

The velocity of transverse waves on a perfectly flexible rope is given by

Tv

where T is the tension in the rope and where μ is its linear mass density (mass per unit length)

Common Test Questions

A common question is to be given an equation for a wave and to be asked what is amplitude

velocity etc Another common type of question is to be given facts about a wave like its

direction velocity and magnitude at a given place and time and to be asked which of several

equations correctly describe the wave

5

Serway Chapter 17

Sound Speed in Solids Liquids and Gases

The speed of sound in a liquid or solid of bulk modulus B and volume mass density ρ

Bv

The speed of sound in air is

331 1 273v m s T C

The speed of sound in air at room temperature is about 343 ms Notice it doesnrsquot depend on

pressure or frequency

Power Intensity and Loudness

The intensity of a sound wave is defined to be the power per unit area ie wattsm2 in the wave

For spherical waves traveling away from a small source emitting waves with average power Рav

the intensity falls off with distance from the source r according to the inverse-square law

av24

Ir

Hence the intensity I2 at distance r2 is related to the intensity I1 at a different distance r1 by

21 2

22 1

I r

I r

Experiments on human hearing have shown that we hear intensity differences logarithmically so

the decibel loudness scale for sound intensity was invented The loudness β of a sound in

decibels is related to its intensity I in Wm2 by the formula

1010logo

I

I

Where Io is a sound intensity near the threshold of hearing defined to be Io = 10-12 Wm2 Note

that on this scale a sound is made 10 times more intense by adding 10 decibels Remember that

intensity is proportional to velocity squared Amplitude is the A is the previous chapterrsquos

equation for a traveling wave If you make the A 10 times as much the intensity will increase by

100 times (20 dB)

Doppler Effect

If sound waves are traveling through a medium and if either the receiver of the waves or the

source of waves is moving then the frequency received is related to the frequency emitted by

6

0

s

v vf f

v v

where fprime is the frequency detected by the observer f is the frequency emitted by the source v is

the speed of the waves vo is the speed of the observer and vs is the speed of the source This

formula assumes that the source and receiver are either moving directly toward each other or

directly away from each other To know which signs to use remember that when observer and

source approach each other the observed frequency is higher while if they move away from each

other it is lower Just examine the signs in the formula and make the answer come out right

For electromagnetic waves (light radio waves X-rays) traveling in vacuum Einsteinrsquos theory of

relativity (and careful experiments) show that the Doppler shift is given by

r

r

c vf f

c v

where vr is the relative speed between the source and the observer

For all kinds of waves (sound light etc) if the relative speed of the source and the observer is

small compared to the speed of the waves then there is a simple approximation to the Doppler

effect For example if the relative speed is 1 of the wave speed then the frequency shifts by

1 Remember that this is only an approximation

Shock Waves

When an object moves through a medium at a speed greater than the speed of waves a V-shaped

shock wave is produced The V-shaped wake behind a speeding boat is a good example of this

effect and the cone of sonic-boom behind a supersonic aircraft is another The angle that V-line

makes with the direction of travel of the source is given by

sins

v

v

where v is the wave speed and vs is the source speed

7

Serway Chapter 18

Principle of Linear Superposition

We say that a system obeys the principle of linear superposition if two or more different motions

of the system can simply be added together to find the net motion of the system Light waves

obey this principle as they propagate through the air as can be seen by shining two flashlights so

that their beams cross The beams propagate along without affecting each other (Light sabers are

a spectacular but unfortunately fictional example of systems that do not obey the principle of

superposition) Wave pulses on an ideal rope also obey this principle two different pulses pass

through each other without change Standing waves are an example of this effect being simply

the linear superposition of two traveling waves of the same frequency but moving in opposite

directions Light waves in matter do not always obey this principle For instance two powerful

laser beams could be made to cross in a piece of glass in such a way that their combined heating

effect in the crossing region could melt the glass and scatter the beams in complicated ways This

is an example of a nonlinear effect

Interference

When two or more waves are present in the same medium at the same time their net effect may

often be obtained simply by adding them at each point in the medium according to the principle

of linear superposition (Note this wonrsquot work if the medium is nonlinear) When this addition

makes the total amplitude be greater than the individual amplitudes of the various waves we say

that the interference is constructive When the addition produces cancellation and an amplitude

less than the amplitudes of the separate waves we have destructive interference

Standing Waves

A standing wave is the superposition of two identical traveling waves moving in opposite

directions Nodes are places where the two waves perfectly destructively interfere to produce

zero amplitude at all times Anti-nodes are places where the two waves perfectly constructively

interfere to produce an amplitude maximum The distance between nodes is λ2 Standing waves

on a string fixed at both ends have nodes at each end of the string Standing waves in an air

column enclosed in a tube have displacement anti-nodes at open ends of the tube and

displacement nodes at closed ends The frequency of the standing wave with the lowest possible

frequency is called the fundamental frequency Standing waves on strings or in air columns all

have frequencies which are integer multiples of the fundamental frequency and are called

8

harmonics (The fundamental is called the ldquofirst harmonicrdquo)

Beats

Beats are heard when two waves with slightly different frequencies f1 and f2 are combined The

waves constructively interfere for a number of cycles then destructively interfere for a number

of cycles We hear a periodic ldquowah-wahrdquo frequency equal to the difference of the two wave

frequencies

1 2bf f f

Musical Instruments

Musical instruments produce tones by exciting standing waves on strings (violins piano) and in

tubes (trumpet organ) The fundamental frequency of the standing wave is called the pitch of the

tone The pitch of concert A is 440 Hz by definition Two tones are an octave apart if one pitch

has twice the frequency of the other In written music there are 12 intervals in each octave with

the ratio between successive intervals equal to 2112 = 105946 The ratios for each tone in an

octave starting at A and ending at the next higher A are

A A B C C D D E F F G G A

1 10595 11225 11892 12599 13348 14142 14983 15874 16818 17818 18877 2

A musical tone is actually a superposition of the fundamental frequency and the higher

harmonics The tone quality of a musical instrument is determined by the amplitudes of the

various harmonics that it produces A violin and a trumpet can play the same pitch but they

donrsquot sound at all alike to our ears The difference between them is in the various amplitudes of

their harmonics

9

Serway Chapter 19

Temperature

Formally temperature is what is measured by a thermometer Roughly high temperature is what

we call hot and low temperature is what we call cold On the atomic level temperature refers to

the kinetic energy of the molecules A collection of molecules is called ldquohotrdquo if the molecules

have rapid random motion while a collection of molecules is called ldquocoldrdquo if the random motion

is slow When two bodies are placed in close contact with each other they exchange molecular

kinetic energy until they come to the same temperature This is the microscopic picture of the

Zeroth Law of Thermodynamics

Absolute Zero

Absolute zero is the lowest possible temperature that any object can have This is the temperature

at which all of the energy than can be removed an object has been removed (This removable

energy we call thermal energy) There is still motion at absolute zero Electrons continue to orbit

around atomic nuclei and even atoms continue to move about with a small amount of kinetic

energy but this small energy cannot be removed from the object For example at absolute zero

helium is a liquid whose atoms still move and slide past each other

Temperature Scales

Kelvin Scale Absolute zero is at T = 0 K water freezes at T = 27315 K room temperature is

around T = 295 K and water boils at T = 373 K Note we donrsquot use a deg symbol Kelvin is

prefered SI Unit

Celsius Scale Absolute zero is at T = -273degC water freezes at T = 0degC room temperature is

around T = 22degC and water boils at T = 100degC

Fahrenheit Scale Absolute zero is at T = -459degF water freezes at TF = 32degF room temperature

is around T = 72degF and water boils at TF = 212degF TF =18TC + 32 Notice that temperature

differences are the same for the Kelvin and Celsius scales

Thermal Expansion

When materials are heated they usually expand and when they are cooled they usually contract

(Water near freezing is a spectacular counterexample it works the other way around) The

coefficient of linear expansion is defined by the relation

1

i

L

L T

10

where Li is the initial length of a rod of the material and ΔL is the change in its length due to a

small temperature change ΔT The coefficient of volume expansion is defined similarly

1

i

V

V T

where Vi is the initial volume of a piece of material and where ΔV is the change in its volume due

to a small temperature change ΔT

Avogadrorsquos Number (N or NA)

One mole of any substance corresponds to 6022 times 1023 molecules The atomic mass given on the

periodic table (p A32-33 in the text) is the mass in grams of one mole of atoms For example

the atomic mass of oxygen is 15999 gmol The mass of a single atom is given by the same

number in units of atomic mass units (u) For example the mass of an oxygen atom is 15999 u

(on the average)

Ideal Gas Law (an example of an equation of state)

When the molecules of a gas are sufficiently inert and widely separated that interactions between

them are negligible we say that it is an ideal gas The pressure P volume V and temperature T

(in kelvins) of such a gas are State Variables and are related by the ideal gas law

Bor PV=NkPV nRT T

where n is the number of moles of the gas where R is the gas constant

8314 Jmol KR

where N is the number of molecules and where kB is Boltzmannrsquos constant

231380 10 J KBk

It works well for air at atmosphere pressure and even better for partial vaccuums The relative

ease of measuring pressure and the linear relationship between pressure and temperature (if V

and n are held fixed) makes an ideal gas an ideal thermometer Thermometers can be based on

properties of solids or liquids but the behavior of these materials with temperature is more

complicated

11

Serway Chapter 20

Heat

Heat is energy that flows between a system and its environment because of a tempera-

ture difference between them The units of heat are Joules as expected for an energy

Unfortunately there are several competing units of energy They are related by

1 cal 4186 J 1 Cal 4186 J 1 Btu 1054 J

Heat Capacity

There is often a simple linear relation between the heat that flows in or out of part of a

system and the temperature change that results from this energy transfer When this

linear relation holds it is convenient to define the heat capacity C and the specific

heat c as follows

For the entire object the heat Q it absorbs is given by Q = C(Tf - Ti)

For a particular material the specific heat is defined by c = Cm which is the heat

capacity per unit mass so that

Q = mc(Tf ndash Ti)

Note C has units of energy (J or Cal)(Kelvin kg)

Heats of Transformation or Latent Heat Q = plusmn mL

When a substance changes phase from solid to liquid or from liquid to gas it absorbs

heat without a change in temperature The latent heat or heat of transformation is

usually given per unit mass of the substance For example for water the heat of fusion

(melting) is L = 333 kJkg while the heat of vaporization (boiling) is L = 2260 kJkg

Note that heat for boiling is considerably bigger than melting for water You have to

be careful with signs heat is given off (negative) if you go down in temperature and

condense steam

Work

In general the small amount of work done on a system as a force Fon is exerted on it

through a vector displacement dx is given by

on xdW d F

12

But if the displacement is done very slowly (as we always assume in thermodynamics)

then the force exerted on the system and the force exerted by the system are in

balance so the force exerted by the system is ndash Fon In thermodynamics it is more

convenient to talk about the force exerted by the system so we change the above

formula for the work done on the system to

xdW d F

where F is the force exerted by the system This has confused students for more than a

century now but this is the way your book and many other books do it so you are

stuck You will need to memorize the minus sign in this definition of the work to be

able to use your textbook

There are many chances to get signs wrong in this and the next two chapters (Mosiah

2 )

When an external agent changes the volume of a gas at pressure P by a small amount

dV the (small amount) of work done on the system is given by

dW PdV

Notice that this minus sign is just what we need to make dW be positive if the external

agent compresses the gas for then dV is negative If on the other hand the external

agent gives way allowing the gas to expand against it then dV is positive and we say

that the work done on the gas is negative

The work done on the system (eg by the gas in a cylinder) in a thermodynamic

process is the area under the curve in a PV diagram It is positive for compressions

and negative for expansions If the volume of gas remains constant in a process then

no work is done by the gas

Cyclic processes are important For cyclic processes represented by PV diagrams the

magnitude of the net work during one cycle is simply the area enclosed by the cycle

on the diagram Be careful to keep track of signs when you are calculating that

enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes

1

Path A-B B-C C-D D-A A to A net

Q

W

ΔU

ΔS

Internal Energy

The energy stored in a substance is called its internal energy Eint This energy may be

stored as random kinetic energy or as potential energy in each molecule (stretched

chemical bonds electrons in excited states etc) For ideal gases all states with the

same temperature will have the same Eint

First Law of Thermodynamics

The change ΔEint in the internal energy of a system is given by

intE Q W

where Q is the heat absorbed by the system and where W is the work done on the

system Hence if a system absorbs heat (and if Wge0) the internal energy increases

Likewise if the system does work (W on the system is negative) and if Qge0 the

internal energy decreases Potential Pitfall Many times people talk about work done

by the system It is the minus of W on the system Donrsquot get tripped up

Processes

Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated

from the environment A process may be approximately adiabatic if it happens so

rapidly that heat does not have time to enter or leave the system Work + or ndash is done

and ΔEint = W

Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing

on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal

energy and temperature does not change (Note the difference between an adiabatic

process and a free expansion is that NO work is done in the adiabatic free expansion)

Isobaric process The pressure is held fixed ΔP = 0 For example usually the

pressure increases when a gas is heated but if it were allowed to expand during the

2

heating process in just the right way its pressure could remain fixed In isobaric

processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)

Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is

then zero and so we have ΔEint = Q

Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint

so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in

an isothermal process The work done on the gas is then given by

lnf

i

Vi

Vf

VW PdV nRT

V

Heat Conduction

The quantity P is defined to be the rate at which heat flows through an object and is a

power having units of watts It is analogous to electric current which is the rate at

which charge flows through an object If the flow of heat through a slab of length L

and cross-sectional area A is steady in time then P is given by the equation

h cT TdQkA

dt L

P =

where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The

heat flows of course because of this temperature difference The quantity k is called

the thermal conductivity and is a constant that is characteristic of the material It is

analogous to the electrical conductivity h cT TL is sometimes called the temperature

gradient and is written dTd dTdx

R-Values

It is common to have the heat-conducting properties of materials described by their R-

values especially for insulating materials like fiberglass batting The connection

between k and R is R = Lk where L is the material thickness In this country R-values

always have units of 2ft F hourBtu

Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11

(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)

The heat flow rate through a slab of area A is given by

3

h cT TA

R

P

in units of Btuhour Note that A must be in square feet and the temperatures must be

in degrees Fahrenheit

Convection

Convection is the transfer of thermal energy by flow of material For instance a home

furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct

throughout the house instead it quickly pumps warm air to all of the rooms

Generally convection is a much faster way to transfer heat than conduction

Radiation

Electromagnetic radiation can also transfer heat When you warm yourself near a

campfire which has burned itself down into a bed of glowing embers you are

receiving radiant heat from the infrared portion of the electromagnetic spectrum The

rate at which an object emits radiant heat is given by Stefanrsquos law

4AeTP

where P is the radiated power in watts σ is a constant

8 2 45696 10 W m K

A is the surface area of the object in m2 and T is the temperature in kelvins The

constant e is called the emissivity and it varies from substance to substance A perfect

absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =

0 Hence black objects radiate very well while shiny ones do not Also an object that

is hotter than its surroundings radiates more energy than it absorbs whereas an object

that is cooler than its surroundings absorbs more energy than it radiates

Terminology

Transfer variables vs state variables

Energy transfer by heat as well as work done depends on the initial final and

intermediate states of the system They are transfer variables But their sum (Q + W =

4

Eint) is a state variable

Figure 205

5

Serway Chapter 21

Kinetic Theory

The ideal gas law works for all atoms and molecules at low pressure It is rather

amazing that it does Kinetic theory explains why The properties of an ideal gas can

be understood by thinking of it as N rapidly moving particles of mass m As these

particles collide with the container walls momentum is imparted to the walls which

we call the force of gas pressure In this picture the pressure is related to the average

of the square of the particle velocity 2v by

22 1( )

3 2

NP mv

V

Using the ideal gas law we obtain the average translational kinetic energy per

molecule

21 3

2 2 Bmv k T

The rms speed is then given by

2rms

3 3Bk T RTv v

m M

where M is the molecular mass in kgmol

Degrees of Freedom

Roughly speaking a degree of freedom is a way in which a molecule can store energy

For instance since there are three different directions in space along which a molecule

can move there are three degrees of freedom for the translational kinetic energy

There are also three different axes of rotation about which a polyatomic molecule can

spin so we say there are three degrees of freedom for the rotational kinetic energy

There are even degrees of freedom associated with the various ways in which a

molecule can vibrate and with the different energy levels in which the electrons of

the molecule can exist

Internal Energy and Degrees of Freedom The internal energy of an ideal gas made

up of molecules with J degrees of freedom is given by

int 2 2 B

J JE nRT Nk T

6

Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of

molar heat capacities CV and CP These are the heat capacities per mole and the

subscript V on CV means that the volume is being held constant while for CP the

pressure is held constant For example to raise the temperature of n moles of a gas

whose pressure is held constant by 10 K we would have to supply an amount of heat Q

= nCP (10) K

Molar Specific Heat of an Ideal Gas at Constant Volume

VQ nC T

3monatomic

2VC R

5diatomic

2VC R

5polyatomic

2VC

Real gases deviate from these formulas because in addition to the translational and ro-

tational degrees of freedom they also have vibrational and electronic degrees of

freedom These are unimportant at low temperatures due to quantum mechanical

effects but become increasingly important at higher temperatures The rough rule is

No of degrees of freedom

2VC R

Molar Specific Heat of an Ideal Gas at Constant Pressure

PQ nC T

P VC C R

The internal energy of an ideal gas depends only on the temperature

int VE nC T

Adiabatic Processes in an Ideal Gas

7

An adiabatic process is one in which no heat is exchanged between the system and the

environment When an ideal gas expands or contracts adiabatically not only does its

pressure change as expected from the ideal gas law but its temperature changes as

well Under these conditions the final pressure Pf can be computed from the initial

pressure Pi and from the final and initial volumes Vf and Vi by

or constantf f i iP V PV PV

where γ = CPCV The quantity γ is called the adiabatic exponent Note that this

doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be

combined with the adiabatic law for pressure given above to obtain the adiabatic law

for temperatures

1 constantTV

Compressions in sound waves are adiabatic because they happen too rapidly for any

appreciable amount of heat to flow This is why the adiabatic exponent appears in the

formula for the speed of sound in an ideal gas

RTv

M

Note that v depends only on T and not on P Because it depends only on the

temperature the speed of sound is the same in Provo as at sea level in spite of the

lower pressure here due to the difference in elevation

Equipartition of Energy

Every kind of molecule has a certain number of degrees of freedom which are

independent ways in which it can store energy Each such degree of freedom has

associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1

2 RT per mole)

(Note since a molecule has so many possible degrees of freedom it would seem that

there should be a lot of 12 sBk T to spread around But because energy is quantized

some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high

enough that 12 Bk T is as big as the lowest quantum of energy

8

Serway Chapter 22

Second Law of Thermodynamics

There are several equivalent forms of this important law

Kelvin It is not possible to change heat completely into work with no other change

taking place Or in other words there are no perfect heat engines

Clausius It is not possible for heat to flow from one body to another body at a higher

temperature with no other change taking place Or in other words there are no

perfect refrigerators

Entropy In any thermodynamic process that proceeds from one equilibrium state to

another the entropy of the system + environment either remains unchanged or

increases The total entropy never decreases This law is a bit of an oddity among the

laws of physics because it is not absolute Things are forbidden by the second law not

because it is impossible for them to happen but because it is extremely unlikely for

them to happen (See below for more information about entropy)

Reversible and Irreversible Processes

A reversible process is one which occurs so slowly that it is in thermal equilibrium (or

very nearly so) at all times A hallmark of such processes is that a motion picture of

them looks perfectly normal whether run forward or backward Imagine for instance

the slow expansion of a gas at constant temperature in a cylinder whose volume is being

increased by a slowly moving piston Run the movie backwards and what do you see

You see the slow compression of a gas at constant temperature which looks perfectly

normal

An irreversible process is one which occurs in such a way that thermal equilibrium is

not maintained throughout the process The mark of this kind of process is that a motion

picture of it looks very odd when run backward Imagine the sudden expansion of a gas

into a previously evacuated chamber because a hole was punched in the wall between a

pressurized chamber and the evacuated one Run the movie backward and what do you

see You see the gas in the soon-to-be-evacuated chamber gather itself together and

stream through a tiny hole into a chamber in which there is already plenty of gas If you

have ever seen this happen get in touch with the support group for those who have

witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW

9

Heat Engines

Because of the vexing difference in sign between work done on and system and work done

by a system we will invent a new work variable Weng Heat engines do work and so the net

W for these engines is negative But in engineering applications hidden minus signs are

regarded as evil so for heat engines we donrsquot talk about W instead we talk about its

magnitude engW W So for heat engines the first law is

int engE Q W

But for heat pumps and refrigerators work is done on the system so we use the usual work

W when we talk about these systems

A heat engine is a machine that absorbs heat converts part of it to work and exhausts the

rest The heat must be absorbed at high temperature and exhausted at low temperature If the

absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then

eng h cW Q Q

and the efficiency of the engine is defined to be

eng

h

We

Q

A perfect engine would convert the heat hQ completely into work Weng giving an effi-

ciency of e = 1 Energy conservation alone allows a perfect engine but the second law

requires e lt 1

Refrigerators and Heat Pumps

A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high

temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine

by some source of power A heat pump is a machine that either works like a refrigerator

keeping a place cold by transferring heat from this cold place to a higher temperature

environment (cooling mode like an air conditioner) or it functions as a heater

transferring heat into a warm place from a cooler one (heating mode like a window unit

that heats a house by extracting thermal energy from the cold outdoors) The coefficient

of performance of a refrigerator or of a heat pump in cooling mode is defined to be

COP cooling modecQ

W

10

For a heat pump in heating mode the coefficient of performance is

COP heating modehQ

W

Note that we donrsquot have to use the engineering work here because in these systems

work is done on the system and W is naturally positive

A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without

doing any work giving an infinite coefficient of performance Energy conservation alone

allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump

in heating mode would transfer Qh into the house without doing any work and so would

also have an infinite coefficient of performance The second law forbids this too A good

coefficient of performance for a real device would be around 5 or 6

Carnot Cycle

The most efficient of all possible engines is one that uses the Carnot cycle This cycle

employs an ideal gas has no friction and operates very slowly so that the gas can be in

thermal equilibrium at all parts of the cycle This means of course that it canrsquot

possibly be built and even if it could be built it would not run fast enough to be useful

Nevertheless this cycle is very important because it gives an upper bound on the

efficiency of real engines There cannot possibly be an engine that is more efficient

than one based on the Carnot cycle This cycle consists of the following four steps

1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its

volume The reason that heat is absorbed is that expansion tends to cool the gas but

thermal contact with the environment at Th keeps the temperature high by heat

conduction into the ideal gas

2 The ideal gas further increases its volume by an adiabatic expansion This expansion

causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc

3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in

volume The reason that heat is exhausted is that compression tends to heat the gas

but thermal contact with the environment at Tc keeps the temperature low by heat

conduction out of the ideal gas

4 The gas is adiabatically compressed back to its original volume (the volume it started

with in step 1) This compression heats the gas from Tc up to Th

11

The efficiency of a Carnot engine is given by the very simple formula

1 cC

h

Te

T

where the temperatures must in be Kelvin No real engine can be more efficient than

this

The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode

is given by

COP cooling mode cC

h c

T

T T

and no real refrigerator can have a coefficient of performance greater than this

The coefficient of performance of a Carnot heat pump in heating mode is

COP heating mode hC

h c

T

T T

So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency

To make the reversible steps in the cycle really reversible they would have to occur

infinitely slowly So the price you pay for making a perfect engine is that it takes

forever to get it to do any work

Entropy

The entropy of a system is defined in terms of its molecular makeup and measures

roughly the disorder of the system If the system is packed into a very small volume

then it is quite ordered and the entropy will be low If it occupies a large volume the

entropy is high (To see what this has to do with disorder note that socks in a drawer

occupy a small volume while socks on the bed in the corner by the door and

hanging from the chandelier occupy a large volume) If the system is very cold then

the molecules hardly move and may even reach out to each other and form a crystal

This is a highly ordered state and therefore has low entropy If the system is very hot

with rapidly speeding molecules crashing into the container walls and bouncing off

each other things are disordered and the entropy is high

It is possible to calculate the entropy of a system in terms of its macroscopic thermody-

namic properties ie pressure volume temperature number of moles etc The key to this

calculation is the concept of a reversible process A reversible process is one that is

12

carried out without friction and so slowly that the process can be reversed at any stage by

making an infinitesimal change in the environment of the system The slow expansion of the

gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment

is reversible If we turned around and began slowly to compress the gas would just slowly

exhaust heat to the environment in the exact reverse way that it absorbed it during expansion

Most processes however are irreversible For example if a gas-filled box were suddenly

increased in size so that the particles were free to wander into the void created by the sudden

expansion then the gas would eventually fill the new volume uniformly at the same

temperature as before the expansion (The temperature is unchanged in this imaginary

process because the kinetic energy of the molecules would be unaffected by such an

instantaneous expansion of the container walls) This imaginary but highly thought-

stimulating process is called a free expansion and it is impossible to reverse it During the

expansion we didnrsquot push on any of the molecules so reversing this process would mean

making them go back into their original volume without pushing on them they simply will

not cooperate to this extent Another way to see that just pushing them back where they came

from does not reverse the free expansion is to think about what would happen if we just

compressed either adiabatically or isothermally An adiabatic compression back to the

original volume would heat the gas above its original temperature and an isothermal

compression would require that heat be exhausted to the environment But the free expansion

involved neither temperature changes nor heat exchanges so neither of these two processes

is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in

a way that takes us back to the initial state

It is possible to calculate the change of entropy for both reversible and irreversible processes

Letrsquos consider a reversible process first In a reversible process the entropy change is given

by the formula

dQS

T

where dQ is the amount of heat added to the system during a small step of the process

The total energy change during the process may then simply be calculated by integration

f f

f i i i

dQS S S dS

T

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11

5

Serway Chapter 17

Sound Speed in Solids Liquids and Gases

The speed of sound in a liquid or solid of bulk modulus B and volume mass density ρ

Bv

The speed of sound in air is

331 1 273v m s T C

The speed of sound in air at room temperature is about 343 ms Notice it doesnrsquot depend on

pressure or frequency

Power Intensity and Loudness

The intensity of a sound wave is defined to be the power per unit area ie wattsm2 in the wave

For spherical waves traveling away from a small source emitting waves with average power Рav

the intensity falls off with distance from the source r according to the inverse-square law

av24

Ir

Hence the intensity I2 at distance r2 is related to the intensity I1 at a different distance r1 by

21 2

22 1

I r

I r

Experiments on human hearing have shown that we hear intensity differences logarithmically so

the decibel loudness scale for sound intensity was invented The loudness β of a sound in

decibels is related to its intensity I in Wm2 by the formula

1010logo

I

I

Where Io is a sound intensity near the threshold of hearing defined to be Io = 10-12 Wm2 Note

that on this scale a sound is made 10 times more intense by adding 10 decibels Remember that

intensity is proportional to velocity squared Amplitude is the A is the previous chapterrsquos

equation for a traveling wave If you make the A 10 times as much the intensity will increase by

100 times (20 dB)

Doppler Effect

If sound waves are traveling through a medium and if either the receiver of the waves or the

source of waves is moving then the frequency received is related to the frequency emitted by

6

0

s

v vf f

v v

where fprime is the frequency detected by the observer f is the frequency emitted by the source v is

the speed of the waves vo is the speed of the observer and vs is the speed of the source This

formula assumes that the source and receiver are either moving directly toward each other or

directly away from each other To know which signs to use remember that when observer and

source approach each other the observed frequency is higher while if they move away from each

other it is lower Just examine the signs in the formula and make the answer come out right

For electromagnetic waves (light radio waves X-rays) traveling in vacuum Einsteinrsquos theory of

relativity (and careful experiments) show that the Doppler shift is given by

r

r

c vf f

c v

where vr is the relative speed between the source and the observer

For all kinds of waves (sound light etc) if the relative speed of the source and the observer is

small compared to the speed of the waves then there is a simple approximation to the Doppler

effect For example if the relative speed is 1 of the wave speed then the frequency shifts by

1 Remember that this is only an approximation

Shock Waves

When an object moves through a medium at a speed greater than the speed of waves a V-shaped

shock wave is produced The V-shaped wake behind a speeding boat is a good example of this

effect and the cone of sonic-boom behind a supersonic aircraft is another The angle that V-line

makes with the direction of travel of the source is given by

sins

v

v

where v is the wave speed and vs is the source speed

7

Serway Chapter 18

Principle of Linear Superposition

We say that a system obeys the principle of linear superposition if two or more different motions

of the system can simply be added together to find the net motion of the system Light waves

obey this principle as they propagate through the air as can be seen by shining two flashlights so

that their beams cross The beams propagate along without affecting each other (Light sabers are

a spectacular but unfortunately fictional example of systems that do not obey the principle of

superposition) Wave pulses on an ideal rope also obey this principle two different pulses pass

through each other without change Standing waves are an example of this effect being simply

the linear superposition of two traveling waves of the same frequency but moving in opposite

directions Light waves in matter do not always obey this principle For instance two powerful

laser beams could be made to cross in a piece of glass in such a way that their combined heating

effect in the crossing region could melt the glass and scatter the beams in complicated ways This

is an example of a nonlinear effect

Interference

When two or more waves are present in the same medium at the same time their net effect may

often be obtained simply by adding them at each point in the medium according to the principle

of linear superposition (Note this wonrsquot work if the medium is nonlinear) When this addition

makes the total amplitude be greater than the individual amplitudes of the various waves we say

that the interference is constructive When the addition produces cancellation and an amplitude

less than the amplitudes of the separate waves we have destructive interference

Standing Waves

A standing wave is the superposition of two identical traveling waves moving in opposite

directions Nodes are places where the two waves perfectly destructively interfere to produce

zero amplitude at all times Anti-nodes are places where the two waves perfectly constructively

interfere to produce an amplitude maximum The distance between nodes is λ2 Standing waves

on a string fixed at both ends have nodes at each end of the string Standing waves in an air

column enclosed in a tube have displacement anti-nodes at open ends of the tube and

displacement nodes at closed ends The frequency of the standing wave with the lowest possible

frequency is called the fundamental frequency Standing waves on strings or in air columns all

have frequencies which are integer multiples of the fundamental frequency and are called

8

harmonics (The fundamental is called the ldquofirst harmonicrdquo)

Beats

Beats are heard when two waves with slightly different frequencies f1 and f2 are combined The

waves constructively interfere for a number of cycles then destructively interfere for a number

of cycles We hear a periodic ldquowah-wahrdquo frequency equal to the difference of the two wave

frequencies

1 2bf f f

Musical Instruments

Musical instruments produce tones by exciting standing waves on strings (violins piano) and in

tubes (trumpet organ) The fundamental frequency of the standing wave is called the pitch of the

tone The pitch of concert A is 440 Hz by definition Two tones are an octave apart if one pitch

has twice the frequency of the other In written music there are 12 intervals in each octave with

the ratio between successive intervals equal to 2112 = 105946 The ratios for each tone in an

octave starting at A and ending at the next higher A are

A A B C C D D E F F G G A

1 10595 11225 11892 12599 13348 14142 14983 15874 16818 17818 18877 2

A musical tone is actually a superposition of the fundamental frequency and the higher

harmonics The tone quality of a musical instrument is determined by the amplitudes of the

various harmonics that it produces A violin and a trumpet can play the same pitch but they

donrsquot sound at all alike to our ears The difference between them is in the various amplitudes of

their harmonics

9

Serway Chapter 19

Temperature

Formally temperature is what is measured by a thermometer Roughly high temperature is what

we call hot and low temperature is what we call cold On the atomic level temperature refers to

the kinetic energy of the molecules A collection of molecules is called ldquohotrdquo if the molecules

have rapid random motion while a collection of molecules is called ldquocoldrdquo if the random motion

is slow When two bodies are placed in close contact with each other they exchange molecular

kinetic energy until they come to the same temperature This is the microscopic picture of the

Zeroth Law of Thermodynamics

Absolute Zero

Absolute zero is the lowest possible temperature that any object can have This is the temperature

at which all of the energy than can be removed an object has been removed (This removable

energy we call thermal energy) There is still motion at absolute zero Electrons continue to orbit

around atomic nuclei and even atoms continue to move about with a small amount of kinetic

energy but this small energy cannot be removed from the object For example at absolute zero

helium is a liquid whose atoms still move and slide past each other

Temperature Scales

Kelvin Scale Absolute zero is at T = 0 K water freezes at T = 27315 K room temperature is

around T = 295 K and water boils at T = 373 K Note we donrsquot use a deg symbol Kelvin is

prefered SI Unit

Celsius Scale Absolute zero is at T = -273degC water freezes at T = 0degC room temperature is

around T = 22degC and water boils at T = 100degC

Fahrenheit Scale Absolute zero is at T = -459degF water freezes at TF = 32degF room temperature

is around T = 72degF and water boils at TF = 212degF TF =18TC + 32 Notice that temperature

differences are the same for the Kelvin and Celsius scales

Thermal Expansion

When materials are heated they usually expand and when they are cooled they usually contract

(Water near freezing is a spectacular counterexample it works the other way around) The

coefficient of linear expansion is defined by the relation

1

i

L

L T

10

where Li is the initial length of a rod of the material and ΔL is the change in its length due to a

small temperature change ΔT The coefficient of volume expansion is defined similarly

1

i

V

V T

where Vi is the initial volume of a piece of material and where ΔV is the change in its volume due

to a small temperature change ΔT

Avogadrorsquos Number (N or NA)

One mole of any substance corresponds to 6022 times 1023 molecules The atomic mass given on the

periodic table (p A32-33 in the text) is the mass in grams of one mole of atoms For example

the atomic mass of oxygen is 15999 gmol The mass of a single atom is given by the same

number in units of atomic mass units (u) For example the mass of an oxygen atom is 15999 u

(on the average)

Ideal Gas Law (an example of an equation of state)

When the molecules of a gas are sufficiently inert and widely separated that interactions between

them are negligible we say that it is an ideal gas The pressure P volume V and temperature T

(in kelvins) of such a gas are State Variables and are related by the ideal gas law

Bor PV=NkPV nRT T

where n is the number of moles of the gas where R is the gas constant

8314 Jmol KR

where N is the number of molecules and where kB is Boltzmannrsquos constant

231380 10 J KBk

It works well for air at atmosphere pressure and even better for partial vaccuums The relative

ease of measuring pressure and the linear relationship between pressure and temperature (if V

and n are held fixed) makes an ideal gas an ideal thermometer Thermometers can be based on

properties of solids or liquids but the behavior of these materials with temperature is more

complicated

11

Serway Chapter 20

Heat

Heat is energy that flows between a system and its environment because of a tempera-

ture difference between them The units of heat are Joules as expected for an energy

Unfortunately there are several competing units of energy They are related by

1 cal 4186 J 1 Cal 4186 J 1 Btu 1054 J

Heat Capacity

There is often a simple linear relation between the heat that flows in or out of part of a

system and the temperature change that results from this energy transfer When this

linear relation holds it is convenient to define the heat capacity C and the specific

heat c as follows

For the entire object the heat Q it absorbs is given by Q = C(Tf - Ti)

For a particular material the specific heat is defined by c = Cm which is the heat

capacity per unit mass so that

Q = mc(Tf ndash Ti)

Note C has units of energy (J or Cal)(Kelvin kg)

Heats of Transformation or Latent Heat Q = plusmn mL

When a substance changes phase from solid to liquid or from liquid to gas it absorbs

heat without a change in temperature The latent heat or heat of transformation is

usually given per unit mass of the substance For example for water the heat of fusion

(melting) is L = 333 kJkg while the heat of vaporization (boiling) is L = 2260 kJkg

Note that heat for boiling is considerably bigger than melting for water You have to

be careful with signs heat is given off (negative) if you go down in temperature and

condense steam

Work

In general the small amount of work done on a system as a force Fon is exerted on it

through a vector displacement dx is given by

on xdW d F

12

But if the displacement is done very slowly (as we always assume in thermodynamics)

then the force exerted on the system and the force exerted by the system are in

balance so the force exerted by the system is ndash Fon In thermodynamics it is more

convenient to talk about the force exerted by the system so we change the above

formula for the work done on the system to

xdW d F

where F is the force exerted by the system This has confused students for more than a

century now but this is the way your book and many other books do it so you are

stuck You will need to memorize the minus sign in this definition of the work to be

able to use your textbook

There are many chances to get signs wrong in this and the next two chapters (Mosiah

2 )

When an external agent changes the volume of a gas at pressure P by a small amount

dV the (small amount) of work done on the system is given by

dW PdV

Notice that this minus sign is just what we need to make dW be positive if the external

agent compresses the gas for then dV is negative If on the other hand the external

agent gives way allowing the gas to expand against it then dV is positive and we say

that the work done on the gas is negative

The work done on the system (eg by the gas in a cylinder) in a thermodynamic

process is the area under the curve in a PV diagram It is positive for compressions

and negative for expansions If the volume of gas remains constant in a process then

no work is done by the gas

Cyclic processes are important For cyclic processes represented by PV diagrams the

magnitude of the net work during one cycle is simply the area enclosed by the cycle

on the diagram Be careful to keep track of signs when you are calculating that

enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes

1

Path A-B B-C C-D D-A A to A net

Q

W

ΔU

ΔS

Internal Energy

The energy stored in a substance is called its internal energy Eint This energy may be

stored as random kinetic energy or as potential energy in each molecule (stretched

chemical bonds electrons in excited states etc) For ideal gases all states with the

same temperature will have the same Eint

First Law of Thermodynamics

The change ΔEint in the internal energy of a system is given by

intE Q W

where Q is the heat absorbed by the system and where W is the work done on the

system Hence if a system absorbs heat (and if Wge0) the internal energy increases

Likewise if the system does work (W on the system is negative) and if Qge0 the

internal energy decreases Potential Pitfall Many times people talk about work done

by the system It is the minus of W on the system Donrsquot get tripped up

Processes

Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated

from the environment A process may be approximately adiabatic if it happens so

rapidly that heat does not have time to enter or leave the system Work + or ndash is done

and ΔEint = W

Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing

on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal

energy and temperature does not change (Note the difference between an adiabatic

process and a free expansion is that NO work is done in the adiabatic free expansion)

Isobaric process The pressure is held fixed ΔP = 0 For example usually the

pressure increases when a gas is heated but if it were allowed to expand during the

2

heating process in just the right way its pressure could remain fixed In isobaric

processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)

Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is

then zero and so we have ΔEint = Q

Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint

so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in

an isothermal process The work done on the gas is then given by

lnf

i

Vi

Vf

VW PdV nRT

V

Heat Conduction

The quantity P is defined to be the rate at which heat flows through an object and is a

power having units of watts It is analogous to electric current which is the rate at

which charge flows through an object If the flow of heat through a slab of length L

and cross-sectional area A is steady in time then P is given by the equation

h cT TdQkA

dt L

P =

where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The

heat flows of course because of this temperature difference The quantity k is called

the thermal conductivity and is a constant that is characteristic of the material It is

analogous to the electrical conductivity h cT TL is sometimes called the temperature

gradient and is written dTd dTdx

R-Values

It is common to have the heat-conducting properties of materials described by their R-

values especially for insulating materials like fiberglass batting The connection

between k and R is R = Lk where L is the material thickness In this country R-values

always have units of 2ft F hourBtu

Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11

(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)

The heat flow rate through a slab of area A is given by

3

h cT TA

R

P

in units of Btuhour Note that A must be in square feet and the temperatures must be

in degrees Fahrenheit

Convection

Convection is the transfer of thermal energy by flow of material For instance a home

furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct

throughout the house instead it quickly pumps warm air to all of the rooms

Generally convection is a much faster way to transfer heat than conduction

Radiation

Electromagnetic radiation can also transfer heat When you warm yourself near a

campfire which has burned itself down into a bed of glowing embers you are

receiving radiant heat from the infrared portion of the electromagnetic spectrum The

rate at which an object emits radiant heat is given by Stefanrsquos law

4AeTP

where P is the radiated power in watts σ is a constant

8 2 45696 10 W m K

A is the surface area of the object in m2 and T is the temperature in kelvins The

constant e is called the emissivity and it varies from substance to substance A perfect

absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =

0 Hence black objects radiate very well while shiny ones do not Also an object that

is hotter than its surroundings radiates more energy than it absorbs whereas an object

that is cooler than its surroundings absorbs more energy than it radiates

Terminology

Transfer variables vs state variables

Energy transfer by heat as well as work done depends on the initial final and

intermediate states of the system They are transfer variables But their sum (Q + W =

4

Eint) is a state variable

Figure 205

5

Serway Chapter 21

Kinetic Theory

The ideal gas law works for all atoms and molecules at low pressure It is rather

amazing that it does Kinetic theory explains why The properties of an ideal gas can

be understood by thinking of it as N rapidly moving particles of mass m As these

particles collide with the container walls momentum is imparted to the walls which

we call the force of gas pressure In this picture the pressure is related to the average

of the square of the particle velocity 2v by

22 1( )

3 2

NP mv

V

Using the ideal gas law we obtain the average translational kinetic energy per

molecule

21 3

2 2 Bmv k T

The rms speed is then given by

2rms

3 3Bk T RTv v

m M

where M is the molecular mass in kgmol

Degrees of Freedom

Roughly speaking a degree of freedom is a way in which a molecule can store energy

For instance since there are three different directions in space along which a molecule

can move there are three degrees of freedom for the translational kinetic energy

There are also three different axes of rotation about which a polyatomic molecule can

spin so we say there are three degrees of freedom for the rotational kinetic energy

There are even degrees of freedom associated with the various ways in which a

molecule can vibrate and with the different energy levels in which the electrons of

the molecule can exist

Internal Energy and Degrees of Freedom The internal energy of an ideal gas made

up of molecules with J degrees of freedom is given by

int 2 2 B

J JE nRT Nk T

6

Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of

molar heat capacities CV and CP These are the heat capacities per mole and the

subscript V on CV means that the volume is being held constant while for CP the

pressure is held constant For example to raise the temperature of n moles of a gas

whose pressure is held constant by 10 K we would have to supply an amount of heat Q

= nCP (10) K

Molar Specific Heat of an Ideal Gas at Constant Volume

VQ nC T

3monatomic

2VC R

5diatomic

2VC R

5polyatomic

2VC

Real gases deviate from these formulas because in addition to the translational and ro-

tational degrees of freedom they also have vibrational and electronic degrees of

freedom These are unimportant at low temperatures due to quantum mechanical

effects but become increasingly important at higher temperatures The rough rule is

No of degrees of freedom

2VC R

Molar Specific Heat of an Ideal Gas at Constant Pressure

PQ nC T

P VC C R

The internal energy of an ideal gas depends only on the temperature

int VE nC T

Adiabatic Processes in an Ideal Gas

7

An adiabatic process is one in which no heat is exchanged between the system and the

environment When an ideal gas expands or contracts adiabatically not only does its

pressure change as expected from the ideal gas law but its temperature changes as

well Under these conditions the final pressure Pf can be computed from the initial

pressure Pi and from the final and initial volumes Vf and Vi by

or constantf f i iP V PV PV

where γ = CPCV The quantity γ is called the adiabatic exponent Note that this

doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be

combined with the adiabatic law for pressure given above to obtain the adiabatic law

for temperatures

1 constantTV

Compressions in sound waves are adiabatic because they happen too rapidly for any

appreciable amount of heat to flow This is why the adiabatic exponent appears in the

formula for the speed of sound in an ideal gas

RTv

M

Note that v depends only on T and not on P Because it depends only on the

temperature the speed of sound is the same in Provo as at sea level in spite of the

lower pressure here due to the difference in elevation

Equipartition of Energy

Every kind of molecule has a certain number of degrees of freedom which are

independent ways in which it can store energy Each such degree of freedom has

associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1

2 RT per mole)

(Note since a molecule has so many possible degrees of freedom it would seem that

there should be a lot of 12 sBk T to spread around But because energy is quantized

some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high

enough that 12 Bk T is as big as the lowest quantum of energy

8

Serway Chapter 22

Second Law of Thermodynamics

There are several equivalent forms of this important law

Kelvin It is not possible to change heat completely into work with no other change

taking place Or in other words there are no perfect heat engines

Clausius It is not possible for heat to flow from one body to another body at a higher

temperature with no other change taking place Or in other words there are no

perfect refrigerators

Entropy In any thermodynamic process that proceeds from one equilibrium state to

another the entropy of the system + environment either remains unchanged or

increases The total entropy never decreases This law is a bit of an oddity among the

laws of physics because it is not absolute Things are forbidden by the second law not

because it is impossible for them to happen but because it is extremely unlikely for

them to happen (See below for more information about entropy)

Reversible and Irreversible Processes

A reversible process is one which occurs so slowly that it is in thermal equilibrium (or

very nearly so) at all times A hallmark of such processes is that a motion picture of

them looks perfectly normal whether run forward or backward Imagine for instance

the slow expansion of a gas at constant temperature in a cylinder whose volume is being

increased by a slowly moving piston Run the movie backwards and what do you see

You see the slow compression of a gas at constant temperature which looks perfectly

normal

An irreversible process is one which occurs in such a way that thermal equilibrium is

not maintained throughout the process The mark of this kind of process is that a motion

picture of it looks very odd when run backward Imagine the sudden expansion of a gas

into a previously evacuated chamber because a hole was punched in the wall between a

pressurized chamber and the evacuated one Run the movie backward and what do you

see You see the gas in the soon-to-be-evacuated chamber gather itself together and

stream through a tiny hole into a chamber in which there is already plenty of gas If you

have ever seen this happen get in touch with the support group for those who have

witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW

9

Heat Engines

Because of the vexing difference in sign between work done on and system and work done

by a system we will invent a new work variable Weng Heat engines do work and so the net

W for these engines is negative But in engineering applications hidden minus signs are

regarded as evil so for heat engines we donrsquot talk about W instead we talk about its

magnitude engW W So for heat engines the first law is

int engE Q W

But for heat pumps and refrigerators work is done on the system so we use the usual work

W when we talk about these systems

A heat engine is a machine that absorbs heat converts part of it to work and exhausts the

rest The heat must be absorbed at high temperature and exhausted at low temperature If the

absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then

eng h cW Q Q

and the efficiency of the engine is defined to be

eng

h

We

Q

A perfect engine would convert the heat hQ completely into work Weng giving an effi-

ciency of e = 1 Energy conservation alone allows a perfect engine but the second law

requires e lt 1

Refrigerators and Heat Pumps

A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high

temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine

by some source of power A heat pump is a machine that either works like a refrigerator

keeping a place cold by transferring heat from this cold place to a higher temperature

environment (cooling mode like an air conditioner) or it functions as a heater

transferring heat into a warm place from a cooler one (heating mode like a window unit

that heats a house by extracting thermal energy from the cold outdoors) The coefficient

of performance of a refrigerator or of a heat pump in cooling mode is defined to be

COP cooling modecQ

W

10

For a heat pump in heating mode the coefficient of performance is

COP heating modehQ

W

Note that we donrsquot have to use the engineering work here because in these systems

work is done on the system and W is naturally positive

A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without

doing any work giving an infinite coefficient of performance Energy conservation alone

allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump

in heating mode would transfer Qh into the house without doing any work and so would

also have an infinite coefficient of performance The second law forbids this too A good

coefficient of performance for a real device would be around 5 or 6

Carnot Cycle

The most efficient of all possible engines is one that uses the Carnot cycle This cycle

employs an ideal gas has no friction and operates very slowly so that the gas can be in

thermal equilibrium at all parts of the cycle This means of course that it canrsquot

possibly be built and even if it could be built it would not run fast enough to be useful

Nevertheless this cycle is very important because it gives an upper bound on the

efficiency of real engines There cannot possibly be an engine that is more efficient

than one based on the Carnot cycle This cycle consists of the following four steps

1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its

volume The reason that heat is absorbed is that expansion tends to cool the gas but

thermal contact with the environment at Th keeps the temperature high by heat

conduction into the ideal gas

2 The ideal gas further increases its volume by an adiabatic expansion This expansion

causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc

3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in

volume The reason that heat is exhausted is that compression tends to heat the gas

but thermal contact with the environment at Tc keeps the temperature low by heat

conduction out of the ideal gas

4 The gas is adiabatically compressed back to its original volume (the volume it started

with in step 1) This compression heats the gas from Tc up to Th

11

The efficiency of a Carnot engine is given by the very simple formula

1 cC

h

Te

T

where the temperatures must in be Kelvin No real engine can be more efficient than

this

The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode

is given by

COP cooling mode cC

h c

T

T T

and no real refrigerator can have a coefficient of performance greater than this

The coefficient of performance of a Carnot heat pump in heating mode is

COP heating mode hC

h c

T

T T

So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency

To make the reversible steps in the cycle really reversible they would have to occur

infinitely slowly So the price you pay for making a perfect engine is that it takes

forever to get it to do any work

Entropy

The entropy of a system is defined in terms of its molecular makeup and measures

roughly the disorder of the system If the system is packed into a very small volume

then it is quite ordered and the entropy will be low If it occupies a large volume the

entropy is high (To see what this has to do with disorder note that socks in a drawer

occupy a small volume while socks on the bed in the corner by the door and

hanging from the chandelier occupy a large volume) If the system is very cold then

the molecules hardly move and may even reach out to each other and form a crystal

This is a highly ordered state and therefore has low entropy If the system is very hot

with rapidly speeding molecules crashing into the container walls and bouncing off

each other things are disordered and the entropy is high

It is possible to calculate the entropy of a system in terms of its macroscopic thermody-

namic properties ie pressure volume temperature number of moles etc The key to this

calculation is the concept of a reversible process A reversible process is one that is

12

carried out without friction and so slowly that the process can be reversed at any stage by

making an infinitesimal change in the environment of the system The slow expansion of the

gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment

is reversible If we turned around and began slowly to compress the gas would just slowly

exhaust heat to the environment in the exact reverse way that it absorbed it during expansion

Most processes however are irreversible For example if a gas-filled box were suddenly

increased in size so that the particles were free to wander into the void created by the sudden

expansion then the gas would eventually fill the new volume uniformly at the same

temperature as before the expansion (The temperature is unchanged in this imaginary

process because the kinetic energy of the molecules would be unaffected by such an

instantaneous expansion of the container walls) This imaginary but highly thought-

stimulating process is called a free expansion and it is impossible to reverse it During the

expansion we didnrsquot push on any of the molecules so reversing this process would mean

making them go back into their original volume without pushing on them they simply will

not cooperate to this extent Another way to see that just pushing them back where they came

from does not reverse the free expansion is to think about what would happen if we just

compressed either adiabatically or isothermally An adiabatic compression back to the

original volume would heat the gas above its original temperature and an isothermal

compression would require that heat be exhausted to the environment But the free expansion

involved neither temperature changes nor heat exchanges so neither of these two processes

is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in

a way that takes us back to the initial state

It is possible to calculate the change of entropy for both reversible and irreversible processes

Letrsquos consider a reversible process first In a reversible process the entropy change is given

by the formula

dQS

T

where dQ is the amount of heat added to the system during a small step of the process

The total energy change during the process may then simply be calculated by integration

f f

f i i i

dQS S S dS

T

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11

6

0

s

v vf f

v v

where fprime is the frequency detected by the observer f is the frequency emitted by the source v is

the speed of the waves vo is the speed of the observer and vs is the speed of the source This

formula assumes that the source and receiver are either moving directly toward each other or

directly away from each other To know which signs to use remember that when observer and

source approach each other the observed frequency is higher while if they move away from each

other it is lower Just examine the signs in the formula and make the answer come out right

For electromagnetic waves (light radio waves X-rays) traveling in vacuum Einsteinrsquos theory of

relativity (and careful experiments) show that the Doppler shift is given by

r

r

c vf f

c v

where vr is the relative speed between the source and the observer

For all kinds of waves (sound light etc) if the relative speed of the source and the observer is

small compared to the speed of the waves then there is a simple approximation to the Doppler

effect For example if the relative speed is 1 of the wave speed then the frequency shifts by

1 Remember that this is only an approximation

Shock Waves

When an object moves through a medium at a speed greater than the speed of waves a V-shaped

shock wave is produced The V-shaped wake behind a speeding boat is a good example of this

effect and the cone of sonic-boom behind a supersonic aircraft is another The angle that V-line

makes with the direction of travel of the source is given by

sins

v

v

where v is the wave speed and vs is the source speed

7

Serway Chapter 18

Principle of Linear Superposition

We say that a system obeys the principle of linear superposition if two or more different motions

of the system can simply be added together to find the net motion of the system Light waves

obey this principle as they propagate through the air as can be seen by shining two flashlights so

that their beams cross The beams propagate along without affecting each other (Light sabers are

a spectacular but unfortunately fictional example of systems that do not obey the principle of

superposition) Wave pulses on an ideal rope also obey this principle two different pulses pass

through each other without change Standing waves are an example of this effect being simply

the linear superposition of two traveling waves of the same frequency but moving in opposite

directions Light waves in matter do not always obey this principle For instance two powerful

laser beams could be made to cross in a piece of glass in such a way that their combined heating

effect in the crossing region could melt the glass and scatter the beams in complicated ways This

is an example of a nonlinear effect

Interference

When two or more waves are present in the same medium at the same time their net effect may

often be obtained simply by adding them at each point in the medium according to the principle

of linear superposition (Note this wonrsquot work if the medium is nonlinear) When this addition

makes the total amplitude be greater than the individual amplitudes of the various waves we say

that the interference is constructive When the addition produces cancellation and an amplitude

less than the amplitudes of the separate waves we have destructive interference

Standing Waves

A standing wave is the superposition of two identical traveling waves moving in opposite

directions Nodes are places where the two waves perfectly destructively interfere to produce

zero amplitude at all times Anti-nodes are places where the two waves perfectly constructively

interfere to produce an amplitude maximum The distance between nodes is λ2 Standing waves

on a string fixed at both ends have nodes at each end of the string Standing waves in an air

column enclosed in a tube have displacement anti-nodes at open ends of the tube and

displacement nodes at closed ends The frequency of the standing wave with the lowest possible

frequency is called the fundamental frequency Standing waves on strings or in air columns all

have frequencies which are integer multiples of the fundamental frequency and are called

8

harmonics (The fundamental is called the ldquofirst harmonicrdquo)

Beats

Beats are heard when two waves with slightly different frequencies f1 and f2 are combined The

waves constructively interfere for a number of cycles then destructively interfere for a number

of cycles We hear a periodic ldquowah-wahrdquo frequency equal to the difference of the two wave

frequencies

1 2bf f f

Musical Instruments

Musical instruments produce tones by exciting standing waves on strings (violins piano) and in

tubes (trumpet organ) The fundamental frequency of the standing wave is called the pitch of the

tone The pitch of concert A is 440 Hz by definition Two tones are an octave apart if one pitch

has twice the frequency of the other In written music there are 12 intervals in each octave with

the ratio between successive intervals equal to 2112 = 105946 The ratios for each tone in an

octave starting at A and ending at the next higher A are

A A B C C D D E F F G G A

1 10595 11225 11892 12599 13348 14142 14983 15874 16818 17818 18877 2

A musical tone is actually a superposition of the fundamental frequency and the higher

harmonics The tone quality of a musical instrument is determined by the amplitudes of the

various harmonics that it produces A violin and a trumpet can play the same pitch but they

donrsquot sound at all alike to our ears The difference between them is in the various amplitudes of

their harmonics

9

Serway Chapter 19

Temperature

Formally temperature is what is measured by a thermometer Roughly high temperature is what

we call hot and low temperature is what we call cold On the atomic level temperature refers to

the kinetic energy of the molecules A collection of molecules is called ldquohotrdquo if the molecules

have rapid random motion while a collection of molecules is called ldquocoldrdquo if the random motion

is slow When two bodies are placed in close contact with each other they exchange molecular

kinetic energy until they come to the same temperature This is the microscopic picture of the

Zeroth Law of Thermodynamics

Absolute Zero

Absolute zero is the lowest possible temperature that any object can have This is the temperature

at which all of the energy than can be removed an object has been removed (This removable

energy we call thermal energy) There is still motion at absolute zero Electrons continue to orbit

around atomic nuclei and even atoms continue to move about with a small amount of kinetic

energy but this small energy cannot be removed from the object For example at absolute zero

helium is a liquid whose atoms still move and slide past each other

Temperature Scales

Kelvin Scale Absolute zero is at T = 0 K water freezes at T = 27315 K room temperature is

around T = 295 K and water boils at T = 373 K Note we donrsquot use a deg symbol Kelvin is

prefered SI Unit

Celsius Scale Absolute zero is at T = -273degC water freezes at T = 0degC room temperature is

around T = 22degC and water boils at T = 100degC

Fahrenheit Scale Absolute zero is at T = -459degF water freezes at TF = 32degF room temperature

is around T = 72degF and water boils at TF = 212degF TF =18TC + 32 Notice that temperature

differences are the same for the Kelvin and Celsius scales

Thermal Expansion

When materials are heated they usually expand and when they are cooled they usually contract

(Water near freezing is a spectacular counterexample it works the other way around) The

coefficient of linear expansion is defined by the relation

1

i

L

L T

10

where Li is the initial length of a rod of the material and ΔL is the change in its length due to a

small temperature change ΔT The coefficient of volume expansion is defined similarly

1

i

V

V T

where Vi is the initial volume of a piece of material and where ΔV is the change in its volume due

to a small temperature change ΔT

Avogadrorsquos Number (N or NA)

One mole of any substance corresponds to 6022 times 1023 molecules The atomic mass given on the

periodic table (p A32-33 in the text) is the mass in grams of one mole of atoms For example

the atomic mass of oxygen is 15999 gmol The mass of a single atom is given by the same

number in units of atomic mass units (u) For example the mass of an oxygen atom is 15999 u

(on the average)

Ideal Gas Law (an example of an equation of state)

When the molecules of a gas are sufficiently inert and widely separated that interactions between

them are negligible we say that it is an ideal gas The pressure P volume V and temperature T

(in kelvins) of such a gas are State Variables and are related by the ideal gas law

Bor PV=NkPV nRT T

where n is the number of moles of the gas where R is the gas constant

8314 Jmol KR

where N is the number of molecules and where kB is Boltzmannrsquos constant

231380 10 J KBk

It works well for air at atmosphere pressure and even better for partial vaccuums The relative

ease of measuring pressure and the linear relationship between pressure and temperature (if V

and n are held fixed) makes an ideal gas an ideal thermometer Thermometers can be based on

properties of solids or liquids but the behavior of these materials with temperature is more

complicated

11

Serway Chapter 20

Heat

Heat is energy that flows between a system and its environment because of a tempera-

ture difference between them The units of heat are Joules as expected for an energy

Unfortunately there are several competing units of energy They are related by

1 cal 4186 J 1 Cal 4186 J 1 Btu 1054 J

Heat Capacity

There is often a simple linear relation between the heat that flows in or out of part of a

system and the temperature change that results from this energy transfer When this

linear relation holds it is convenient to define the heat capacity C and the specific

heat c as follows

For the entire object the heat Q it absorbs is given by Q = C(Tf - Ti)

For a particular material the specific heat is defined by c = Cm which is the heat

capacity per unit mass so that

Q = mc(Tf ndash Ti)

Note C has units of energy (J or Cal)(Kelvin kg)

Heats of Transformation or Latent Heat Q = plusmn mL

When a substance changes phase from solid to liquid or from liquid to gas it absorbs

heat without a change in temperature The latent heat or heat of transformation is

usually given per unit mass of the substance For example for water the heat of fusion

(melting) is L = 333 kJkg while the heat of vaporization (boiling) is L = 2260 kJkg

Note that heat for boiling is considerably bigger than melting for water You have to

be careful with signs heat is given off (negative) if you go down in temperature and

condense steam

Work

In general the small amount of work done on a system as a force Fon is exerted on it

through a vector displacement dx is given by

on xdW d F

12

But if the displacement is done very slowly (as we always assume in thermodynamics)

then the force exerted on the system and the force exerted by the system are in

balance so the force exerted by the system is ndash Fon In thermodynamics it is more

convenient to talk about the force exerted by the system so we change the above

formula for the work done on the system to

xdW d F

where F is the force exerted by the system This has confused students for more than a

century now but this is the way your book and many other books do it so you are

stuck You will need to memorize the minus sign in this definition of the work to be

able to use your textbook

There are many chances to get signs wrong in this and the next two chapters (Mosiah

2 )

When an external agent changes the volume of a gas at pressure P by a small amount

dV the (small amount) of work done on the system is given by

dW PdV

Notice that this minus sign is just what we need to make dW be positive if the external

agent compresses the gas for then dV is negative If on the other hand the external

agent gives way allowing the gas to expand against it then dV is positive and we say

that the work done on the gas is negative

The work done on the system (eg by the gas in a cylinder) in a thermodynamic

process is the area under the curve in a PV diagram It is positive for compressions

and negative for expansions If the volume of gas remains constant in a process then

no work is done by the gas

Cyclic processes are important For cyclic processes represented by PV diagrams the

magnitude of the net work during one cycle is simply the area enclosed by the cycle

on the diagram Be careful to keep track of signs when you are calculating that

enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes

1

Path A-B B-C C-D D-A A to A net

Q

W

ΔU

ΔS

Internal Energy

The energy stored in a substance is called its internal energy Eint This energy may be

stored as random kinetic energy or as potential energy in each molecule (stretched

chemical bonds electrons in excited states etc) For ideal gases all states with the

same temperature will have the same Eint

First Law of Thermodynamics

The change ΔEint in the internal energy of a system is given by

intE Q W

where Q is the heat absorbed by the system and where W is the work done on the

system Hence if a system absorbs heat (and if Wge0) the internal energy increases

Likewise if the system does work (W on the system is negative) and if Qge0 the

internal energy decreases Potential Pitfall Many times people talk about work done

by the system It is the minus of W on the system Donrsquot get tripped up

Processes

Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated

from the environment A process may be approximately adiabatic if it happens so

rapidly that heat does not have time to enter or leave the system Work + or ndash is done

and ΔEint = W

Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing

on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal

energy and temperature does not change (Note the difference between an adiabatic

process and a free expansion is that NO work is done in the adiabatic free expansion)

Isobaric process The pressure is held fixed ΔP = 0 For example usually the

pressure increases when a gas is heated but if it were allowed to expand during the

2

heating process in just the right way its pressure could remain fixed In isobaric

processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)

Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is

then zero and so we have ΔEint = Q

Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint

so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in

an isothermal process The work done on the gas is then given by

lnf

i

Vi

Vf

VW PdV nRT

V

Heat Conduction

The quantity P is defined to be the rate at which heat flows through an object and is a

power having units of watts It is analogous to electric current which is the rate at

which charge flows through an object If the flow of heat through a slab of length L

and cross-sectional area A is steady in time then P is given by the equation

h cT TdQkA

dt L

P =

where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The

heat flows of course because of this temperature difference The quantity k is called

the thermal conductivity and is a constant that is characteristic of the material It is

analogous to the electrical conductivity h cT TL is sometimes called the temperature

gradient and is written dTd dTdx

R-Values

It is common to have the heat-conducting properties of materials described by their R-

values especially for insulating materials like fiberglass batting The connection

between k and R is R = Lk where L is the material thickness In this country R-values

always have units of 2ft F hourBtu

Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11

(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)

The heat flow rate through a slab of area A is given by

3

h cT TA

R

P

in units of Btuhour Note that A must be in square feet and the temperatures must be

in degrees Fahrenheit

Convection

Convection is the transfer of thermal energy by flow of material For instance a home

furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct

throughout the house instead it quickly pumps warm air to all of the rooms

Generally convection is a much faster way to transfer heat than conduction

Radiation

Electromagnetic radiation can also transfer heat When you warm yourself near a

campfire which has burned itself down into a bed of glowing embers you are

receiving radiant heat from the infrared portion of the electromagnetic spectrum The

rate at which an object emits radiant heat is given by Stefanrsquos law

4AeTP

where P is the radiated power in watts σ is a constant

8 2 45696 10 W m K

A is the surface area of the object in m2 and T is the temperature in kelvins The

constant e is called the emissivity and it varies from substance to substance A perfect

absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =

0 Hence black objects radiate very well while shiny ones do not Also an object that

is hotter than its surroundings radiates more energy than it absorbs whereas an object

that is cooler than its surroundings absorbs more energy than it radiates

Terminology

Transfer variables vs state variables

Energy transfer by heat as well as work done depends on the initial final and

intermediate states of the system They are transfer variables But their sum (Q + W =

4

Eint) is a state variable

Figure 205

5

Serway Chapter 21

Kinetic Theory

The ideal gas law works for all atoms and molecules at low pressure It is rather

amazing that it does Kinetic theory explains why The properties of an ideal gas can

be understood by thinking of it as N rapidly moving particles of mass m As these

particles collide with the container walls momentum is imparted to the walls which

we call the force of gas pressure In this picture the pressure is related to the average

of the square of the particle velocity 2v by

22 1( )

3 2

NP mv

V

Using the ideal gas law we obtain the average translational kinetic energy per

molecule

21 3

2 2 Bmv k T

The rms speed is then given by

2rms

3 3Bk T RTv v

m M

where M is the molecular mass in kgmol

Degrees of Freedom

Roughly speaking a degree of freedom is a way in which a molecule can store energy

For instance since there are three different directions in space along which a molecule

can move there are three degrees of freedom for the translational kinetic energy

There are also three different axes of rotation about which a polyatomic molecule can

spin so we say there are three degrees of freedom for the rotational kinetic energy

There are even degrees of freedom associated with the various ways in which a

molecule can vibrate and with the different energy levels in which the electrons of

the molecule can exist

Internal Energy and Degrees of Freedom The internal energy of an ideal gas made

up of molecules with J degrees of freedom is given by

int 2 2 B

J JE nRT Nk T

6

Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of

molar heat capacities CV and CP These are the heat capacities per mole and the

subscript V on CV means that the volume is being held constant while for CP the

pressure is held constant For example to raise the temperature of n moles of a gas

whose pressure is held constant by 10 K we would have to supply an amount of heat Q

= nCP (10) K

Molar Specific Heat of an Ideal Gas at Constant Volume

VQ nC T

3monatomic

2VC R

5diatomic

2VC R

5polyatomic

2VC

Real gases deviate from these formulas because in addition to the translational and ro-

tational degrees of freedom they also have vibrational and electronic degrees of

freedom These are unimportant at low temperatures due to quantum mechanical

effects but become increasingly important at higher temperatures The rough rule is

No of degrees of freedom

2VC R

Molar Specific Heat of an Ideal Gas at Constant Pressure

PQ nC T

P VC C R

The internal energy of an ideal gas depends only on the temperature

int VE nC T

Adiabatic Processes in an Ideal Gas

7

An adiabatic process is one in which no heat is exchanged between the system and the

environment When an ideal gas expands or contracts adiabatically not only does its

pressure change as expected from the ideal gas law but its temperature changes as

well Under these conditions the final pressure Pf can be computed from the initial

pressure Pi and from the final and initial volumes Vf and Vi by

or constantf f i iP V PV PV

where γ = CPCV The quantity γ is called the adiabatic exponent Note that this

doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be

combined with the adiabatic law for pressure given above to obtain the adiabatic law

for temperatures

1 constantTV

Compressions in sound waves are adiabatic because they happen too rapidly for any

appreciable amount of heat to flow This is why the adiabatic exponent appears in the

formula for the speed of sound in an ideal gas

RTv

M

Note that v depends only on T and not on P Because it depends only on the

temperature the speed of sound is the same in Provo as at sea level in spite of the

lower pressure here due to the difference in elevation

Equipartition of Energy

Every kind of molecule has a certain number of degrees of freedom which are

independent ways in which it can store energy Each such degree of freedom has

associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1

2 RT per mole)

(Note since a molecule has so many possible degrees of freedom it would seem that

there should be a lot of 12 sBk T to spread around But because energy is quantized

some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high

enough that 12 Bk T is as big as the lowest quantum of energy

8

Serway Chapter 22

Second Law of Thermodynamics

There are several equivalent forms of this important law

Kelvin It is not possible to change heat completely into work with no other change

taking place Or in other words there are no perfect heat engines

Clausius It is not possible for heat to flow from one body to another body at a higher

temperature with no other change taking place Or in other words there are no

perfect refrigerators

Entropy In any thermodynamic process that proceeds from one equilibrium state to

another the entropy of the system + environment either remains unchanged or

increases The total entropy never decreases This law is a bit of an oddity among the

laws of physics because it is not absolute Things are forbidden by the second law not

because it is impossible for them to happen but because it is extremely unlikely for

them to happen (See below for more information about entropy)

Reversible and Irreversible Processes

A reversible process is one which occurs so slowly that it is in thermal equilibrium (or

very nearly so) at all times A hallmark of such processes is that a motion picture of

them looks perfectly normal whether run forward or backward Imagine for instance

the slow expansion of a gas at constant temperature in a cylinder whose volume is being

increased by a slowly moving piston Run the movie backwards and what do you see

You see the slow compression of a gas at constant temperature which looks perfectly

normal

An irreversible process is one which occurs in such a way that thermal equilibrium is

not maintained throughout the process The mark of this kind of process is that a motion

picture of it looks very odd when run backward Imagine the sudden expansion of a gas

into a previously evacuated chamber because a hole was punched in the wall between a

pressurized chamber and the evacuated one Run the movie backward and what do you

see You see the gas in the soon-to-be-evacuated chamber gather itself together and

stream through a tiny hole into a chamber in which there is already plenty of gas If you

have ever seen this happen get in touch with the support group for those who have

witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW

9

Heat Engines

Because of the vexing difference in sign between work done on and system and work done

by a system we will invent a new work variable Weng Heat engines do work and so the net

W for these engines is negative But in engineering applications hidden minus signs are

regarded as evil so for heat engines we donrsquot talk about W instead we talk about its

magnitude engW W So for heat engines the first law is

int engE Q W

But for heat pumps and refrigerators work is done on the system so we use the usual work

W when we talk about these systems

A heat engine is a machine that absorbs heat converts part of it to work and exhausts the

rest The heat must be absorbed at high temperature and exhausted at low temperature If the

absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then

eng h cW Q Q

and the efficiency of the engine is defined to be

eng

h

We

Q

A perfect engine would convert the heat hQ completely into work Weng giving an effi-

ciency of e = 1 Energy conservation alone allows a perfect engine but the second law

requires e lt 1

Refrigerators and Heat Pumps

A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high

temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine

by some source of power A heat pump is a machine that either works like a refrigerator

keeping a place cold by transferring heat from this cold place to a higher temperature

environment (cooling mode like an air conditioner) or it functions as a heater

transferring heat into a warm place from a cooler one (heating mode like a window unit

that heats a house by extracting thermal energy from the cold outdoors) The coefficient

of performance of a refrigerator or of a heat pump in cooling mode is defined to be

COP cooling modecQ

W

10

For a heat pump in heating mode the coefficient of performance is

COP heating modehQ

W

Note that we donrsquot have to use the engineering work here because in these systems

work is done on the system and W is naturally positive

A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without

doing any work giving an infinite coefficient of performance Energy conservation alone

allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump

in heating mode would transfer Qh into the house without doing any work and so would

also have an infinite coefficient of performance The second law forbids this too A good

coefficient of performance for a real device would be around 5 or 6

Carnot Cycle

The most efficient of all possible engines is one that uses the Carnot cycle This cycle

employs an ideal gas has no friction and operates very slowly so that the gas can be in

thermal equilibrium at all parts of the cycle This means of course that it canrsquot

possibly be built and even if it could be built it would not run fast enough to be useful

Nevertheless this cycle is very important because it gives an upper bound on the

efficiency of real engines There cannot possibly be an engine that is more efficient

than one based on the Carnot cycle This cycle consists of the following four steps

1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its

volume The reason that heat is absorbed is that expansion tends to cool the gas but

thermal contact with the environment at Th keeps the temperature high by heat

conduction into the ideal gas

2 The ideal gas further increases its volume by an adiabatic expansion This expansion

causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc

3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in

volume The reason that heat is exhausted is that compression tends to heat the gas

but thermal contact with the environment at Tc keeps the temperature low by heat

conduction out of the ideal gas

4 The gas is adiabatically compressed back to its original volume (the volume it started

with in step 1) This compression heats the gas from Tc up to Th

11

The efficiency of a Carnot engine is given by the very simple formula

1 cC

h

Te

T

where the temperatures must in be Kelvin No real engine can be more efficient than

this

The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode

is given by

COP cooling mode cC

h c

T

T T

and no real refrigerator can have a coefficient of performance greater than this

The coefficient of performance of a Carnot heat pump in heating mode is

COP heating mode hC

h c

T

T T

So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency

To make the reversible steps in the cycle really reversible they would have to occur

infinitely slowly So the price you pay for making a perfect engine is that it takes

forever to get it to do any work

Entropy

The entropy of a system is defined in terms of its molecular makeup and measures

roughly the disorder of the system If the system is packed into a very small volume

then it is quite ordered and the entropy will be low If it occupies a large volume the

entropy is high (To see what this has to do with disorder note that socks in a drawer

occupy a small volume while socks on the bed in the corner by the door and

hanging from the chandelier occupy a large volume) If the system is very cold then

the molecules hardly move and may even reach out to each other and form a crystal

This is a highly ordered state and therefore has low entropy If the system is very hot

with rapidly speeding molecules crashing into the container walls and bouncing off

each other things are disordered and the entropy is high

It is possible to calculate the entropy of a system in terms of its macroscopic thermody-

namic properties ie pressure volume temperature number of moles etc The key to this

calculation is the concept of a reversible process A reversible process is one that is

12

carried out without friction and so slowly that the process can be reversed at any stage by

making an infinitesimal change in the environment of the system The slow expansion of the

gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment

is reversible If we turned around and began slowly to compress the gas would just slowly

exhaust heat to the environment in the exact reverse way that it absorbed it during expansion

Most processes however are irreversible For example if a gas-filled box were suddenly

increased in size so that the particles were free to wander into the void created by the sudden

expansion then the gas would eventually fill the new volume uniformly at the same

temperature as before the expansion (The temperature is unchanged in this imaginary

process because the kinetic energy of the molecules would be unaffected by such an

instantaneous expansion of the container walls) This imaginary but highly thought-

stimulating process is called a free expansion and it is impossible to reverse it During the

expansion we didnrsquot push on any of the molecules so reversing this process would mean

making them go back into their original volume without pushing on them they simply will

not cooperate to this extent Another way to see that just pushing them back where they came

from does not reverse the free expansion is to think about what would happen if we just

compressed either adiabatically or isothermally An adiabatic compression back to the

original volume would heat the gas above its original temperature and an isothermal

compression would require that heat be exhausted to the environment But the free expansion

involved neither temperature changes nor heat exchanges so neither of these two processes

is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in

a way that takes us back to the initial state

It is possible to calculate the change of entropy for both reversible and irreversible processes

Letrsquos consider a reversible process first In a reversible process the entropy change is given

by the formula

dQS

T

where dQ is the amount of heat added to the system during a small step of the process

The total energy change during the process may then simply be calculated by integration

f f

f i i i

dQS S S dS

T

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11

7

Serway Chapter 18

Principle of Linear Superposition

We say that a system obeys the principle of linear superposition if two or more different motions

of the system can simply be added together to find the net motion of the system Light waves

obey this principle as they propagate through the air as can be seen by shining two flashlights so

that their beams cross The beams propagate along without affecting each other (Light sabers are

a spectacular but unfortunately fictional example of systems that do not obey the principle of

superposition) Wave pulses on an ideal rope also obey this principle two different pulses pass

through each other without change Standing waves are an example of this effect being simply

the linear superposition of two traveling waves of the same frequency but moving in opposite

directions Light waves in matter do not always obey this principle For instance two powerful

laser beams could be made to cross in a piece of glass in such a way that their combined heating

effect in the crossing region could melt the glass and scatter the beams in complicated ways This

is an example of a nonlinear effect

Interference

When two or more waves are present in the same medium at the same time their net effect may

often be obtained simply by adding them at each point in the medium according to the principle

of linear superposition (Note this wonrsquot work if the medium is nonlinear) When this addition

makes the total amplitude be greater than the individual amplitudes of the various waves we say

that the interference is constructive When the addition produces cancellation and an amplitude

less than the amplitudes of the separate waves we have destructive interference

Standing Waves

A standing wave is the superposition of two identical traveling waves moving in opposite

directions Nodes are places where the two waves perfectly destructively interfere to produce

zero amplitude at all times Anti-nodes are places where the two waves perfectly constructively

interfere to produce an amplitude maximum The distance between nodes is λ2 Standing waves

on a string fixed at both ends have nodes at each end of the string Standing waves in an air

column enclosed in a tube have displacement anti-nodes at open ends of the tube and

displacement nodes at closed ends The frequency of the standing wave with the lowest possible

frequency is called the fundamental frequency Standing waves on strings or in air columns all

have frequencies which are integer multiples of the fundamental frequency and are called

8

harmonics (The fundamental is called the ldquofirst harmonicrdquo)

Beats

Beats are heard when two waves with slightly different frequencies f1 and f2 are combined The

waves constructively interfere for a number of cycles then destructively interfere for a number

of cycles We hear a periodic ldquowah-wahrdquo frequency equal to the difference of the two wave

frequencies

1 2bf f f

Musical Instruments

Musical instruments produce tones by exciting standing waves on strings (violins piano) and in

tubes (trumpet organ) The fundamental frequency of the standing wave is called the pitch of the

tone The pitch of concert A is 440 Hz by definition Two tones are an octave apart if one pitch

has twice the frequency of the other In written music there are 12 intervals in each octave with

the ratio between successive intervals equal to 2112 = 105946 The ratios for each tone in an

octave starting at A and ending at the next higher A are

A A B C C D D E F F G G A

1 10595 11225 11892 12599 13348 14142 14983 15874 16818 17818 18877 2

A musical tone is actually a superposition of the fundamental frequency and the higher

harmonics The tone quality of a musical instrument is determined by the amplitudes of the

various harmonics that it produces A violin and a trumpet can play the same pitch but they

donrsquot sound at all alike to our ears The difference between them is in the various amplitudes of

their harmonics

9

Serway Chapter 19

Temperature

Formally temperature is what is measured by a thermometer Roughly high temperature is what

we call hot and low temperature is what we call cold On the atomic level temperature refers to

the kinetic energy of the molecules A collection of molecules is called ldquohotrdquo if the molecules

have rapid random motion while a collection of molecules is called ldquocoldrdquo if the random motion

is slow When two bodies are placed in close contact with each other they exchange molecular

kinetic energy until they come to the same temperature This is the microscopic picture of the

Zeroth Law of Thermodynamics

Absolute Zero

Absolute zero is the lowest possible temperature that any object can have This is the temperature

at which all of the energy than can be removed an object has been removed (This removable

energy we call thermal energy) There is still motion at absolute zero Electrons continue to orbit

around atomic nuclei and even atoms continue to move about with a small amount of kinetic

energy but this small energy cannot be removed from the object For example at absolute zero

helium is a liquid whose atoms still move and slide past each other

Temperature Scales

Kelvin Scale Absolute zero is at T = 0 K water freezes at T = 27315 K room temperature is

around T = 295 K and water boils at T = 373 K Note we donrsquot use a deg symbol Kelvin is

prefered SI Unit

Celsius Scale Absolute zero is at T = -273degC water freezes at T = 0degC room temperature is

around T = 22degC and water boils at T = 100degC

Fahrenheit Scale Absolute zero is at T = -459degF water freezes at TF = 32degF room temperature

is around T = 72degF and water boils at TF = 212degF TF =18TC + 32 Notice that temperature

differences are the same for the Kelvin and Celsius scales

Thermal Expansion

When materials are heated they usually expand and when they are cooled they usually contract

(Water near freezing is a spectacular counterexample it works the other way around) The

coefficient of linear expansion is defined by the relation

1

i

L

L T

10

where Li is the initial length of a rod of the material and ΔL is the change in its length due to a

small temperature change ΔT The coefficient of volume expansion is defined similarly

1

i

V

V T

where Vi is the initial volume of a piece of material and where ΔV is the change in its volume due

to a small temperature change ΔT

Avogadrorsquos Number (N or NA)

One mole of any substance corresponds to 6022 times 1023 molecules The atomic mass given on the

periodic table (p A32-33 in the text) is the mass in grams of one mole of atoms For example

the atomic mass of oxygen is 15999 gmol The mass of a single atom is given by the same

number in units of atomic mass units (u) For example the mass of an oxygen atom is 15999 u

(on the average)

Ideal Gas Law (an example of an equation of state)

When the molecules of a gas are sufficiently inert and widely separated that interactions between

them are negligible we say that it is an ideal gas The pressure P volume V and temperature T

(in kelvins) of such a gas are State Variables and are related by the ideal gas law

Bor PV=NkPV nRT T

where n is the number of moles of the gas where R is the gas constant

8314 Jmol KR

where N is the number of molecules and where kB is Boltzmannrsquos constant

231380 10 J KBk

It works well for air at atmosphere pressure and even better for partial vaccuums The relative

ease of measuring pressure and the linear relationship between pressure and temperature (if V

and n are held fixed) makes an ideal gas an ideal thermometer Thermometers can be based on

properties of solids or liquids but the behavior of these materials with temperature is more

complicated

11

Serway Chapter 20

Heat

Heat is energy that flows between a system and its environment because of a tempera-

ture difference between them The units of heat are Joules as expected for an energy

Unfortunately there are several competing units of energy They are related by

1 cal 4186 J 1 Cal 4186 J 1 Btu 1054 J

Heat Capacity

There is often a simple linear relation between the heat that flows in or out of part of a

system and the temperature change that results from this energy transfer When this

linear relation holds it is convenient to define the heat capacity C and the specific

heat c as follows

For the entire object the heat Q it absorbs is given by Q = C(Tf - Ti)

For a particular material the specific heat is defined by c = Cm which is the heat

capacity per unit mass so that

Q = mc(Tf ndash Ti)

Note C has units of energy (J or Cal)(Kelvin kg)

Heats of Transformation or Latent Heat Q = plusmn mL

When a substance changes phase from solid to liquid or from liquid to gas it absorbs

heat without a change in temperature The latent heat or heat of transformation is

usually given per unit mass of the substance For example for water the heat of fusion

(melting) is L = 333 kJkg while the heat of vaporization (boiling) is L = 2260 kJkg

Note that heat for boiling is considerably bigger than melting for water You have to

be careful with signs heat is given off (negative) if you go down in temperature and

condense steam

Work

In general the small amount of work done on a system as a force Fon is exerted on it

through a vector displacement dx is given by

on xdW d F

12

But if the displacement is done very slowly (as we always assume in thermodynamics)

then the force exerted on the system and the force exerted by the system are in

balance so the force exerted by the system is ndash Fon In thermodynamics it is more

convenient to talk about the force exerted by the system so we change the above

formula for the work done on the system to

xdW d F

where F is the force exerted by the system This has confused students for more than a

century now but this is the way your book and many other books do it so you are

stuck You will need to memorize the minus sign in this definition of the work to be

able to use your textbook

There are many chances to get signs wrong in this and the next two chapters (Mosiah

2 )

When an external agent changes the volume of a gas at pressure P by a small amount

dV the (small amount) of work done on the system is given by

dW PdV

Notice that this minus sign is just what we need to make dW be positive if the external

agent compresses the gas for then dV is negative If on the other hand the external

agent gives way allowing the gas to expand against it then dV is positive and we say

that the work done on the gas is negative

The work done on the system (eg by the gas in a cylinder) in a thermodynamic

process is the area under the curve in a PV diagram It is positive for compressions

and negative for expansions If the volume of gas remains constant in a process then

no work is done by the gas

Cyclic processes are important For cyclic processes represented by PV diagrams the

magnitude of the net work during one cycle is simply the area enclosed by the cycle

on the diagram Be careful to keep track of signs when you are calculating that

enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes

1

Path A-B B-C C-D D-A A to A net

Q

W

ΔU

ΔS

Internal Energy

The energy stored in a substance is called its internal energy Eint This energy may be

stored as random kinetic energy or as potential energy in each molecule (stretched

chemical bonds electrons in excited states etc) For ideal gases all states with the

same temperature will have the same Eint

First Law of Thermodynamics

The change ΔEint in the internal energy of a system is given by

intE Q W

where Q is the heat absorbed by the system and where W is the work done on the

system Hence if a system absorbs heat (and if Wge0) the internal energy increases

Likewise if the system does work (W on the system is negative) and if Qge0 the

internal energy decreases Potential Pitfall Many times people talk about work done

by the system It is the minus of W on the system Donrsquot get tripped up

Processes

Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated

from the environment A process may be approximately adiabatic if it happens so

rapidly that heat does not have time to enter or leave the system Work + or ndash is done

and ΔEint = W

Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing

on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal

energy and temperature does not change (Note the difference between an adiabatic

process and a free expansion is that NO work is done in the adiabatic free expansion)

Isobaric process The pressure is held fixed ΔP = 0 For example usually the

pressure increases when a gas is heated but if it were allowed to expand during the

2

heating process in just the right way its pressure could remain fixed In isobaric

processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)

Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is

then zero and so we have ΔEint = Q

Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint

so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in

an isothermal process The work done on the gas is then given by

lnf

i

Vi

Vf

VW PdV nRT

V

Heat Conduction

The quantity P is defined to be the rate at which heat flows through an object and is a

power having units of watts It is analogous to electric current which is the rate at

which charge flows through an object If the flow of heat through a slab of length L

and cross-sectional area A is steady in time then P is given by the equation

h cT TdQkA

dt L

P =

where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The

heat flows of course because of this temperature difference The quantity k is called

the thermal conductivity and is a constant that is characteristic of the material It is

analogous to the electrical conductivity h cT TL is sometimes called the temperature

gradient and is written dTd dTdx

R-Values

It is common to have the heat-conducting properties of materials described by their R-

values especially for insulating materials like fiberglass batting The connection

between k and R is R = Lk where L is the material thickness In this country R-values

always have units of 2ft F hourBtu

Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11

(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)

The heat flow rate through a slab of area A is given by

3

h cT TA

R

P

in units of Btuhour Note that A must be in square feet and the temperatures must be

in degrees Fahrenheit

Convection

Convection is the transfer of thermal energy by flow of material For instance a home

furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct

throughout the house instead it quickly pumps warm air to all of the rooms

Generally convection is a much faster way to transfer heat than conduction

Radiation

Electromagnetic radiation can also transfer heat When you warm yourself near a

campfire which has burned itself down into a bed of glowing embers you are

receiving radiant heat from the infrared portion of the electromagnetic spectrum The

rate at which an object emits radiant heat is given by Stefanrsquos law

4AeTP

where P is the radiated power in watts σ is a constant

8 2 45696 10 W m K

A is the surface area of the object in m2 and T is the temperature in kelvins The

constant e is called the emissivity and it varies from substance to substance A perfect

absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =

0 Hence black objects radiate very well while shiny ones do not Also an object that

is hotter than its surroundings radiates more energy than it absorbs whereas an object

that is cooler than its surroundings absorbs more energy than it radiates

Terminology

Transfer variables vs state variables

Energy transfer by heat as well as work done depends on the initial final and

intermediate states of the system They are transfer variables But their sum (Q + W =

4

Eint) is a state variable

Figure 205

5

Serway Chapter 21

Kinetic Theory

The ideal gas law works for all atoms and molecules at low pressure It is rather

amazing that it does Kinetic theory explains why The properties of an ideal gas can

be understood by thinking of it as N rapidly moving particles of mass m As these

particles collide with the container walls momentum is imparted to the walls which

we call the force of gas pressure In this picture the pressure is related to the average

of the square of the particle velocity 2v by

22 1( )

3 2

NP mv

V

Using the ideal gas law we obtain the average translational kinetic energy per

molecule

21 3

2 2 Bmv k T

The rms speed is then given by

2rms

3 3Bk T RTv v

m M

where M is the molecular mass in kgmol

Degrees of Freedom

Roughly speaking a degree of freedom is a way in which a molecule can store energy

For instance since there are three different directions in space along which a molecule

can move there are three degrees of freedom for the translational kinetic energy

There are also three different axes of rotation about which a polyatomic molecule can

spin so we say there are three degrees of freedom for the rotational kinetic energy

There are even degrees of freedom associated with the various ways in which a

molecule can vibrate and with the different energy levels in which the electrons of

the molecule can exist

Internal Energy and Degrees of Freedom The internal energy of an ideal gas made

up of molecules with J degrees of freedom is given by

int 2 2 B

J JE nRT Nk T

6

Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of

molar heat capacities CV and CP These are the heat capacities per mole and the

subscript V on CV means that the volume is being held constant while for CP the

pressure is held constant For example to raise the temperature of n moles of a gas

whose pressure is held constant by 10 K we would have to supply an amount of heat Q

= nCP (10) K

Molar Specific Heat of an Ideal Gas at Constant Volume

VQ nC T

3monatomic

2VC R

5diatomic

2VC R

5polyatomic

2VC

Real gases deviate from these formulas because in addition to the translational and ro-

tational degrees of freedom they also have vibrational and electronic degrees of

freedom These are unimportant at low temperatures due to quantum mechanical

effects but become increasingly important at higher temperatures The rough rule is

No of degrees of freedom

2VC R

Molar Specific Heat of an Ideal Gas at Constant Pressure

PQ nC T

P VC C R

The internal energy of an ideal gas depends only on the temperature

int VE nC T

Adiabatic Processes in an Ideal Gas

7

An adiabatic process is one in which no heat is exchanged between the system and the

environment When an ideal gas expands or contracts adiabatically not only does its

pressure change as expected from the ideal gas law but its temperature changes as

well Under these conditions the final pressure Pf can be computed from the initial

pressure Pi and from the final and initial volumes Vf and Vi by

or constantf f i iP V PV PV

where γ = CPCV The quantity γ is called the adiabatic exponent Note that this

doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be

combined with the adiabatic law for pressure given above to obtain the adiabatic law

for temperatures

1 constantTV

Compressions in sound waves are adiabatic because they happen too rapidly for any

appreciable amount of heat to flow This is why the adiabatic exponent appears in the

formula for the speed of sound in an ideal gas

RTv

M

Note that v depends only on T and not on P Because it depends only on the

temperature the speed of sound is the same in Provo as at sea level in spite of the

lower pressure here due to the difference in elevation

Equipartition of Energy

Every kind of molecule has a certain number of degrees of freedom which are

independent ways in which it can store energy Each such degree of freedom has

associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1

2 RT per mole)

(Note since a molecule has so many possible degrees of freedom it would seem that

there should be a lot of 12 sBk T to spread around But because energy is quantized

some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high

enough that 12 Bk T is as big as the lowest quantum of energy

8

Serway Chapter 22

Second Law of Thermodynamics

There are several equivalent forms of this important law

Kelvin It is not possible to change heat completely into work with no other change

taking place Or in other words there are no perfect heat engines

Clausius It is not possible for heat to flow from one body to another body at a higher

temperature with no other change taking place Or in other words there are no

perfect refrigerators

Entropy In any thermodynamic process that proceeds from one equilibrium state to

another the entropy of the system + environment either remains unchanged or

increases The total entropy never decreases This law is a bit of an oddity among the

laws of physics because it is not absolute Things are forbidden by the second law not

because it is impossible for them to happen but because it is extremely unlikely for

them to happen (See below for more information about entropy)

Reversible and Irreversible Processes

A reversible process is one which occurs so slowly that it is in thermal equilibrium (or

very nearly so) at all times A hallmark of such processes is that a motion picture of

them looks perfectly normal whether run forward or backward Imagine for instance

the slow expansion of a gas at constant temperature in a cylinder whose volume is being

increased by a slowly moving piston Run the movie backwards and what do you see

You see the slow compression of a gas at constant temperature which looks perfectly

normal

An irreversible process is one which occurs in such a way that thermal equilibrium is

not maintained throughout the process The mark of this kind of process is that a motion

picture of it looks very odd when run backward Imagine the sudden expansion of a gas

into a previously evacuated chamber because a hole was punched in the wall between a

pressurized chamber and the evacuated one Run the movie backward and what do you

see You see the gas in the soon-to-be-evacuated chamber gather itself together and

stream through a tiny hole into a chamber in which there is already plenty of gas If you

have ever seen this happen get in touch with the support group for those who have

witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW

9

Heat Engines

Because of the vexing difference in sign between work done on and system and work done

by a system we will invent a new work variable Weng Heat engines do work and so the net

W for these engines is negative But in engineering applications hidden minus signs are

regarded as evil so for heat engines we donrsquot talk about W instead we talk about its

magnitude engW W So for heat engines the first law is

int engE Q W

But for heat pumps and refrigerators work is done on the system so we use the usual work

W when we talk about these systems

A heat engine is a machine that absorbs heat converts part of it to work and exhausts the

rest The heat must be absorbed at high temperature and exhausted at low temperature If the

absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then

eng h cW Q Q

and the efficiency of the engine is defined to be

eng

h

We

Q

A perfect engine would convert the heat hQ completely into work Weng giving an effi-

ciency of e = 1 Energy conservation alone allows a perfect engine but the second law

requires e lt 1

Refrigerators and Heat Pumps

A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high

temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine

by some source of power A heat pump is a machine that either works like a refrigerator

keeping a place cold by transferring heat from this cold place to a higher temperature

environment (cooling mode like an air conditioner) or it functions as a heater

transferring heat into a warm place from a cooler one (heating mode like a window unit

that heats a house by extracting thermal energy from the cold outdoors) The coefficient

of performance of a refrigerator or of a heat pump in cooling mode is defined to be

COP cooling modecQ

W

10

For a heat pump in heating mode the coefficient of performance is

COP heating modehQ

W

Note that we donrsquot have to use the engineering work here because in these systems

work is done on the system and W is naturally positive

A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without

doing any work giving an infinite coefficient of performance Energy conservation alone

allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump

in heating mode would transfer Qh into the house without doing any work and so would

also have an infinite coefficient of performance The second law forbids this too A good

coefficient of performance for a real device would be around 5 or 6

Carnot Cycle

The most efficient of all possible engines is one that uses the Carnot cycle This cycle

employs an ideal gas has no friction and operates very slowly so that the gas can be in

thermal equilibrium at all parts of the cycle This means of course that it canrsquot

possibly be built and even if it could be built it would not run fast enough to be useful

Nevertheless this cycle is very important because it gives an upper bound on the

efficiency of real engines There cannot possibly be an engine that is more efficient

than one based on the Carnot cycle This cycle consists of the following four steps

1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its

volume The reason that heat is absorbed is that expansion tends to cool the gas but

thermal contact with the environment at Th keeps the temperature high by heat

conduction into the ideal gas

2 The ideal gas further increases its volume by an adiabatic expansion This expansion

causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc

3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in

volume The reason that heat is exhausted is that compression tends to heat the gas

but thermal contact with the environment at Tc keeps the temperature low by heat

conduction out of the ideal gas

4 The gas is adiabatically compressed back to its original volume (the volume it started

with in step 1) This compression heats the gas from Tc up to Th

11

The efficiency of a Carnot engine is given by the very simple formula

1 cC

h

Te

T

where the temperatures must in be Kelvin No real engine can be more efficient than

this

The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode

is given by

COP cooling mode cC

h c

T

T T

and no real refrigerator can have a coefficient of performance greater than this

The coefficient of performance of a Carnot heat pump in heating mode is

COP heating mode hC

h c

T

T T

So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency

To make the reversible steps in the cycle really reversible they would have to occur

infinitely slowly So the price you pay for making a perfect engine is that it takes

forever to get it to do any work

Entropy

The entropy of a system is defined in terms of its molecular makeup and measures

roughly the disorder of the system If the system is packed into a very small volume

then it is quite ordered and the entropy will be low If it occupies a large volume the

entropy is high (To see what this has to do with disorder note that socks in a drawer

occupy a small volume while socks on the bed in the corner by the door and

hanging from the chandelier occupy a large volume) If the system is very cold then

the molecules hardly move and may even reach out to each other and form a crystal

This is a highly ordered state and therefore has low entropy If the system is very hot

with rapidly speeding molecules crashing into the container walls and bouncing off

each other things are disordered and the entropy is high

It is possible to calculate the entropy of a system in terms of its macroscopic thermody-

namic properties ie pressure volume temperature number of moles etc The key to this

calculation is the concept of a reversible process A reversible process is one that is

12

carried out without friction and so slowly that the process can be reversed at any stage by

making an infinitesimal change in the environment of the system The slow expansion of the

gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment

is reversible If we turned around and began slowly to compress the gas would just slowly

exhaust heat to the environment in the exact reverse way that it absorbed it during expansion

Most processes however are irreversible For example if a gas-filled box were suddenly

increased in size so that the particles were free to wander into the void created by the sudden

expansion then the gas would eventually fill the new volume uniformly at the same

temperature as before the expansion (The temperature is unchanged in this imaginary

process because the kinetic energy of the molecules would be unaffected by such an

instantaneous expansion of the container walls) This imaginary but highly thought-

stimulating process is called a free expansion and it is impossible to reverse it During the

expansion we didnrsquot push on any of the molecules so reversing this process would mean

making them go back into their original volume without pushing on them they simply will

not cooperate to this extent Another way to see that just pushing them back where they came

from does not reverse the free expansion is to think about what would happen if we just

compressed either adiabatically or isothermally An adiabatic compression back to the

original volume would heat the gas above its original temperature and an isothermal

compression would require that heat be exhausted to the environment But the free expansion

involved neither temperature changes nor heat exchanges so neither of these two processes

is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in

a way that takes us back to the initial state

It is possible to calculate the change of entropy for both reversible and irreversible processes

Letrsquos consider a reversible process first In a reversible process the entropy change is given

by the formula

dQS

T

where dQ is the amount of heat added to the system during a small step of the process

The total energy change during the process may then simply be calculated by integration

f f

f i i i

dQS S S dS

T

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11

8

harmonics (The fundamental is called the ldquofirst harmonicrdquo)

Beats

Beats are heard when two waves with slightly different frequencies f1 and f2 are combined The

waves constructively interfere for a number of cycles then destructively interfere for a number

of cycles We hear a periodic ldquowah-wahrdquo frequency equal to the difference of the two wave

frequencies

1 2bf f f

Musical Instruments

Musical instruments produce tones by exciting standing waves on strings (violins piano) and in

tubes (trumpet organ) The fundamental frequency of the standing wave is called the pitch of the

tone The pitch of concert A is 440 Hz by definition Two tones are an octave apart if one pitch

has twice the frequency of the other In written music there are 12 intervals in each octave with

the ratio between successive intervals equal to 2112 = 105946 The ratios for each tone in an

octave starting at A and ending at the next higher A are

A A B C C D D E F F G G A

1 10595 11225 11892 12599 13348 14142 14983 15874 16818 17818 18877 2

A musical tone is actually a superposition of the fundamental frequency and the higher

harmonics The tone quality of a musical instrument is determined by the amplitudes of the

various harmonics that it produces A violin and a trumpet can play the same pitch but they

donrsquot sound at all alike to our ears The difference between them is in the various amplitudes of

their harmonics

9

Serway Chapter 19

Temperature

Formally temperature is what is measured by a thermometer Roughly high temperature is what

we call hot and low temperature is what we call cold On the atomic level temperature refers to

the kinetic energy of the molecules A collection of molecules is called ldquohotrdquo if the molecules

have rapid random motion while a collection of molecules is called ldquocoldrdquo if the random motion

is slow When two bodies are placed in close contact with each other they exchange molecular

kinetic energy until they come to the same temperature This is the microscopic picture of the

Zeroth Law of Thermodynamics

Absolute Zero

Absolute zero is the lowest possible temperature that any object can have This is the temperature

at which all of the energy than can be removed an object has been removed (This removable

energy we call thermal energy) There is still motion at absolute zero Electrons continue to orbit

around atomic nuclei and even atoms continue to move about with a small amount of kinetic

energy but this small energy cannot be removed from the object For example at absolute zero

helium is a liquid whose atoms still move and slide past each other

Temperature Scales

Kelvin Scale Absolute zero is at T = 0 K water freezes at T = 27315 K room temperature is

around T = 295 K and water boils at T = 373 K Note we donrsquot use a deg symbol Kelvin is

prefered SI Unit

Celsius Scale Absolute zero is at T = -273degC water freezes at T = 0degC room temperature is

around T = 22degC and water boils at T = 100degC

Fahrenheit Scale Absolute zero is at T = -459degF water freezes at TF = 32degF room temperature

is around T = 72degF and water boils at TF = 212degF TF =18TC + 32 Notice that temperature

differences are the same for the Kelvin and Celsius scales

Thermal Expansion

When materials are heated they usually expand and when they are cooled they usually contract

(Water near freezing is a spectacular counterexample it works the other way around) The

coefficient of linear expansion is defined by the relation

1

i

L

L T

10

where Li is the initial length of a rod of the material and ΔL is the change in its length due to a

small temperature change ΔT The coefficient of volume expansion is defined similarly

1

i

V

V T

where Vi is the initial volume of a piece of material and where ΔV is the change in its volume due

to a small temperature change ΔT

Avogadrorsquos Number (N or NA)

One mole of any substance corresponds to 6022 times 1023 molecules The atomic mass given on the

periodic table (p A32-33 in the text) is the mass in grams of one mole of atoms For example

the atomic mass of oxygen is 15999 gmol The mass of a single atom is given by the same

number in units of atomic mass units (u) For example the mass of an oxygen atom is 15999 u

(on the average)

Ideal Gas Law (an example of an equation of state)

When the molecules of a gas are sufficiently inert and widely separated that interactions between

them are negligible we say that it is an ideal gas The pressure P volume V and temperature T

(in kelvins) of such a gas are State Variables and are related by the ideal gas law

Bor PV=NkPV nRT T

where n is the number of moles of the gas where R is the gas constant

8314 Jmol KR

where N is the number of molecules and where kB is Boltzmannrsquos constant

231380 10 J KBk

It works well for air at atmosphere pressure and even better for partial vaccuums The relative

ease of measuring pressure and the linear relationship between pressure and temperature (if V

and n are held fixed) makes an ideal gas an ideal thermometer Thermometers can be based on

properties of solids or liquids but the behavior of these materials with temperature is more

complicated

11

Serway Chapter 20

Heat

Heat is energy that flows between a system and its environment because of a tempera-

ture difference between them The units of heat are Joules as expected for an energy

Unfortunately there are several competing units of energy They are related by

1 cal 4186 J 1 Cal 4186 J 1 Btu 1054 J

Heat Capacity

There is often a simple linear relation between the heat that flows in or out of part of a

system and the temperature change that results from this energy transfer When this

linear relation holds it is convenient to define the heat capacity C and the specific

heat c as follows

For the entire object the heat Q it absorbs is given by Q = C(Tf - Ti)

For a particular material the specific heat is defined by c = Cm which is the heat

capacity per unit mass so that

Q = mc(Tf ndash Ti)

Note C has units of energy (J or Cal)(Kelvin kg)

Heats of Transformation or Latent Heat Q = plusmn mL

When a substance changes phase from solid to liquid or from liquid to gas it absorbs

heat without a change in temperature The latent heat or heat of transformation is

usually given per unit mass of the substance For example for water the heat of fusion

(melting) is L = 333 kJkg while the heat of vaporization (boiling) is L = 2260 kJkg

Note that heat for boiling is considerably bigger than melting for water You have to

be careful with signs heat is given off (negative) if you go down in temperature and

condense steam

Work

In general the small amount of work done on a system as a force Fon is exerted on it

through a vector displacement dx is given by

on xdW d F

12

But if the displacement is done very slowly (as we always assume in thermodynamics)

then the force exerted on the system and the force exerted by the system are in

balance so the force exerted by the system is ndash Fon In thermodynamics it is more

convenient to talk about the force exerted by the system so we change the above

formula for the work done on the system to

xdW d F

where F is the force exerted by the system This has confused students for more than a

century now but this is the way your book and many other books do it so you are

stuck You will need to memorize the minus sign in this definition of the work to be

able to use your textbook

There are many chances to get signs wrong in this and the next two chapters (Mosiah

2 )

When an external agent changes the volume of a gas at pressure P by a small amount

dV the (small amount) of work done on the system is given by

dW PdV

Notice that this minus sign is just what we need to make dW be positive if the external

agent compresses the gas for then dV is negative If on the other hand the external

agent gives way allowing the gas to expand against it then dV is positive and we say

that the work done on the gas is negative

The work done on the system (eg by the gas in a cylinder) in a thermodynamic

process is the area under the curve in a PV diagram It is positive for compressions

and negative for expansions If the volume of gas remains constant in a process then

no work is done by the gas

Cyclic processes are important For cyclic processes represented by PV diagrams the

magnitude of the net work during one cycle is simply the area enclosed by the cycle

on the diagram Be careful to keep track of signs when you are calculating that

enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes

1

Path A-B B-C C-D D-A A to A net

Q

W

ΔU

ΔS

Internal Energy

The energy stored in a substance is called its internal energy Eint This energy may be

stored as random kinetic energy or as potential energy in each molecule (stretched

chemical bonds electrons in excited states etc) For ideal gases all states with the

same temperature will have the same Eint

First Law of Thermodynamics

The change ΔEint in the internal energy of a system is given by

intE Q W

where Q is the heat absorbed by the system and where W is the work done on the

system Hence if a system absorbs heat (and if Wge0) the internal energy increases

Likewise if the system does work (W on the system is negative) and if Qge0 the

internal energy decreases Potential Pitfall Many times people talk about work done

by the system It is the minus of W on the system Donrsquot get tripped up

Processes

Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated

from the environment A process may be approximately adiabatic if it happens so

rapidly that heat does not have time to enter or leave the system Work + or ndash is done

and ΔEint = W

Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing

on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal

energy and temperature does not change (Note the difference between an adiabatic

process and a free expansion is that NO work is done in the adiabatic free expansion)

Isobaric process The pressure is held fixed ΔP = 0 For example usually the

pressure increases when a gas is heated but if it were allowed to expand during the

2

heating process in just the right way its pressure could remain fixed In isobaric

processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)

Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is

then zero and so we have ΔEint = Q

Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint

so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in

an isothermal process The work done on the gas is then given by

lnf

i

Vi

Vf

VW PdV nRT

V

Heat Conduction

The quantity P is defined to be the rate at which heat flows through an object and is a

power having units of watts It is analogous to electric current which is the rate at

which charge flows through an object If the flow of heat through a slab of length L

and cross-sectional area A is steady in time then P is given by the equation

h cT TdQkA

dt L

P =

where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The

heat flows of course because of this temperature difference The quantity k is called

the thermal conductivity and is a constant that is characteristic of the material It is

analogous to the electrical conductivity h cT TL is sometimes called the temperature

gradient and is written dTd dTdx

R-Values

It is common to have the heat-conducting properties of materials described by their R-

values especially for insulating materials like fiberglass batting The connection

between k and R is R = Lk where L is the material thickness In this country R-values

always have units of 2ft F hourBtu

Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11

(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)

The heat flow rate through a slab of area A is given by

3

h cT TA

R

P

in units of Btuhour Note that A must be in square feet and the temperatures must be

in degrees Fahrenheit

Convection

Convection is the transfer of thermal energy by flow of material For instance a home

furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct

throughout the house instead it quickly pumps warm air to all of the rooms

Generally convection is a much faster way to transfer heat than conduction

Radiation

Electromagnetic radiation can also transfer heat When you warm yourself near a

campfire which has burned itself down into a bed of glowing embers you are

receiving radiant heat from the infrared portion of the electromagnetic spectrum The

rate at which an object emits radiant heat is given by Stefanrsquos law

4AeTP

where P is the radiated power in watts σ is a constant

8 2 45696 10 W m K

A is the surface area of the object in m2 and T is the temperature in kelvins The

constant e is called the emissivity and it varies from substance to substance A perfect

absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =

0 Hence black objects radiate very well while shiny ones do not Also an object that

is hotter than its surroundings radiates more energy than it absorbs whereas an object

that is cooler than its surroundings absorbs more energy than it radiates

Terminology

Transfer variables vs state variables

Energy transfer by heat as well as work done depends on the initial final and

intermediate states of the system They are transfer variables But their sum (Q + W =

4

Eint) is a state variable

Figure 205

5

Serway Chapter 21

Kinetic Theory

The ideal gas law works for all atoms and molecules at low pressure It is rather

amazing that it does Kinetic theory explains why The properties of an ideal gas can

be understood by thinking of it as N rapidly moving particles of mass m As these

particles collide with the container walls momentum is imparted to the walls which

we call the force of gas pressure In this picture the pressure is related to the average

of the square of the particle velocity 2v by

22 1( )

3 2

NP mv

V

Using the ideal gas law we obtain the average translational kinetic energy per

molecule

21 3

2 2 Bmv k T

The rms speed is then given by

2rms

3 3Bk T RTv v

m M

where M is the molecular mass in kgmol

Degrees of Freedom

Roughly speaking a degree of freedom is a way in which a molecule can store energy

For instance since there are three different directions in space along which a molecule

can move there are three degrees of freedom for the translational kinetic energy

There are also three different axes of rotation about which a polyatomic molecule can

spin so we say there are three degrees of freedom for the rotational kinetic energy

There are even degrees of freedom associated with the various ways in which a

molecule can vibrate and with the different energy levels in which the electrons of

the molecule can exist

Internal Energy and Degrees of Freedom The internal energy of an ideal gas made

up of molecules with J degrees of freedom is given by

int 2 2 B

J JE nRT Nk T

6

Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of

molar heat capacities CV and CP These are the heat capacities per mole and the

subscript V on CV means that the volume is being held constant while for CP the

pressure is held constant For example to raise the temperature of n moles of a gas

whose pressure is held constant by 10 K we would have to supply an amount of heat Q

= nCP (10) K

Molar Specific Heat of an Ideal Gas at Constant Volume

VQ nC T

3monatomic

2VC R

5diatomic

2VC R

5polyatomic

2VC

Real gases deviate from these formulas because in addition to the translational and ro-

tational degrees of freedom they also have vibrational and electronic degrees of

freedom These are unimportant at low temperatures due to quantum mechanical

effects but become increasingly important at higher temperatures The rough rule is

No of degrees of freedom

2VC R

Molar Specific Heat of an Ideal Gas at Constant Pressure

PQ nC T

P VC C R

The internal energy of an ideal gas depends only on the temperature

int VE nC T

Adiabatic Processes in an Ideal Gas

7

An adiabatic process is one in which no heat is exchanged between the system and the

environment When an ideal gas expands or contracts adiabatically not only does its

pressure change as expected from the ideal gas law but its temperature changes as

well Under these conditions the final pressure Pf can be computed from the initial

pressure Pi and from the final and initial volumes Vf and Vi by

or constantf f i iP V PV PV

where γ = CPCV The quantity γ is called the adiabatic exponent Note that this

doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be

combined with the adiabatic law for pressure given above to obtain the adiabatic law

for temperatures

1 constantTV

Compressions in sound waves are adiabatic because they happen too rapidly for any

appreciable amount of heat to flow This is why the adiabatic exponent appears in the

formula for the speed of sound in an ideal gas

RTv

M

Note that v depends only on T and not on P Because it depends only on the

temperature the speed of sound is the same in Provo as at sea level in spite of the

lower pressure here due to the difference in elevation

Equipartition of Energy

Every kind of molecule has a certain number of degrees of freedom which are

independent ways in which it can store energy Each such degree of freedom has

associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1

2 RT per mole)

(Note since a molecule has so many possible degrees of freedom it would seem that

there should be a lot of 12 sBk T to spread around But because energy is quantized

some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high

enough that 12 Bk T is as big as the lowest quantum of energy

8

Serway Chapter 22

Second Law of Thermodynamics

There are several equivalent forms of this important law

Kelvin It is not possible to change heat completely into work with no other change

taking place Or in other words there are no perfect heat engines

Clausius It is not possible for heat to flow from one body to another body at a higher

temperature with no other change taking place Or in other words there are no

perfect refrigerators

Entropy In any thermodynamic process that proceeds from one equilibrium state to

another the entropy of the system + environment either remains unchanged or

increases The total entropy never decreases This law is a bit of an oddity among the

laws of physics because it is not absolute Things are forbidden by the second law not

because it is impossible for them to happen but because it is extremely unlikely for

them to happen (See below for more information about entropy)

Reversible and Irreversible Processes

A reversible process is one which occurs so slowly that it is in thermal equilibrium (or

very nearly so) at all times A hallmark of such processes is that a motion picture of

them looks perfectly normal whether run forward or backward Imagine for instance

the slow expansion of a gas at constant temperature in a cylinder whose volume is being

increased by a slowly moving piston Run the movie backwards and what do you see

You see the slow compression of a gas at constant temperature which looks perfectly

normal

An irreversible process is one which occurs in such a way that thermal equilibrium is

not maintained throughout the process The mark of this kind of process is that a motion

picture of it looks very odd when run backward Imagine the sudden expansion of a gas

into a previously evacuated chamber because a hole was punched in the wall between a

pressurized chamber and the evacuated one Run the movie backward and what do you

see You see the gas in the soon-to-be-evacuated chamber gather itself together and

stream through a tiny hole into a chamber in which there is already plenty of gas If you

have ever seen this happen get in touch with the support group for those who have

witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW

9

Heat Engines

Because of the vexing difference in sign between work done on and system and work done

by a system we will invent a new work variable Weng Heat engines do work and so the net

W for these engines is negative But in engineering applications hidden minus signs are

regarded as evil so for heat engines we donrsquot talk about W instead we talk about its

magnitude engW W So for heat engines the first law is

int engE Q W

But for heat pumps and refrigerators work is done on the system so we use the usual work

W when we talk about these systems

A heat engine is a machine that absorbs heat converts part of it to work and exhausts the

rest The heat must be absorbed at high temperature and exhausted at low temperature If the

absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then

eng h cW Q Q

and the efficiency of the engine is defined to be

eng

h

We

Q

A perfect engine would convert the heat hQ completely into work Weng giving an effi-

ciency of e = 1 Energy conservation alone allows a perfect engine but the second law

requires e lt 1

Refrigerators and Heat Pumps

A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high

temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine

by some source of power A heat pump is a machine that either works like a refrigerator

keeping a place cold by transferring heat from this cold place to a higher temperature

environment (cooling mode like an air conditioner) or it functions as a heater

transferring heat into a warm place from a cooler one (heating mode like a window unit

that heats a house by extracting thermal energy from the cold outdoors) The coefficient

of performance of a refrigerator or of a heat pump in cooling mode is defined to be

COP cooling modecQ

W

10

For a heat pump in heating mode the coefficient of performance is

COP heating modehQ

W

Note that we donrsquot have to use the engineering work here because in these systems

work is done on the system and W is naturally positive

A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without

doing any work giving an infinite coefficient of performance Energy conservation alone

allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump

in heating mode would transfer Qh into the house without doing any work and so would

also have an infinite coefficient of performance The second law forbids this too A good

coefficient of performance for a real device would be around 5 or 6

Carnot Cycle

The most efficient of all possible engines is one that uses the Carnot cycle This cycle

employs an ideal gas has no friction and operates very slowly so that the gas can be in

thermal equilibrium at all parts of the cycle This means of course that it canrsquot

possibly be built and even if it could be built it would not run fast enough to be useful

Nevertheless this cycle is very important because it gives an upper bound on the

efficiency of real engines There cannot possibly be an engine that is more efficient

than one based on the Carnot cycle This cycle consists of the following four steps

1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its

volume The reason that heat is absorbed is that expansion tends to cool the gas but

thermal contact with the environment at Th keeps the temperature high by heat

conduction into the ideal gas

2 The ideal gas further increases its volume by an adiabatic expansion This expansion

causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc

3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in

volume The reason that heat is exhausted is that compression tends to heat the gas

but thermal contact with the environment at Tc keeps the temperature low by heat

conduction out of the ideal gas

4 The gas is adiabatically compressed back to its original volume (the volume it started

with in step 1) This compression heats the gas from Tc up to Th

11

The efficiency of a Carnot engine is given by the very simple formula

1 cC

h

Te

T

where the temperatures must in be Kelvin No real engine can be more efficient than

this

The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode

is given by

COP cooling mode cC

h c

T

T T

and no real refrigerator can have a coefficient of performance greater than this

The coefficient of performance of a Carnot heat pump in heating mode is

COP heating mode hC

h c

T

T T

So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency

To make the reversible steps in the cycle really reversible they would have to occur

infinitely slowly So the price you pay for making a perfect engine is that it takes

forever to get it to do any work

Entropy

The entropy of a system is defined in terms of its molecular makeup and measures

roughly the disorder of the system If the system is packed into a very small volume

then it is quite ordered and the entropy will be low If it occupies a large volume the

entropy is high (To see what this has to do with disorder note that socks in a drawer

occupy a small volume while socks on the bed in the corner by the door and

hanging from the chandelier occupy a large volume) If the system is very cold then

the molecules hardly move and may even reach out to each other and form a crystal

This is a highly ordered state and therefore has low entropy If the system is very hot

with rapidly speeding molecules crashing into the container walls and bouncing off

each other things are disordered and the entropy is high

It is possible to calculate the entropy of a system in terms of its macroscopic thermody-

namic properties ie pressure volume temperature number of moles etc The key to this

calculation is the concept of a reversible process A reversible process is one that is

12

carried out without friction and so slowly that the process can be reversed at any stage by

making an infinitesimal change in the environment of the system The slow expansion of the

gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment

is reversible If we turned around and began slowly to compress the gas would just slowly

exhaust heat to the environment in the exact reverse way that it absorbed it during expansion

Most processes however are irreversible For example if a gas-filled box were suddenly

increased in size so that the particles were free to wander into the void created by the sudden

expansion then the gas would eventually fill the new volume uniformly at the same

temperature as before the expansion (The temperature is unchanged in this imaginary

process because the kinetic energy of the molecules would be unaffected by such an

instantaneous expansion of the container walls) This imaginary but highly thought-

stimulating process is called a free expansion and it is impossible to reverse it During the

expansion we didnrsquot push on any of the molecules so reversing this process would mean

making them go back into their original volume without pushing on them they simply will

not cooperate to this extent Another way to see that just pushing them back where they came

from does not reverse the free expansion is to think about what would happen if we just

compressed either adiabatically or isothermally An adiabatic compression back to the

original volume would heat the gas above its original temperature and an isothermal

compression would require that heat be exhausted to the environment But the free expansion

involved neither temperature changes nor heat exchanges so neither of these two processes

is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in

a way that takes us back to the initial state

It is possible to calculate the change of entropy for both reversible and irreversible processes

Letrsquos consider a reversible process first In a reversible process the entropy change is given

by the formula

dQS

T

where dQ is the amount of heat added to the system during a small step of the process

The total energy change during the process may then simply be calculated by integration

f f

f i i i

dQS S S dS

T

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11

9

Serway Chapter 19

Temperature

Formally temperature is what is measured by a thermometer Roughly high temperature is what

we call hot and low temperature is what we call cold On the atomic level temperature refers to

the kinetic energy of the molecules A collection of molecules is called ldquohotrdquo if the molecules

have rapid random motion while a collection of molecules is called ldquocoldrdquo if the random motion

is slow When two bodies are placed in close contact with each other they exchange molecular

kinetic energy until they come to the same temperature This is the microscopic picture of the

Zeroth Law of Thermodynamics

Absolute Zero

Absolute zero is the lowest possible temperature that any object can have This is the temperature

at which all of the energy than can be removed an object has been removed (This removable

energy we call thermal energy) There is still motion at absolute zero Electrons continue to orbit

around atomic nuclei and even atoms continue to move about with a small amount of kinetic

energy but this small energy cannot be removed from the object For example at absolute zero

helium is a liquid whose atoms still move and slide past each other

Temperature Scales

Kelvin Scale Absolute zero is at T = 0 K water freezes at T = 27315 K room temperature is

around T = 295 K and water boils at T = 373 K Note we donrsquot use a deg symbol Kelvin is

prefered SI Unit

Celsius Scale Absolute zero is at T = -273degC water freezes at T = 0degC room temperature is

around T = 22degC and water boils at T = 100degC

Fahrenheit Scale Absolute zero is at T = -459degF water freezes at TF = 32degF room temperature

is around T = 72degF and water boils at TF = 212degF TF =18TC + 32 Notice that temperature

differences are the same for the Kelvin and Celsius scales

Thermal Expansion

When materials are heated they usually expand and when they are cooled they usually contract

(Water near freezing is a spectacular counterexample it works the other way around) The

coefficient of linear expansion is defined by the relation

1

i

L

L T

10

where Li is the initial length of a rod of the material and ΔL is the change in its length due to a

small temperature change ΔT The coefficient of volume expansion is defined similarly

1

i

V

V T

where Vi is the initial volume of a piece of material and where ΔV is the change in its volume due

to a small temperature change ΔT

Avogadrorsquos Number (N or NA)

One mole of any substance corresponds to 6022 times 1023 molecules The atomic mass given on the

periodic table (p A32-33 in the text) is the mass in grams of one mole of atoms For example

the atomic mass of oxygen is 15999 gmol The mass of a single atom is given by the same

number in units of atomic mass units (u) For example the mass of an oxygen atom is 15999 u

(on the average)

Ideal Gas Law (an example of an equation of state)

When the molecules of a gas are sufficiently inert and widely separated that interactions between

them are negligible we say that it is an ideal gas The pressure P volume V and temperature T

(in kelvins) of such a gas are State Variables and are related by the ideal gas law

Bor PV=NkPV nRT T

where n is the number of moles of the gas where R is the gas constant

8314 Jmol KR

where N is the number of molecules and where kB is Boltzmannrsquos constant

231380 10 J KBk

It works well for air at atmosphere pressure and even better for partial vaccuums The relative

ease of measuring pressure and the linear relationship between pressure and temperature (if V

and n are held fixed) makes an ideal gas an ideal thermometer Thermometers can be based on

properties of solids or liquids but the behavior of these materials with temperature is more

complicated

11

Serway Chapter 20

Heat

Heat is energy that flows between a system and its environment because of a tempera-

ture difference between them The units of heat are Joules as expected for an energy

Unfortunately there are several competing units of energy They are related by

1 cal 4186 J 1 Cal 4186 J 1 Btu 1054 J

Heat Capacity

There is often a simple linear relation between the heat that flows in or out of part of a

system and the temperature change that results from this energy transfer When this

linear relation holds it is convenient to define the heat capacity C and the specific

heat c as follows

For the entire object the heat Q it absorbs is given by Q = C(Tf - Ti)

For a particular material the specific heat is defined by c = Cm which is the heat

capacity per unit mass so that

Q = mc(Tf ndash Ti)

Note C has units of energy (J or Cal)(Kelvin kg)

Heats of Transformation or Latent Heat Q = plusmn mL

When a substance changes phase from solid to liquid or from liquid to gas it absorbs

heat without a change in temperature The latent heat or heat of transformation is

usually given per unit mass of the substance For example for water the heat of fusion

(melting) is L = 333 kJkg while the heat of vaporization (boiling) is L = 2260 kJkg

Note that heat for boiling is considerably bigger than melting for water You have to

be careful with signs heat is given off (negative) if you go down in temperature and

condense steam

Work

In general the small amount of work done on a system as a force Fon is exerted on it

through a vector displacement dx is given by

on xdW d F

12

But if the displacement is done very slowly (as we always assume in thermodynamics)

then the force exerted on the system and the force exerted by the system are in

balance so the force exerted by the system is ndash Fon In thermodynamics it is more

convenient to talk about the force exerted by the system so we change the above

formula for the work done on the system to

xdW d F

where F is the force exerted by the system This has confused students for more than a

century now but this is the way your book and many other books do it so you are

stuck You will need to memorize the minus sign in this definition of the work to be

able to use your textbook

There are many chances to get signs wrong in this and the next two chapters (Mosiah

2 )

When an external agent changes the volume of a gas at pressure P by a small amount

dV the (small amount) of work done on the system is given by

dW PdV

Notice that this minus sign is just what we need to make dW be positive if the external

agent compresses the gas for then dV is negative If on the other hand the external

agent gives way allowing the gas to expand against it then dV is positive and we say

that the work done on the gas is negative

The work done on the system (eg by the gas in a cylinder) in a thermodynamic

process is the area under the curve in a PV diagram It is positive for compressions

and negative for expansions If the volume of gas remains constant in a process then

no work is done by the gas

Cyclic processes are important For cyclic processes represented by PV diagrams the

magnitude of the net work during one cycle is simply the area enclosed by the cycle

on the diagram Be careful to keep track of signs when you are calculating that

enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes

1

Path A-B B-C C-D D-A A to A net

Q

W

ΔU

ΔS

Internal Energy

The energy stored in a substance is called its internal energy Eint This energy may be

stored as random kinetic energy or as potential energy in each molecule (stretched

chemical bonds electrons in excited states etc) For ideal gases all states with the

same temperature will have the same Eint

First Law of Thermodynamics

The change ΔEint in the internal energy of a system is given by

intE Q W

where Q is the heat absorbed by the system and where W is the work done on the

system Hence if a system absorbs heat (and if Wge0) the internal energy increases

Likewise if the system does work (W on the system is negative) and if Qge0 the

internal energy decreases Potential Pitfall Many times people talk about work done

by the system It is the minus of W on the system Donrsquot get tripped up

Processes

Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated

from the environment A process may be approximately adiabatic if it happens so

rapidly that heat does not have time to enter or leave the system Work + or ndash is done

and ΔEint = W

Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing

on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal

energy and temperature does not change (Note the difference between an adiabatic

process and a free expansion is that NO work is done in the adiabatic free expansion)

Isobaric process The pressure is held fixed ΔP = 0 For example usually the

pressure increases when a gas is heated but if it were allowed to expand during the

2

heating process in just the right way its pressure could remain fixed In isobaric

processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)

Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is

then zero and so we have ΔEint = Q

Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint

so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in

an isothermal process The work done on the gas is then given by

lnf

i

Vi

Vf

VW PdV nRT

V

Heat Conduction

The quantity P is defined to be the rate at which heat flows through an object and is a

power having units of watts It is analogous to electric current which is the rate at

which charge flows through an object If the flow of heat through a slab of length L

and cross-sectional area A is steady in time then P is given by the equation

h cT TdQkA

dt L

P =

where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The

heat flows of course because of this temperature difference The quantity k is called

the thermal conductivity and is a constant that is characteristic of the material It is

analogous to the electrical conductivity h cT TL is sometimes called the temperature

gradient and is written dTd dTdx

R-Values

It is common to have the heat-conducting properties of materials described by their R-

values especially for insulating materials like fiberglass batting The connection

between k and R is R = Lk where L is the material thickness In this country R-values

always have units of 2ft F hourBtu

Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11

(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)

The heat flow rate through a slab of area A is given by

3

h cT TA

R

P

in units of Btuhour Note that A must be in square feet and the temperatures must be

in degrees Fahrenheit

Convection

Convection is the transfer of thermal energy by flow of material For instance a home

furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct

throughout the house instead it quickly pumps warm air to all of the rooms

Generally convection is a much faster way to transfer heat than conduction

Radiation

Electromagnetic radiation can also transfer heat When you warm yourself near a

campfire which has burned itself down into a bed of glowing embers you are

receiving radiant heat from the infrared portion of the electromagnetic spectrum The

rate at which an object emits radiant heat is given by Stefanrsquos law

4AeTP

where P is the radiated power in watts σ is a constant

8 2 45696 10 W m K

A is the surface area of the object in m2 and T is the temperature in kelvins The

constant e is called the emissivity and it varies from substance to substance A perfect

absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =

0 Hence black objects radiate very well while shiny ones do not Also an object that

is hotter than its surroundings radiates more energy than it absorbs whereas an object

that is cooler than its surroundings absorbs more energy than it radiates

Terminology

Transfer variables vs state variables

Energy transfer by heat as well as work done depends on the initial final and

intermediate states of the system They are transfer variables But their sum (Q + W =

4

Eint) is a state variable

Figure 205

5

Serway Chapter 21

Kinetic Theory

The ideal gas law works for all atoms and molecules at low pressure It is rather

amazing that it does Kinetic theory explains why The properties of an ideal gas can

be understood by thinking of it as N rapidly moving particles of mass m As these

particles collide with the container walls momentum is imparted to the walls which

we call the force of gas pressure In this picture the pressure is related to the average

of the square of the particle velocity 2v by

22 1( )

3 2

NP mv

V

Using the ideal gas law we obtain the average translational kinetic energy per

molecule

21 3

2 2 Bmv k T

The rms speed is then given by

2rms

3 3Bk T RTv v

m M

where M is the molecular mass in kgmol

Degrees of Freedom

Roughly speaking a degree of freedom is a way in which a molecule can store energy

For instance since there are three different directions in space along which a molecule

can move there are three degrees of freedom for the translational kinetic energy

There are also three different axes of rotation about which a polyatomic molecule can

spin so we say there are three degrees of freedom for the rotational kinetic energy

There are even degrees of freedom associated with the various ways in which a

molecule can vibrate and with the different energy levels in which the electrons of

the molecule can exist

Internal Energy and Degrees of Freedom The internal energy of an ideal gas made

up of molecules with J degrees of freedom is given by

int 2 2 B

J JE nRT Nk T

6

Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of

molar heat capacities CV and CP These are the heat capacities per mole and the

subscript V on CV means that the volume is being held constant while for CP the

pressure is held constant For example to raise the temperature of n moles of a gas

whose pressure is held constant by 10 K we would have to supply an amount of heat Q

= nCP (10) K

Molar Specific Heat of an Ideal Gas at Constant Volume

VQ nC T

3monatomic

2VC R

5diatomic

2VC R

5polyatomic

2VC

Real gases deviate from these formulas because in addition to the translational and ro-

tational degrees of freedom they also have vibrational and electronic degrees of

freedom These are unimportant at low temperatures due to quantum mechanical

effects but become increasingly important at higher temperatures The rough rule is

No of degrees of freedom

2VC R

Molar Specific Heat of an Ideal Gas at Constant Pressure

PQ nC T

P VC C R

The internal energy of an ideal gas depends only on the temperature

int VE nC T

Adiabatic Processes in an Ideal Gas

7

An adiabatic process is one in which no heat is exchanged between the system and the

environment When an ideal gas expands or contracts adiabatically not only does its

pressure change as expected from the ideal gas law but its temperature changes as

well Under these conditions the final pressure Pf can be computed from the initial

pressure Pi and from the final and initial volumes Vf and Vi by

or constantf f i iP V PV PV

where γ = CPCV The quantity γ is called the adiabatic exponent Note that this

doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be

combined with the adiabatic law for pressure given above to obtain the adiabatic law

for temperatures

1 constantTV

Compressions in sound waves are adiabatic because they happen too rapidly for any

appreciable amount of heat to flow This is why the adiabatic exponent appears in the

formula for the speed of sound in an ideal gas

RTv

M

Note that v depends only on T and not on P Because it depends only on the

temperature the speed of sound is the same in Provo as at sea level in spite of the

lower pressure here due to the difference in elevation

Equipartition of Energy

Every kind of molecule has a certain number of degrees of freedom which are

independent ways in which it can store energy Each such degree of freedom has

associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1

2 RT per mole)

(Note since a molecule has so many possible degrees of freedom it would seem that

there should be a lot of 12 sBk T to spread around But because energy is quantized

some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high

enough that 12 Bk T is as big as the lowest quantum of energy

8

Serway Chapter 22

Second Law of Thermodynamics

There are several equivalent forms of this important law

Kelvin It is not possible to change heat completely into work with no other change

taking place Or in other words there are no perfect heat engines

Clausius It is not possible for heat to flow from one body to another body at a higher

temperature with no other change taking place Or in other words there are no

perfect refrigerators

Entropy In any thermodynamic process that proceeds from one equilibrium state to

another the entropy of the system + environment either remains unchanged or

increases The total entropy never decreases This law is a bit of an oddity among the

laws of physics because it is not absolute Things are forbidden by the second law not

because it is impossible for them to happen but because it is extremely unlikely for

them to happen (See below for more information about entropy)

Reversible and Irreversible Processes

A reversible process is one which occurs so slowly that it is in thermal equilibrium (or

very nearly so) at all times A hallmark of such processes is that a motion picture of

them looks perfectly normal whether run forward or backward Imagine for instance

the slow expansion of a gas at constant temperature in a cylinder whose volume is being

increased by a slowly moving piston Run the movie backwards and what do you see

You see the slow compression of a gas at constant temperature which looks perfectly

normal

An irreversible process is one which occurs in such a way that thermal equilibrium is

not maintained throughout the process The mark of this kind of process is that a motion

picture of it looks very odd when run backward Imagine the sudden expansion of a gas

into a previously evacuated chamber because a hole was punched in the wall between a

pressurized chamber and the evacuated one Run the movie backward and what do you

see You see the gas in the soon-to-be-evacuated chamber gather itself together and

stream through a tiny hole into a chamber in which there is already plenty of gas If you

have ever seen this happen get in touch with the support group for those who have

witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW

9

Heat Engines

Because of the vexing difference in sign between work done on and system and work done

by a system we will invent a new work variable Weng Heat engines do work and so the net

W for these engines is negative But in engineering applications hidden minus signs are

regarded as evil so for heat engines we donrsquot talk about W instead we talk about its

magnitude engW W So for heat engines the first law is

int engE Q W

But for heat pumps and refrigerators work is done on the system so we use the usual work

W when we talk about these systems

A heat engine is a machine that absorbs heat converts part of it to work and exhausts the

rest The heat must be absorbed at high temperature and exhausted at low temperature If the

absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then

eng h cW Q Q

and the efficiency of the engine is defined to be

eng

h

We

Q

A perfect engine would convert the heat hQ completely into work Weng giving an effi-

ciency of e = 1 Energy conservation alone allows a perfect engine but the second law

requires e lt 1

Refrigerators and Heat Pumps

A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high

temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine

by some source of power A heat pump is a machine that either works like a refrigerator

keeping a place cold by transferring heat from this cold place to a higher temperature

environment (cooling mode like an air conditioner) or it functions as a heater

transferring heat into a warm place from a cooler one (heating mode like a window unit

that heats a house by extracting thermal energy from the cold outdoors) The coefficient

of performance of a refrigerator or of a heat pump in cooling mode is defined to be

COP cooling modecQ

W

10

For a heat pump in heating mode the coefficient of performance is

COP heating modehQ

W

Note that we donrsquot have to use the engineering work here because in these systems

work is done on the system and W is naturally positive

A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without

doing any work giving an infinite coefficient of performance Energy conservation alone

allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump

in heating mode would transfer Qh into the house without doing any work and so would

also have an infinite coefficient of performance The second law forbids this too A good

coefficient of performance for a real device would be around 5 or 6

Carnot Cycle

The most efficient of all possible engines is one that uses the Carnot cycle This cycle

employs an ideal gas has no friction and operates very slowly so that the gas can be in

thermal equilibrium at all parts of the cycle This means of course that it canrsquot

possibly be built and even if it could be built it would not run fast enough to be useful

Nevertheless this cycle is very important because it gives an upper bound on the

efficiency of real engines There cannot possibly be an engine that is more efficient

than one based on the Carnot cycle This cycle consists of the following four steps

1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its

volume The reason that heat is absorbed is that expansion tends to cool the gas but

thermal contact with the environment at Th keeps the temperature high by heat

conduction into the ideal gas

2 The ideal gas further increases its volume by an adiabatic expansion This expansion

causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc

3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in

volume The reason that heat is exhausted is that compression tends to heat the gas

but thermal contact with the environment at Tc keeps the temperature low by heat

conduction out of the ideal gas

4 The gas is adiabatically compressed back to its original volume (the volume it started

with in step 1) This compression heats the gas from Tc up to Th

11

The efficiency of a Carnot engine is given by the very simple formula

1 cC

h

Te

T

where the temperatures must in be Kelvin No real engine can be more efficient than

this

The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode

is given by

COP cooling mode cC

h c

T

T T

and no real refrigerator can have a coefficient of performance greater than this

The coefficient of performance of a Carnot heat pump in heating mode is

COP heating mode hC

h c

T

T T

So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency

To make the reversible steps in the cycle really reversible they would have to occur

infinitely slowly So the price you pay for making a perfect engine is that it takes

forever to get it to do any work

Entropy

The entropy of a system is defined in terms of its molecular makeup and measures

roughly the disorder of the system If the system is packed into a very small volume

then it is quite ordered and the entropy will be low If it occupies a large volume the

entropy is high (To see what this has to do with disorder note that socks in a drawer

occupy a small volume while socks on the bed in the corner by the door and

hanging from the chandelier occupy a large volume) If the system is very cold then

the molecules hardly move and may even reach out to each other and form a crystal

This is a highly ordered state and therefore has low entropy If the system is very hot

with rapidly speeding molecules crashing into the container walls and bouncing off

each other things are disordered and the entropy is high

It is possible to calculate the entropy of a system in terms of its macroscopic thermody-

namic properties ie pressure volume temperature number of moles etc The key to this

calculation is the concept of a reversible process A reversible process is one that is

12

carried out without friction and so slowly that the process can be reversed at any stage by

making an infinitesimal change in the environment of the system The slow expansion of the

gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment

is reversible If we turned around and began slowly to compress the gas would just slowly

exhaust heat to the environment in the exact reverse way that it absorbed it during expansion

Most processes however are irreversible For example if a gas-filled box were suddenly

increased in size so that the particles were free to wander into the void created by the sudden

expansion then the gas would eventually fill the new volume uniformly at the same

temperature as before the expansion (The temperature is unchanged in this imaginary

process because the kinetic energy of the molecules would be unaffected by such an

instantaneous expansion of the container walls) This imaginary but highly thought-

stimulating process is called a free expansion and it is impossible to reverse it During the

expansion we didnrsquot push on any of the molecules so reversing this process would mean

making them go back into their original volume without pushing on them they simply will

not cooperate to this extent Another way to see that just pushing them back where they came

from does not reverse the free expansion is to think about what would happen if we just

compressed either adiabatically or isothermally An adiabatic compression back to the

original volume would heat the gas above its original temperature and an isothermal

compression would require that heat be exhausted to the environment But the free expansion

involved neither temperature changes nor heat exchanges so neither of these two processes

is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in

a way that takes us back to the initial state

It is possible to calculate the change of entropy for both reversible and irreversible processes

Letrsquos consider a reversible process first In a reversible process the entropy change is given

by the formula

dQS

T

where dQ is the amount of heat added to the system during a small step of the process

The total energy change during the process may then simply be calculated by integration

f f

f i i i

dQS S S dS

T

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11

10

where Li is the initial length of a rod of the material and ΔL is the change in its length due to a

small temperature change ΔT The coefficient of volume expansion is defined similarly

1

i

V

V T

where Vi is the initial volume of a piece of material and where ΔV is the change in its volume due

to a small temperature change ΔT

Avogadrorsquos Number (N or NA)

One mole of any substance corresponds to 6022 times 1023 molecules The atomic mass given on the

periodic table (p A32-33 in the text) is the mass in grams of one mole of atoms For example

the atomic mass of oxygen is 15999 gmol The mass of a single atom is given by the same

number in units of atomic mass units (u) For example the mass of an oxygen atom is 15999 u

(on the average)

Ideal Gas Law (an example of an equation of state)

When the molecules of a gas are sufficiently inert and widely separated that interactions between

them are negligible we say that it is an ideal gas The pressure P volume V and temperature T

(in kelvins) of such a gas are State Variables and are related by the ideal gas law

Bor PV=NkPV nRT T

where n is the number of moles of the gas where R is the gas constant

8314 Jmol KR

where N is the number of molecules and where kB is Boltzmannrsquos constant

231380 10 J KBk

It works well for air at atmosphere pressure and even better for partial vaccuums The relative

ease of measuring pressure and the linear relationship between pressure and temperature (if V

and n are held fixed) makes an ideal gas an ideal thermometer Thermometers can be based on

properties of solids or liquids but the behavior of these materials with temperature is more

complicated

11

Serway Chapter 20

Heat

Heat is energy that flows between a system and its environment because of a tempera-

ture difference between them The units of heat are Joules as expected for an energy

Unfortunately there are several competing units of energy They are related by

1 cal 4186 J 1 Cal 4186 J 1 Btu 1054 J

Heat Capacity

There is often a simple linear relation between the heat that flows in or out of part of a

system and the temperature change that results from this energy transfer When this

linear relation holds it is convenient to define the heat capacity C and the specific

heat c as follows

For the entire object the heat Q it absorbs is given by Q = C(Tf - Ti)

For a particular material the specific heat is defined by c = Cm which is the heat

capacity per unit mass so that

Q = mc(Tf ndash Ti)

Note C has units of energy (J or Cal)(Kelvin kg)

Heats of Transformation or Latent Heat Q = plusmn mL

When a substance changes phase from solid to liquid or from liquid to gas it absorbs

heat without a change in temperature The latent heat or heat of transformation is

usually given per unit mass of the substance For example for water the heat of fusion

(melting) is L = 333 kJkg while the heat of vaporization (boiling) is L = 2260 kJkg

Note that heat for boiling is considerably bigger than melting for water You have to

be careful with signs heat is given off (negative) if you go down in temperature and

condense steam

Work

In general the small amount of work done on a system as a force Fon is exerted on it

through a vector displacement dx is given by

on xdW d F

12

But if the displacement is done very slowly (as we always assume in thermodynamics)

then the force exerted on the system and the force exerted by the system are in

balance so the force exerted by the system is ndash Fon In thermodynamics it is more

convenient to talk about the force exerted by the system so we change the above

formula for the work done on the system to

xdW d F

where F is the force exerted by the system This has confused students for more than a

century now but this is the way your book and many other books do it so you are

stuck You will need to memorize the minus sign in this definition of the work to be

able to use your textbook

There are many chances to get signs wrong in this and the next two chapters (Mosiah

2 )

When an external agent changes the volume of a gas at pressure P by a small amount

dV the (small amount) of work done on the system is given by

dW PdV

Notice that this minus sign is just what we need to make dW be positive if the external

agent compresses the gas for then dV is negative If on the other hand the external

agent gives way allowing the gas to expand against it then dV is positive and we say

that the work done on the gas is negative

The work done on the system (eg by the gas in a cylinder) in a thermodynamic

process is the area under the curve in a PV diagram It is positive for compressions

and negative for expansions If the volume of gas remains constant in a process then

no work is done by the gas

Cyclic processes are important For cyclic processes represented by PV diagrams the

magnitude of the net work during one cycle is simply the area enclosed by the cycle

on the diagram Be careful to keep track of signs when you are calculating that

enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes

1

Path A-B B-C C-D D-A A to A net

Q

W

ΔU

ΔS

Internal Energy

The energy stored in a substance is called its internal energy Eint This energy may be

stored as random kinetic energy or as potential energy in each molecule (stretched

chemical bonds electrons in excited states etc) For ideal gases all states with the

same temperature will have the same Eint

First Law of Thermodynamics

The change ΔEint in the internal energy of a system is given by

intE Q W

where Q is the heat absorbed by the system and where W is the work done on the

system Hence if a system absorbs heat (and if Wge0) the internal energy increases

Likewise if the system does work (W on the system is negative) and if Qge0 the

internal energy decreases Potential Pitfall Many times people talk about work done

by the system It is the minus of W on the system Donrsquot get tripped up

Processes

Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated

from the environment A process may be approximately adiabatic if it happens so

rapidly that heat does not have time to enter or leave the system Work + or ndash is done

and ΔEint = W

Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing

on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal

energy and temperature does not change (Note the difference between an adiabatic

process and a free expansion is that NO work is done in the adiabatic free expansion)

Isobaric process The pressure is held fixed ΔP = 0 For example usually the

pressure increases when a gas is heated but if it were allowed to expand during the

2

heating process in just the right way its pressure could remain fixed In isobaric

processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)

Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is

then zero and so we have ΔEint = Q

Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint

so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in

an isothermal process The work done on the gas is then given by

lnf

i

Vi

Vf

VW PdV nRT

V

Heat Conduction

The quantity P is defined to be the rate at which heat flows through an object and is a

power having units of watts It is analogous to electric current which is the rate at

which charge flows through an object If the flow of heat through a slab of length L

and cross-sectional area A is steady in time then P is given by the equation

h cT TdQkA

dt L

P =

where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The

heat flows of course because of this temperature difference The quantity k is called

the thermal conductivity and is a constant that is characteristic of the material It is

analogous to the electrical conductivity h cT TL is sometimes called the temperature

gradient and is written dTd dTdx

R-Values

It is common to have the heat-conducting properties of materials described by their R-

values especially for insulating materials like fiberglass batting The connection

between k and R is R = Lk where L is the material thickness In this country R-values

always have units of 2ft F hourBtu

Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11

(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)

The heat flow rate through a slab of area A is given by

3

h cT TA

R

P

in units of Btuhour Note that A must be in square feet and the temperatures must be

in degrees Fahrenheit

Convection

Convection is the transfer of thermal energy by flow of material For instance a home

furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct

throughout the house instead it quickly pumps warm air to all of the rooms

Generally convection is a much faster way to transfer heat than conduction

Radiation

Electromagnetic radiation can also transfer heat When you warm yourself near a

campfire which has burned itself down into a bed of glowing embers you are

receiving radiant heat from the infrared portion of the electromagnetic spectrum The

rate at which an object emits radiant heat is given by Stefanrsquos law

4AeTP

where P is the radiated power in watts σ is a constant

8 2 45696 10 W m K

A is the surface area of the object in m2 and T is the temperature in kelvins The

constant e is called the emissivity and it varies from substance to substance A perfect

absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =

0 Hence black objects radiate very well while shiny ones do not Also an object that

is hotter than its surroundings radiates more energy than it absorbs whereas an object

that is cooler than its surroundings absorbs more energy than it radiates

Terminology

Transfer variables vs state variables

Energy transfer by heat as well as work done depends on the initial final and

intermediate states of the system They are transfer variables But their sum (Q + W =

4

Eint) is a state variable

Figure 205

5

Serway Chapter 21

Kinetic Theory

The ideal gas law works for all atoms and molecules at low pressure It is rather

amazing that it does Kinetic theory explains why The properties of an ideal gas can

be understood by thinking of it as N rapidly moving particles of mass m As these

particles collide with the container walls momentum is imparted to the walls which

we call the force of gas pressure In this picture the pressure is related to the average

of the square of the particle velocity 2v by

22 1( )

3 2

NP mv

V

Using the ideal gas law we obtain the average translational kinetic energy per

molecule

21 3

2 2 Bmv k T

The rms speed is then given by

2rms

3 3Bk T RTv v

m M

where M is the molecular mass in kgmol

Degrees of Freedom

Roughly speaking a degree of freedom is a way in which a molecule can store energy

For instance since there are three different directions in space along which a molecule

can move there are three degrees of freedom for the translational kinetic energy

There are also three different axes of rotation about which a polyatomic molecule can

spin so we say there are three degrees of freedom for the rotational kinetic energy

There are even degrees of freedom associated with the various ways in which a

molecule can vibrate and with the different energy levels in which the electrons of

the molecule can exist

Internal Energy and Degrees of Freedom The internal energy of an ideal gas made

up of molecules with J degrees of freedom is given by

int 2 2 B

J JE nRT Nk T

6

Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of

molar heat capacities CV and CP These are the heat capacities per mole and the

subscript V on CV means that the volume is being held constant while for CP the

pressure is held constant For example to raise the temperature of n moles of a gas

whose pressure is held constant by 10 K we would have to supply an amount of heat Q

= nCP (10) K

Molar Specific Heat of an Ideal Gas at Constant Volume

VQ nC T

3monatomic

2VC R

5diatomic

2VC R

5polyatomic

2VC

Real gases deviate from these formulas because in addition to the translational and ro-

tational degrees of freedom they also have vibrational and electronic degrees of

freedom These are unimportant at low temperatures due to quantum mechanical

effects but become increasingly important at higher temperatures The rough rule is

No of degrees of freedom

2VC R

Molar Specific Heat of an Ideal Gas at Constant Pressure

PQ nC T

P VC C R

The internal energy of an ideal gas depends only on the temperature

int VE nC T

Adiabatic Processes in an Ideal Gas

7

An adiabatic process is one in which no heat is exchanged between the system and the

environment When an ideal gas expands or contracts adiabatically not only does its

pressure change as expected from the ideal gas law but its temperature changes as

well Under these conditions the final pressure Pf can be computed from the initial

pressure Pi and from the final and initial volumes Vf and Vi by

or constantf f i iP V PV PV

where γ = CPCV The quantity γ is called the adiabatic exponent Note that this

doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be

combined with the adiabatic law for pressure given above to obtain the adiabatic law

for temperatures

1 constantTV

Compressions in sound waves are adiabatic because they happen too rapidly for any

appreciable amount of heat to flow This is why the adiabatic exponent appears in the

formula for the speed of sound in an ideal gas

RTv

M

Note that v depends only on T and not on P Because it depends only on the

temperature the speed of sound is the same in Provo as at sea level in spite of the

lower pressure here due to the difference in elevation

Equipartition of Energy

Every kind of molecule has a certain number of degrees of freedom which are

independent ways in which it can store energy Each such degree of freedom has

associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1

2 RT per mole)

(Note since a molecule has so many possible degrees of freedom it would seem that

there should be a lot of 12 sBk T to spread around But because energy is quantized

some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high

enough that 12 Bk T is as big as the lowest quantum of energy

8

Serway Chapter 22

Second Law of Thermodynamics

There are several equivalent forms of this important law

Kelvin It is not possible to change heat completely into work with no other change

taking place Or in other words there are no perfect heat engines

Clausius It is not possible for heat to flow from one body to another body at a higher

temperature with no other change taking place Or in other words there are no

perfect refrigerators

Entropy In any thermodynamic process that proceeds from one equilibrium state to

another the entropy of the system + environment either remains unchanged or

increases The total entropy never decreases This law is a bit of an oddity among the

laws of physics because it is not absolute Things are forbidden by the second law not

because it is impossible for them to happen but because it is extremely unlikely for

them to happen (See below for more information about entropy)

Reversible and Irreversible Processes

A reversible process is one which occurs so slowly that it is in thermal equilibrium (or

very nearly so) at all times A hallmark of such processes is that a motion picture of

them looks perfectly normal whether run forward or backward Imagine for instance

the slow expansion of a gas at constant temperature in a cylinder whose volume is being

increased by a slowly moving piston Run the movie backwards and what do you see

You see the slow compression of a gas at constant temperature which looks perfectly

normal

An irreversible process is one which occurs in such a way that thermal equilibrium is

not maintained throughout the process The mark of this kind of process is that a motion

picture of it looks very odd when run backward Imagine the sudden expansion of a gas

into a previously evacuated chamber because a hole was punched in the wall between a

pressurized chamber and the evacuated one Run the movie backward and what do you

see You see the gas in the soon-to-be-evacuated chamber gather itself together and

stream through a tiny hole into a chamber in which there is already plenty of gas If you

have ever seen this happen get in touch with the support group for those who have

witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW

9

Heat Engines

Because of the vexing difference in sign between work done on and system and work done

by a system we will invent a new work variable Weng Heat engines do work and so the net

W for these engines is negative But in engineering applications hidden minus signs are

regarded as evil so for heat engines we donrsquot talk about W instead we talk about its

magnitude engW W So for heat engines the first law is

int engE Q W

But for heat pumps and refrigerators work is done on the system so we use the usual work

W when we talk about these systems

A heat engine is a machine that absorbs heat converts part of it to work and exhausts the

rest The heat must be absorbed at high temperature and exhausted at low temperature If the

absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then

eng h cW Q Q

and the efficiency of the engine is defined to be

eng

h

We

Q

A perfect engine would convert the heat hQ completely into work Weng giving an effi-

ciency of e = 1 Energy conservation alone allows a perfect engine but the second law

requires e lt 1

Refrigerators and Heat Pumps

A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high

temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine

by some source of power A heat pump is a machine that either works like a refrigerator

keeping a place cold by transferring heat from this cold place to a higher temperature

environment (cooling mode like an air conditioner) or it functions as a heater

transferring heat into a warm place from a cooler one (heating mode like a window unit

that heats a house by extracting thermal energy from the cold outdoors) The coefficient

of performance of a refrigerator or of a heat pump in cooling mode is defined to be

COP cooling modecQ

W

10

For a heat pump in heating mode the coefficient of performance is

COP heating modehQ

W

Note that we donrsquot have to use the engineering work here because in these systems

work is done on the system and W is naturally positive

A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without

doing any work giving an infinite coefficient of performance Energy conservation alone

allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump

in heating mode would transfer Qh into the house without doing any work and so would

also have an infinite coefficient of performance The second law forbids this too A good

coefficient of performance for a real device would be around 5 or 6

Carnot Cycle

The most efficient of all possible engines is one that uses the Carnot cycle This cycle

employs an ideal gas has no friction and operates very slowly so that the gas can be in

thermal equilibrium at all parts of the cycle This means of course that it canrsquot

possibly be built and even if it could be built it would not run fast enough to be useful

Nevertheless this cycle is very important because it gives an upper bound on the

efficiency of real engines There cannot possibly be an engine that is more efficient

than one based on the Carnot cycle This cycle consists of the following four steps

1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its

volume The reason that heat is absorbed is that expansion tends to cool the gas but

thermal contact with the environment at Th keeps the temperature high by heat

conduction into the ideal gas

2 The ideal gas further increases its volume by an adiabatic expansion This expansion

causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc

3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in

volume The reason that heat is exhausted is that compression tends to heat the gas

but thermal contact with the environment at Tc keeps the temperature low by heat

conduction out of the ideal gas

4 The gas is adiabatically compressed back to its original volume (the volume it started

with in step 1) This compression heats the gas from Tc up to Th

11

The efficiency of a Carnot engine is given by the very simple formula

1 cC

h

Te

T

where the temperatures must in be Kelvin No real engine can be more efficient than

this

The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode

is given by

COP cooling mode cC

h c

T

T T

and no real refrigerator can have a coefficient of performance greater than this

The coefficient of performance of a Carnot heat pump in heating mode is

COP heating mode hC

h c

T

T T

So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency

To make the reversible steps in the cycle really reversible they would have to occur

infinitely slowly So the price you pay for making a perfect engine is that it takes

forever to get it to do any work

Entropy

The entropy of a system is defined in terms of its molecular makeup and measures

roughly the disorder of the system If the system is packed into a very small volume

then it is quite ordered and the entropy will be low If it occupies a large volume the

entropy is high (To see what this has to do with disorder note that socks in a drawer

occupy a small volume while socks on the bed in the corner by the door and

hanging from the chandelier occupy a large volume) If the system is very cold then

the molecules hardly move and may even reach out to each other and form a crystal

This is a highly ordered state and therefore has low entropy If the system is very hot

with rapidly speeding molecules crashing into the container walls and bouncing off

each other things are disordered and the entropy is high

It is possible to calculate the entropy of a system in terms of its macroscopic thermody-

namic properties ie pressure volume temperature number of moles etc The key to this

calculation is the concept of a reversible process A reversible process is one that is

12

carried out without friction and so slowly that the process can be reversed at any stage by

making an infinitesimal change in the environment of the system The slow expansion of the

gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment

is reversible If we turned around and began slowly to compress the gas would just slowly

exhaust heat to the environment in the exact reverse way that it absorbed it during expansion

Most processes however are irreversible For example if a gas-filled box were suddenly

increased in size so that the particles were free to wander into the void created by the sudden

expansion then the gas would eventually fill the new volume uniformly at the same

temperature as before the expansion (The temperature is unchanged in this imaginary

process because the kinetic energy of the molecules would be unaffected by such an

instantaneous expansion of the container walls) This imaginary but highly thought-

stimulating process is called a free expansion and it is impossible to reverse it During the

expansion we didnrsquot push on any of the molecules so reversing this process would mean

making them go back into their original volume without pushing on them they simply will

not cooperate to this extent Another way to see that just pushing them back where they came

from does not reverse the free expansion is to think about what would happen if we just

compressed either adiabatically or isothermally An adiabatic compression back to the

original volume would heat the gas above its original temperature and an isothermal

compression would require that heat be exhausted to the environment But the free expansion

involved neither temperature changes nor heat exchanges so neither of these two processes

is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in

a way that takes us back to the initial state

It is possible to calculate the change of entropy for both reversible and irreversible processes

Letrsquos consider a reversible process first In a reversible process the entropy change is given

by the formula

dQS

T

where dQ is the amount of heat added to the system during a small step of the process

The total energy change during the process may then simply be calculated by integration

f f

f i i i

dQS S S dS

T

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11

11

Serway Chapter 20

Heat

Heat is energy that flows between a system and its environment because of a tempera-

ture difference between them The units of heat are Joules as expected for an energy

Unfortunately there are several competing units of energy They are related by

1 cal 4186 J 1 Cal 4186 J 1 Btu 1054 J

Heat Capacity

There is often a simple linear relation between the heat that flows in or out of part of a

system and the temperature change that results from this energy transfer When this

linear relation holds it is convenient to define the heat capacity C and the specific

heat c as follows

For the entire object the heat Q it absorbs is given by Q = C(Tf - Ti)

For a particular material the specific heat is defined by c = Cm which is the heat

capacity per unit mass so that

Q = mc(Tf ndash Ti)

Note C has units of energy (J or Cal)(Kelvin kg)

Heats of Transformation or Latent Heat Q = plusmn mL

When a substance changes phase from solid to liquid or from liquid to gas it absorbs

heat without a change in temperature The latent heat or heat of transformation is

usually given per unit mass of the substance For example for water the heat of fusion

(melting) is L = 333 kJkg while the heat of vaporization (boiling) is L = 2260 kJkg

Note that heat for boiling is considerably bigger than melting for water You have to

be careful with signs heat is given off (negative) if you go down in temperature and

condense steam

Work

In general the small amount of work done on a system as a force Fon is exerted on it

through a vector displacement dx is given by

on xdW d F

12

But if the displacement is done very slowly (as we always assume in thermodynamics)

then the force exerted on the system and the force exerted by the system are in

balance so the force exerted by the system is ndash Fon In thermodynamics it is more

convenient to talk about the force exerted by the system so we change the above

formula for the work done on the system to

xdW d F

where F is the force exerted by the system This has confused students for more than a

century now but this is the way your book and many other books do it so you are

stuck You will need to memorize the minus sign in this definition of the work to be

able to use your textbook

There are many chances to get signs wrong in this and the next two chapters (Mosiah

2 )

When an external agent changes the volume of a gas at pressure P by a small amount

dV the (small amount) of work done on the system is given by

dW PdV

Notice that this minus sign is just what we need to make dW be positive if the external

agent compresses the gas for then dV is negative If on the other hand the external

agent gives way allowing the gas to expand against it then dV is positive and we say

that the work done on the gas is negative

The work done on the system (eg by the gas in a cylinder) in a thermodynamic

process is the area under the curve in a PV diagram It is positive for compressions

and negative for expansions If the volume of gas remains constant in a process then

no work is done by the gas

Cyclic processes are important For cyclic processes represented by PV diagrams the

magnitude of the net work during one cycle is simply the area enclosed by the cycle

on the diagram Be careful to keep track of signs when you are calculating that

enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes

1

Path A-B B-C C-D D-A A to A net

Q

W

ΔU

ΔS

Internal Energy

The energy stored in a substance is called its internal energy Eint This energy may be

stored as random kinetic energy or as potential energy in each molecule (stretched

chemical bonds electrons in excited states etc) For ideal gases all states with the

same temperature will have the same Eint

First Law of Thermodynamics

The change ΔEint in the internal energy of a system is given by

intE Q W

where Q is the heat absorbed by the system and where W is the work done on the

system Hence if a system absorbs heat (and if Wge0) the internal energy increases

Likewise if the system does work (W on the system is negative) and if Qge0 the

internal energy decreases Potential Pitfall Many times people talk about work done

by the system It is the minus of W on the system Donrsquot get tripped up

Processes

Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated

from the environment A process may be approximately adiabatic if it happens so

rapidly that heat does not have time to enter or leave the system Work + or ndash is done

and ΔEint = W

Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing

on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal

energy and temperature does not change (Note the difference between an adiabatic

process and a free expansion is that NO work is done in the adiabatic free expansion)

Isobaric process The pressure is held fixed ΔP = 0 For example usually the

pressure increases when a gas is heated but if it were allowed to expand during the

2

heating process in just the right way its pressure could remain fixed In isobaric

processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)

Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is

then zero and so we have ΔEint = Q

Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint

so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in

an isothermal process The work done on the gas is then given by

lnf

i

Vi

Vf

VW PdV nRT

V

Heat Conduction

The quantity P is defined to be the rate at which heat flows through an object and is a

power having units of watts It is analogous to electric current which is the rate at

which charge flows through an object If the flow of heat through a slab of length L

and cross-sectional area A is steady in time then P is given by the equation

h cT TdQkA

dt L

P =

where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The

heat flows of course because of this temperature difference The quantity k is called

the thermal conductivity and is a constant that is characteristic of the material It is

analogous to the electrical conductivity h cT TL is sometimes called the temperature

gradient and is written dTd dTdx

R-Values

It is common to have the heat-conducting properties of materials described by their R-

values especially for insulating materials like fiberglass batting The connection

between k and R is R = Lk where L is the material thickness In this country R-values

always have units of 2ft F hourBtu

Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11

(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)

The heat flow rate through a slab of area A is given by

3

h cT TA

R

P

in units of Btuhour Note that A must be in square feet and the temperatures must be

in degrees Fahrenheit

Convection

Convection is the transfer of thermal energy by flow of material For instance a home

furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct

throughout the house instead it quickly pumps warm air to all of the rooms

Generally convection is a much faster way to transfer heat than conduction

Radiation

Electromagnetic radiation can also transfer heat When you warm yourself near a

campfire which has burned itself down into a bed of glowing embers you are

receiving radiant heat from the infrared portion of the electromagnetic spectrum The

rate at which an object emits radiant heat is given by Stefanrsquos law

4AeTP

where P is the radiated power in watts σ is a constant

8 2 45696 10 W m K

A is the surface area of the object in m2 and T is the temperature in kelvins The

constant e is called the emissivity and it varies from substance to substance A perfect

absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =

0 Hence black objects radiate very well while shiny ones do not Also an object that

is hotter than its surroundings radiates more energy than it absorbs whereas an object

that is cooler than its surroundings absorbs more energy than it radiates

Terminology

Transfer variables vs state variables

Energy transfer by heat as well as work done depends on the initial final and

intermediate states of the system They are transfer variables But their sum (Q + W =

4

Eint) is a state variable

Figure 205

5

Serway Chapter 21

Kinetic Theory

The ideal gas law works for all atoms and molecules at low pressure It is rather

amazing that it does Kinetic theory explains why The properties of an ideal gas can

be understood by thinking of it as N rapidly moving particles of mass m As these

particles collide with the container walls momentum is imparted to the walls which

we call the force of gas pressure In this picture the pressure is related to the average

of the square of the particle velocity 2v by

22 1( )

3 2

NP mv

V

Using the ideal gas law we obtain the average translational kinetic energy per

molecule

21 3

2 2 Bmv k T

The rms speed is then given by

2rms

3 3Bk T RTv v

m M

where M is the molecular mass in kgmol

Degrees of Freedom

Roughly speaking a degree of freedom is a way in which a molecule can store energy

For instance since there are three different directions in space along which a molecule

can move there are three degrees of freedom for the translational kinetic energy

There are also three different axes of rotation about which a polyatomic molecule can

spin so we say there are three degrees of freedom for the rotational kinetic energy

There are even degrees of freedom associated with the various ways in which a

molecule can vibrate and with the different energy levels in which the electrons of

the molecule can exist

Internal Energy and Degrees of Freedom The internal energy of an ideal gas made

up of molecules with J degrees of freedom is given by

int 2 2 B

J JE nRT Nk T

6

Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of

molar heat capacities CV and CP These are the heat capacities per mole and the

subscript V on CV means that the volume is being held constant while for CP the

pressure is held constant For example to raise the temperature of n moles of a gas

whose pressure is held constant by 10 K we would have to supply an amount of heat Q

= nCP (10) K

Molar Specific Heat of an Ideal Gas at Constant Volume

VQ nC T

3monatomic

2VC R

5diatomic

2VC R

5polyatomic

2VC

Real gases deviate from these formulas because in addition to the translational and ro-

tational degrees of freedom they also have vibrational and electronic degrees of

freedom These are unimportant at low temperatures due to quantum mechanical

effects but become increasingly important at higher temperatures The rough rule is

No of degrees of freedom

2VC R

Molar Specific Heat of an Ideal Gas at Constant Pressure

PQ nC T

P VC C R

The internal energy of an ideal gas depends only on the temperature

int VE nC T

Adiabatic Processes in an Ideal Gas

7

An adiabatic process is one in which no heat is exchanged between the system and the

environment When an ideal gas expands or contracts adiabatically not only does its

pressure change as expected from the ideal gas law but its temperature changes as

well Under these conditions the final pressure Pf can be computed from the initial

pressure Pi and from the final and initial volumes Vf and Vi by

or constantf f i iP V PV PV

where γ = CPCV The quantity γ is called the adiabatic exponent Note that this

doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be

combined with the adiabatic law for pressure given above to obtain the adiabatic law

for temperatures

1 constantTV

Compressions in sound waves are adiabatic because they happen too rapidly for any

appreciable amount of heat to flow This is why the adiabatic exponent appears in the

formula for the speed of sound in an ideal gas

RTv

M

Note that v depends only on T and not on P Because it depends only on the

temperature the speed of sound is the same in Provo as at sea level in spite of the

lower pressure here due to the difference in elevation

Equipartition of Energy

Every kind of molecule has a certain number of degrees of freedom which are

independent ways in which it can store energy Each such degree of freedom has

associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1

2 RT per mole)

(Note since a molecule has so many possible degrees of freedom it would seem that

there should be a lot of 12 sBk T to spread around But because energy is quantized

some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high

enough that 12 Bk T is as big as the lowest quantum of energy

8

Serway Chapter 22

Second Law of Thermodynamics

There are several equivalent forms of this important law

Kelvin It is not possible to change heat completely into work with no other change

taking place Or in other words there are no perfect heat engines

Clausius It is not possible for heat to flow from one body to another body at a higher

temperature with no other change taking place Or in other words there are no

perfect refrigerators

Entropy In any thermodynamic process that proceeds from one equilibrium state to

another the entropy of the system + environment either remains unchanged or

increases The total entropy never decreases This law is a bit of an oddity among the

laws of physics because it is not absolute Things are forbidden by the second law not

because it is impossible for them to happen but because it is extremely unlikely for

them to happen (See below for more information about entropy)

Reversible and Irreversible Processes

A reversible process is one which occurs so slowly that it is in thermal equilibrium (or

very nearly so) at all times A hallmark of such processes is that a motion picture of

them looks perfectly normal whether run forward or backward Imagine for instance

the slow expansion of a gas at constant temperature in a cylinder whose volume is being

increased by a slowly moving piston Run the movie backwards and what do you see

You see the slow compression of a gas at constant temperature which looks perfectly

normal

An irreversible process is one which occurs in such a way that thermal equilibrium is

not maintained throughout the process The mark of this kind of process is that a motion

picture of it looks very odd when run backward Imagine the sudden expansion of a gas

into a previously evacuated chamber because a hole was punched in the wall between a

pressurized chamber and the evacuated one Run the movie backward and what do you

see You see the gas in the soon-to-be-evacuated chamber gather itself together and

stream through a tiny hole into a chamber in which there is already plenty of gas If you

have ever seen this happen get in touch with the support group for those who have

witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW

9

Heat Engines

Because of the vexing difference in sign between work done on and system and work done

by a system we will invent a new work variable Weng Heat engines do work and so the net

W for these engines is negative But in engineering applications hidden minus signs are

regarded as evil so for heat engines we donrsquot talk about W instead we talk about its

magnitude engW W So for heat engines the first law is

int engE Q W

But for heat pumps and refrigerators work is done on the system so we use the usual work

W when we talk about these systems

A heat engine is a machine that absorbs heat converts part of it to work and exhausts the

rest The heat must be absorbed at high temperature and exhausted at low temperature If the

absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then

eng h cW Q Q

and the efficiency of the engine is defined to be

eng

h

We

Q

A perfect engine would convert the heat hQ completely into work Weng giving an effi-

ciency of e = 1 Energy conservation alone allows a perfect engine but the second law

requires e lt 1

Refrigerators and Heat Pumps

A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high

temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine

by some source of power A heat pump is a machine that either works like a refrigerator

keeping a place cold by transferring heat from this cold place to a higher temperature

environment (cooling mode like an air conditioner) or it functions as a heater

transferring heat into a warm place from a cooler one (heating mode like a window unit

that heats a house by extracting thermal energy from the cold outdoors) The coefficient

of performance of a refrigerator or of a heat pump in cooling mode is defined to be

COP cooling modecQ

W

10

For a heat pump in heating mode the coefficient of performance is

COP heating modehQ

W

Note that we donrsquot have to use the engineering work here because in these systems

work is done on the system and W is naturally positive

A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without

doing any work giving an infinite coefficient of performance Energy conservation alone

allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump

in heating mode would transfer Qh into the house without doing any work and so would

also have an infinite coefficient of performance The second law forbids this too A good

coefficient of performance for a real device would be around 5 or 6

Carnot Cycle

The most efficient of all possible engines is one that uses the Carnot cycle This cycle

employs an ideal gas has no friction and operates very slowly so that the gas can be in

thermal equilibrium at all parts of the cycle This means of course that it canrsquot

possibly be built and even if it could be built it would not run fast enough to be useful

Nevertheless this cycle is very important because it gives an upper bound on the

efficiency of real engines There cannot possibly be an engine that is more efficient

than one based on the Carnot cycle This cycle consists of the following four steps

1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its

volume The reason that heat is absorbed is that expansion tends to cool the gas but

thermal contact with the environment at Th keeps the temperature high by heat

conduction into the ideal gas

2 The ideal gas further increases its volume by an adiabatic expansion This expansion

causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc

3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in

volume The reason that heat is exhausted is that compression tends to heat the gas

but thermal contact with the environment at Tc keeps the temperature low by heat

conduction out of the ideal gas

4 The gas is adiabatically compressed back to its original volume (the volume it started

with in step 1) This compression heats the gas from Tc up to Th

11

The efficiency of a Carnot engine is given by the very simple formula

1 cC

h

Te

T

where the temperatures must in be Kelvin No real engine can be more efficient than

this

The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode

is given by

COP cooling mode cC

h c

T

T T

and no real refrigerator can have a coefficient of performance greater than this

The coefficient of performance of a Carnot heat pump in heating mode is

COP heating mode hC

h c

T

T T

So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency

To make the reversible steps in the cycle really reversible they would have to occur

infinitely slowly So the price you pay for making a perfect engine is that it takes

forever to get it to do any work

Entropy

The entropy of a system is defined in terms of its molecular makeup and measures

roughly the disorder of the system If the system is packed into a very small volume

then it is quite ordered and the entropy will be low If it occupies a large volume the

entropy is high (To see what this has to do with disorder note that socks in a drawer

occupy a small volume while socks on the bed in the corner by the door and

hanging from the chandelier occupy a large volume) If the system is very cold then

the molecules hardly move and may even reach out to each other and form a crystal

This is a highly ordered state and therefore has low entropy If the system is very hot

with rapidly speeding molecules crashing into the container walls and bouncing off

each other things are disordered and the entropy is high

It is possible to calculate the entropy of a system in terms of its macroscopic thermody-

namic properties ie pressure volume temperature number of moles etc The key to this

calculation is the concept of a reversible process A reversible process is one that is

12

carried out without friction and so slowly that the process can be reversed at any stage by

making an infinitesimal change in the environment of the system The slow expansion of the

gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment

is reversible If we turned around and began slowly to compress the gas would just slowly

exhaust heat to the environment in the exact reverse way that it absorbed it during expansion

Most processes however are irreversible For example if a gas-filled box were suddenly

increased in size so that the particles were free to wander into the void created by the sudden

expansion then the gas would eventually fill the new volume uniformly at the same

temperature as before the expansion (The temperature is unchanged in this imaginary

process because the kinetic energy of the molecules would be unaffected by such an

instantaneous expansion of the container walls) This imaginary but highly thought-

stimulating process is called a free expansion and it is impossible to reverse it During the

expansion we didnrsquot push on any of the molecules so reversing this process would mean

making them go back into their original volume without pushing on them they simply will

not cooperate to this extent Another way to see that just pushing them back where they came

from does not reverse the free expansion is to think about what would happen if we just

compressed either adiabatically or isothermally An adiabatic compression back to the

original volume would heat the gas above its original temperature and an isothermal

compression would require that heat be exhausted to the environment But the free expansion

involved neither temperature changes nor heat exchanges so neither of these two processes

is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in

a way that takes us back to the initial state

It is possible to calculate the change of entropy for both reversible and irreversible processes

Letrsquos consider a reversible process first In a reversible process the entropy change is given

by the formula

dQS

T

where dQ is the amount of heat added to the system during a small step of the process

The total energy change during the process may then simply be calculated by integration

f f

f i i i

dQS S S dS

T

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11

12

But if the displacement is done very slowly (as we always assume in thermodynamics)

then the force exerted on the system and the force exerted by the system are in

balance so the force exerted by the system is ndash Fon In thermodynamics it is more

convenient to talk about the force exerted by the system so we change the above

formula for the work done on the system to

xdW d F

where F is the force exerted by the system This has confused students for more than a

century now but this is the way your book and many other books do it so you are

stuck You will need to memorize the minus sign in this definition of the work to be

able to use your textbook

There are many chances to get signs wrong in this and the next two chapters (Mosiah

2 )

When an external agent changes the volume of a gas at pressure P by a small amount

dV the (small amount) of work done on the system is given by

dW PdV

Notice that this minus sign is just what we need to make dW be positive if the external

agent compresses the gas for then dV is negative If on the other hand the external

agent gives way allowing the gas to expand against it then dV is positive and we say

that the work done on the gas is negative

The work done on the system (eg by the gas in a cylinder) in a thermodynamic

process is the area under the curve in a PV diagram It is positive for compressions

and negative for expansions If the volume of gas remains constant in a process then

no work is done by the gas

Cyclic processes are important For cyclic processes represented by PV diagrams the

magnitude of the net work during one cycle is simply the area enclosed by the cycle

on the diagram Be careful to keep track of signs when you are calculating that

enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes

1

Path A-B B-C C-D D-A A to A net

Q

W

ΔU

ΔS

Internal Energy

The energy stored in a substance is called its internal energy Eint This energy may be

stored as random kinetic energy or as potential energy in each molecule (stretched

chemical bonds electrons in excited states etc) For ideal gases all states with the

same temperature will have the same Eint

First Law of Thermodynamics

The change ΔEint in the internal energy of a system is given by

intE Q W

where Q is the heat absorbed by the system and where W is the work done on the

system Hence if a system absorbs heat (and if Wge0) the internal energy increases

Likewise if the system does work (W on the system is negative) and if Qge0 the

internal energy decreases Potential Pitfall Many times people talk about work done

by the system It is the minus of W on the system Donrsquot get tripped up

Processes

Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated

from the environment A process may be approximately adiabatic if it happens so

rapidly that heat does not have time to enter or leave the system Work + or ndash is done

and ΔEint = W

Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing

on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal

energy and temperature does not change (Note the difference between an adiabatic

process and a free expansion is that NO work is done in the adiabatic free expansion)

Isobaric process The pressure is held fixed ΔP = 0 For example usually the

pressure increases when a gas is heated but if it were allowed to expand during the

2

heating process in just the right way its pressure could remain fixed In isobaric

processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)

Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is

then zero and so we have ΔEint = Q

Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint

so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in

an isothermal process The work done on the gas is then given by

lnf

i

Vi

Vf

VW PdV nRT

V

Heat Conduction

The quantity P is defined to be the rate at which heat flows through an object and is a

power having units of watts It is analogous to electric current which is the rate at

which charge flows through an object If the flow of heat through a slab of length L

and cross-sectional area A is steady in time then P is given by the equation

h cT TdQkA

dt L

P =

where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The

heat flows of course because of this temperature difference The quantity k is called

the thermal conductivity and is a constant that is characteristic of the material It is

analogous to the electrical conductivity h cT TL is sometimes called the temperature

gradient and is written dTd dTdx

R-Values

It is common to have the heat-conducting properties of materials described by their R-

values especially for insulating materials like fiberglass batting The connection

between k and R is R = Lk where L is the material thickness In this country R-values

always have units of 2ft F hourBtu

Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11

(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)

The heat flow rate through a slab of area A is given by

3

h cT TA

R

P

in units of Btuhour Note that A must be in square feet and the temperatures must be

in degrees Fahrenheit

Convection

Convection is the transfer of thermal energy by flow of material For instance a home

furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct

throughout the house instead it quickly pumps warm air to all of the rooms

Generally convection is a much faster way to transfer heat than conduction

Radiation

Electromagnetic radiation can also transfer heat When you warm yourself near a

campfire which has burned itself down into a bed of glowing embers you are

receiving radiant heat from the infrared portion of the electromagnetic spectrum The

rate at which an object emits radiant heat is given by Stefanrsquos law

4AeTP

where P is the radiated power in watts σ is a constant

8 2 45696 10 W m K

A is the surface area of the object in m2 and T is the temperature in kelvins The

constant e is called the emissivity and it varies from substance to substance A perfect

absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =

0 Hence black objects radiate very well while shiny ones do not Also an object that

is hotter than its surroundings radiates more energy than it absorbs whereas an object

that is cooler than its surroundings absorbs more energy than it radiates

Terminology

Transfer variables vs state variables

Energy transfer by heat as well as work done depends on the initial final and

intermediate states of the system They are transfer variables But their sum (Q + W =

4

Eint) is a state variable

Figure 205

5

Serway Chapter 21

Kinetic Theory

The ideal gas law works for all atoms and molecules at low pressure It is rather

amazing that it does Kinetic theory explains why The properties of an ideal gas can

be understood by thinking of it as N rapidly moving particles of mass m As these

particles collide with the container walls momentum is imparted to the walls which

we call the force of gas pressure In this picture the pressure is related to the average

of the square of the particle velocity 2v by

22 1( )

3 2

NP mv

V

Using the ideal gas law we obtain the average translational kinetic energy per

molecule

21 3

2 2 Bmv k T

The rms speed is then given by

2rms

3 3Bk T RTv v

m M

where M is the molecular mass in kgmol

Degrees of Freedom

Roughly speaking a degree of freedom is a way in which a molecule can store energy

For instance since there are three different directions in space along which a molecule

can move there are three degrees of freedom for the translational kinetic energy

There are also three different axes of rotation about which a polyatomic molecule can

spin so we say there are three degrees of freedom for the rotational kinetic energy

There are even degrees of freedom associated with the various ways in which a

molecule can vibrate and with the different energy levels in which the electrons of

the molecule can exist

Internal Energy and Degrees of Freedom The internal energy of an ideal gas made

up of molecules with J degrees of freedom is given by

int 2 2 B

J JE nRT Nk T

6

Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of

molar heat capacities CV and CP These are the heat capacities per mole and the

subscript V on CV means that the volume is being held constant while for CP the

pressure is held constant For example to raise the temperature of n moles of a gas

whose pressure is held constant by 10 K we would have to supply an amount of heat Q

= nCP (10) K

Molar Specific Heat of an Ideal Gas at Constant Volume

VQ nC T

3monatomic

2VC R

5diatomic

2VC R

5polyatomic

2VC

Real gases deviate from these formulas because in addition to the translational and ro-

tational degrees of freedom they also have vibrational and electronic degrees of

freedom These are unimportant at low temperatures due to quantum mechanical

effects but become increasingly important at higher temperatures The rough rule is

No of degrees of freedom

2VC R

Molar Specific Heat of an Ideal Gas at Constant Pressure

PQ nC T

P VC C R

The internal energy of an ideal gas depends only on the temperature

int VE nC T

Adiabatic Processes in an Ideal Gas

7

An adiabatic process is one in which no heat is exchanged between the system and the

environment When an ideal gas expands or contracts adiabatically not only does its

pressure change as expected from the ideal gas law but its temperature changes as

well Under these conditions the final pressure Pf can be computed from the initial

pressure Pi and from the final and initial volumes Vf and Vi by

or constantf f i iP V PV PV

where γ = CPCV The quantity γ is called the adiabatic exponent Note that this

doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be

combined with the adiabatic law for pressure given above to obtain the adiabatic law

for temperatures

1 constantTV

Compressions in sound waves are adiabatic because they happen too rapidly for any

appreciable amount of heat to flow This is why the adiabatic exponent appears in the

formula for the speed of sound in an ideal gas

RTv

M

Note that v depends only on T and not on P Because it depends only on the

temperature the speed of sound is the same in Provo as at sea level in spite of the

lower pressure here due to the difference in elevation

Equipartition of Energy

Every kind of molecule has a certain number of degrees of freedom which are

independent ways in which it can store energy Each such degree of freedom has

associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1

2 RT per mole)

(Note since a molecule has so many possible degrees of freedom it would seem that

there should be a lot of 12 sBk T to spread around But because energy is quantized

some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high

enough that 12 Bk T is as big as the lowest quantum of energy

8

Serway Chapter 22

Second Law of Thermodynamics

There are several equivalent forms of this important law

Kelvin It is not possible to change heat completely into work with no other change

taking place Or in other words there are no perfect heat engines

Clausius It is not possible for heat to flow from one body to another body at a higher

temperature with no other change taking place Or in other words there are no

perfect refrigerators

Entropy In any thermodynamic process that proceeds from one equilibrium state to

another the entropy of the system + environment either remains unchanged or

increases The total entropy never decreases This law is a bit of an oddity among the

laws of physics because it is not absolute Things are forbidden by the second law not

because it is impossible for them to happen but because it is extremely unlikely for

them to happen (See below for more information about entropy)

Reversible and Irreversible Processes

A reversible process is one which occurs so slowly that it is in thermal equilibrium (or

very nearly so) at all times A hallmark of such processes is that a motion picture of

them looks perfectly normal whether run forward or backward Imagine for instance

the slow expansion of a gas at constant temperature in a cylinder whose volume is being

increased by a slowly moving piston Run the movie backwards and what do you see

You see the slow compression of a gas at constant temperature which looks perfectly

normal

An irreversible process is one which occurs in such a way that thermal equilibrium is

not maintained throughout the process The mark of this kind of process is that a motion

picture of it looks very odd when run backward Imagine the sudden expansion of a gas

into a previously evacuated chamber because a hole was punched in the wall between a

pressurized chamber and the evacuated one Run the movie backward and what do you

see You see the gas in the soon-to-be-evacuated chamber gather itself together and

stream through a tiny hole into a chamber in which there is already plenty of gas If you

have ever seen this happen get in touch with the support group for those who have

witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW

9

Heat Engines

Because of the vexing difference in sign between work done on and system and work done

by a system we will invent a new work variable Weng Heat engines do work and so the net

W for these engines is negative But in engineering applications hidden minus signs are

regarded as evil so for heat engines we donrsquot talk about W instead we talk about its

magnitude engW W So for heat engines the first law is

int engE Q W

But for heat pumps and refrigerators work is done on the system so we use the usual work

W when we talk about these systems

A heat engine is a machine that absorbs heat converts part of it to work and exhausts the

rest The heat must be absorbed at high temperature and exhausted at low temperature If the

absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then

eng h cW Q Q

and the efficiency of the engine is defined to be

eng

h

We

Q

A perfect engine would convert the heat hQ completely into work Weng giving an effi-

ciency of e = 1 Energy conservation alone allows a perfect engine but the second law

requires e lt 1

Refrigerators and Heat Pumps

A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high

temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine

by some source of power A heat pump is a machine that either works like a refrigerator

keeping a place cold by transferring heat from this cold place to a higher temperature

environment (cooling mode like an air conditioner) or it functions as a heater

transferring heat into a warm place from a cooler one (heating mode like a window unit

that heats a house by extracting thermal energy from the cold outdoors) The coefficient

of performance of a refrigerator or of a heat pump in cooling mode is defined to be

COP cooling modecQ

W

10

For a heat pump in heating mode the coefficient of performance is

COP heating modehQ

W

Note that we donrsquot have to use the engineering work here because in these systems

work is done on the system and W is naturally positive

A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without

doing any work giving an infinite coefficient of performance Energy conservation alone

allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump

in heating mode would transfer Qh into the house without doing any work and so would

also have an infinite coefficient of performance The second law forbids this too A good

coefficient of performance for a real device would be around 5 or 6

Carnot Cycle

The most efficient of all possible engines is one that uses the Carnot cycle This cycle

employs an ideal gas has no friction and operates very slowly so that the gas can be in

thermal equilibrium at all parts of the cycle This means of course that it canrsquot

possibly be built and even if it could be built it would not run fast enough to be useful

Nevertheless this cycle is very important because it gives an upper bound on the

efficiency of real engines There cannot possibly be an engine that is more efficient

than one based on the Carnot cycle This cycle consists of the following four steps

1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its

volume The reason that heat is absorbed is that expansion tends to cool the gas but

thermal contact with the environment at Th keeps the temperature high by heat

conduction into the ideal gas

2 The ideal gas further increases its volume by an adiabatic expansion This expansion

causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc

3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in

volume The reason that heat is exhausted is that compression tends to heat the gas

but thermal contact with the environment at Tc keeps the temperature low by heat

conduction out of the ideal gas

4 The gas is adiabatically compressed back to its original volume (the volume it started

with in step 1) This compression heats the gas from Tc up to Th

11

The efficiency of a Carnot engine is given by the very simple formula

1 cC

h

Te

T

where the temperatures must in be Kelvin No real engine can be more efficient than

this

The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode

is given by

COP cooling mode cC

h c

T

T T

and no real refrigerator can have a coefficient of performance greater than this

The coefficient of performance of a Carnot heat pump in heating mode is

COP heating mode hC

h c

T

T T

So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency

To make the reversible steps in the cycle really reversible they would have to occur

infinitely slowly So the price you pay for making a perfect engine is that it takes

forever to get it to do any work

Entropy

The entropy of a system is defined in terms of its molecular makeup and measures

roughly the disorder of the system If the system is packed into a very small volume

then it is quite ordered and the entropy will be low If it occupies a large volume the

entropy is high (To see what this has to do with disorder note that socks in a drawer

occupy a small volume while socks on the bed in the corner by the door and

hanging from the chandelier occupy a large volume) If the system is very cold then

the molecules hardly move and may even reach out to each other and form a crystal

This is a highly ordered state and therefore has low entropy If the system is very hot

with rapidly speeding molecules crashing into the container walls and bouncing off

each other things are disordered and the entropy is high

It is possible to calculate the entropy of a system in terms of its macroscopic thermody-

namic properties ie pressure volume temperature number of moles etc The key to this

calculation is the concept of a reversible process A reversible process is one that is

12

carried out without friction and so slowly that the process can be reversed at any stage by

making an infinitesimal change in the environment of the system The slow expansion of the

gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment

is reversible If we turned around and began slowly to compress the gas would just slowly

exhaust heat to the environment in the exact reverse way that it absorbed it during expansion

Most processes however are irreversible For example if a gas-filled box were suddenly

increased in size so that the particles were free to wander into the void created by the sudden

expansion then the gas would eventually fill the new volume uniformly at the same

temperature as before the expansion (The temperature is unchanged in this imaginary

process because the kinetic energy of the molecules would be unaffected by such an

instantaneous expansion of the container walls) This imaginary but highly thought-

stimulating process is called a free expansion and it is impossible to reverse it During the

expansion we didnrsquot push on any of the molecules so reversing this process would mean

making them go back into their original volume without pushing on them they simply will

not cooperate to this extent Another way to see that just pushing them back where they came

from does not reverse the free expansion is to think about what would happen if we just

compressed either adiabatically or isothermally An adiabatic compression back to the

original volume would heat the gas above its original temperature and an isothermal

compression would require that heat be exhausted to the environment But the free expansion

involved neither temperature changes nor heat exchanges so neither of these two processes

is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in

a way that takes us back to the initial state

It is possible to calculate the change of entropy for both reversible and irreversible processes

Letrsquos consider a reversible process first In a reversible process the entropy change is given

by the formula

dQS

T

where dQ is the amount of heat added to the system during a small step of the process

The total energy change during the process may then simply be calculated by integration

f f

f i i i

dQS S S dS

T

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11

1

Path A-B B-C C-D D-A A to A net

Q

W

ΔU

ΔS

Internal Energy

The energy stored in a substance is called its internal energy Eint This energy may be

stored as random kinetic energy or as potential energy in each molecule (stretched

chemical bonds electrons in excited states etc) For ideal gases all states with the

same temperature will have the same Eint

First Law of Thermodynamics

The change ΔEint in the internal energy of a system is given by

intE Q W

where Q is the heat absorbed by the system and where W is the work done on the

system Hence if a system absorbs heat (and if Wge0) the internal energy increases

Likewise if the system does work (W on the system is negative) and if Qge0 the

internal energy decreases Potential Pitfall Many times people talk about work done

by the system It is the minus of W on the system Donrsquot get tripped up

Processes

Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated

from the environment A process may be approximately adiabatic if it happens so

rapidly that heat does not have time to enter or leave the system Work + or ndash is done

and ΔEint = W

Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing

on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal

energy and temperature does not change (Note the difference between an adiabatic

process and a free expansion is that NO work is done in the adiabatic free expansion)

Isobaric process The pressure is held fixed ΔP = 0 For example usually the

pressure increases when a gas is heated but if it were allowed to expand during the

2

heating process in just the right way its pressure could remain fixed In isobaric

processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)

Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is

then zero and so we have ΔEint = Q

Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint

so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in

an isothermal process The work done on the gas is then given by

lnf

i

Vi

Vf

VW PdV nRT

V

Heat Conduction

The quantity P is defined to be the rate at which heat flows through an object and is a

power having units of watts It is analogous to electric current which is the rate at

which charge flows through an object If the flow of heat through a slab of length L

and cross-sectional area A is steady in time then P is given by the equation

h cT TdQkA

dt L

P =

where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The

heat flows of course because of this temperature difference The quantity k is called

the thermal conductivity and is a constant that is characteristic of the material It is

analogous to the electrical conductivity h cT TL is sometimes called the temperature

gradient and is written dTd dTdx

R-Values

It is common to have the heat-conducting properties of materials described by their R-

values especially for insulating materials like fiberglass batting The connection

between k and R is R = Lk where L is the material thickness In this country R-values

always have units of 2ft F hourBtu

Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11

(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)

The heat flow rate through a slab of area A is given by

3

h cT TA

R

P

in units of Btuhour Note that A must be in square feet and the temperatures must be

in degrees Fahrenheit

Convection

Convection is the transfer of thermal energy by flow of material For instance a home

furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct

throughout the house instead it quickly pumps warm air to all of the rooms

Generally convection is a much faster way to transfer heat than conduction

Radiation

Electromagnetic radiation can also transfer heat When you warm yourself near a

campfire which has burned itself down into a bed of glowing embers you are

receiving radiant heat from the infrared portion of the electromagnetic spectrum The

rate at which an object emits radiant heat is given by Stefanrsquos law

4AeTP

where P is the radiated power in watts σ is a constant

8 2 45696 10 W m K

A is the surface area of the object in m2 and T is the temperature in kelvins The

constant e is called the emissivity and it varies from substance to substance A perfect

absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =

0 Hence black objects radiate very well while shiny ones do not Also an object that

is hotter than its surroundings radiates more energy than it absorbs whereas an object

that is cooler than its surroundings absorbs more energy than it radiates

Terminology

Transfer variables vs state variables

Energy transfer by heat as well as work done depends on the initial final and

intermediate states of the system They are transfer variables But their sum (Q + W =

4

Eint) is a state variable

Figure 205

5

Serway Chapter 21

Kinetic Theory

The ideal gas law works for all atoms and molecules at low pressure It is rather

amazing that it does Kinetic theory explains why The properties of an ideal gas can

be understood by thinking of it as N rapidly moving particles of mass m As these

particles collide with the container walls momentum is imparted to the walls which

we call the force of gas pressure In this picture the pressure is related to the average

of the square of the particle velocity 2v by

22 1( )

3 2

NP mv

V

Using the ideal gas law we obtain the average translational kinetic energy per

molecule

21 3

2 2 Bmv k T

The rms speed is then given by

2rms

3 3Bk T RTv v

m M

where M is the molecular mass in kgmol

Degrees of Freedom

Roughly speaking a degree of freedom is a way in which a molecule can store energy

For instance since there are three different directions in space along which a molecule

can move there are three degrees of freedom for the translational kinetic energy

There are also three different axes of rotation about which a polyatomic molecule can

spin so we say there are three degrees of freedom for the rotational kinetic energy

There are even degrees of freedom associated with the various ways in which a

molecule can vibrate and with the different energy levels in which the electrons of

the molecule can exist

Internal Energy and Degrees of Freedom The internal energy of an ideal gas made

up of molecules with J degrees of freedom is given by

int 2 2 B

J JE nRT Nk T

6

Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of

molar heat capacities CV and CP These are the heat capacities per mole and the

subscript V on CV means that the volume is being held constant while for CP the

pressure is held constant For example to raise the temperature of n moles of a gas

whose pressure is held constant by 10 K we would have to supply an amount of heat Q

= nCP (10) K

Molar Specific Heat of an Ideal Gas at Constant Volume

VQ nC T

3monatomic

2VC R

5diatomic

2VC R

5polyatomic

2VC

Real gases deviate from these formulas because in addition to the translational and ro-

tational degrees of freedom they also have vibrational and electronic degrees of

freedom These are unimportant at low temperatures due to quantum mechanical

effects but become increasingly important at higher temperatures The rough rule is

No of degrees of freedom

2VC R

Molar Specific Heat of an Ideal Gas at Constant Pressure

PQ nC T

P VC C R

The internal energy of an ideal gas depends only on the temperature

int VE nC T

Adiabatic Processes in an Ideal Gas

7

An adiabatic process is one in which no heat is exchanged between the system and the

environment When an ideal gas expands or contracts adiabatically not only does its

pressure change as expected from the ideal gas law but its temperature changes as

well Under these conditions the final pressure Pf can be computed from the initial

pressure Pi and from the final and initial volumes Vf and Vi by

or constantf f i iP V PV PV

where γ = CPCV The quantity γ is called the adiabatic exponent Note that this

doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be

combined with the adiabatic law for pressure given above to obtain the adiabatic law

for temperatures

1 constantTV

Compressions in sound waves are adiabatic because they happen too rapidly for any

appreciable amount of heat to flow This is why the adiabatic exponent appears in the

formula for the speed of sound in an ideal gas

RTv

M

Note that v depends only on T and not on P Because it depends only on the

temperature the speed of sound is the same in Provo as at sea level in spite of the

lower pressure here due to the difference in elevation

Equipartition of Energy

Every kind of molecule has a certain number of degrees of freedom which are

independent ways in which it can store energy Each such degree of freedom has

associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1

2 RT per mole)

(Note since a molecule has so many possible degrees of freedom it would seem that

there should be a lot of 12 sBk T to spread around But because energy is quantized

some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high

enough that 12 Bk T is as big as the lowest quantum of energy

8

Serway Chapter 22

Second Law of Thermodynamics

There are several equivalent forms of this important law

Kelvin It is not possible to change heat completely into work with no other change

taking place Or in other words there are no perfect heat engines

Clausius It is not possible for heat to flow from one body to another body at a higher

temperature with no other change taking place Or in other words there are no

perfect refrigerators

Entropy In any thermodynamic process that proceeds from one equilibrium state to

another the entropy of the system + environment either remains unchanged or

increases The total entropy never decreases This law is a bit of an oddity among the

laws of physics because it is not absolute Things are forbidden by the second law not

because it is impossible for them to happen but because it is extremely unlikely for

them to happen (See below for more information about entropy)

Reversible and Irreversible Processes

A reversible process is one which occurs so slowly that it is in thermal equilibrium (or

very nearly so) at all times A hallmark of such processes is that a motion picture of

them looks perfectly normal whether run forward or backward Imagine for instance

the slow expansion of a gas at constant temperature in a cylinder whose volume is being

increased by a slowly moving piston Run the movie backwards and what do you see

You see the slow compression of a gas at constant temperature which looks perfectly

normal

An irreversible process is one which occurs in such a way that thermal equilibrium is

not maintained throughout the process The mark of this kind of process is that a motion

picture of it looks very odd when run backward Imagine the sudden expansion of a gas

into a previously evacuated chamber because a hole was punched in the wall between a

pressurized chamber and the evacuated one Run the movie backward and what do you

see You see the gas in the soon-to-be-evacuated chamber gather itself together and

stream through a tiny hole into a chamber in which there is already plenty of gas If you

have ever seen this happen get in touch with the support group for those who have

witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW

9

Heat Engines

Because of the vexing difference in sign between work done on and system and work done

by a system we will invent a new work variable Weng Heat engines do work and so the net

W for these engines is negative But in engineering applications hidden minus signs are

regarded as evil so for heat engines we donrsquot talk about W instead we talk about its

magnitude engW W So for heat engines the first law is

int engE Q W

But for heat pumps and refrigerators work is done on the system so we use the usual work

W when we talk about these systems

A heat engine is a machine that absorbs heat converts part of it to work and exhausts the

rest The heat must be absorbed at high temperature and exhausted at low temperature If the

absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then

eng h cW Q Q

and the efficiency of the engine is defined to be

eng

h

We

Q

A perfect engine would convert the heat hQ completely into work Weng giving an effi-

ciency of e = 1 Energy conservation alone allows a perfect engine but the second law

requires e lt 1

Refrigerators and Heat Pumps

A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high

temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine

by some source of power A heat pump is a machine that either works like a refrigerator

keeping a place cold by transferring heat from this cold place to a higher temperature

environment (cooling mode like an air conditioner) or it functions as a heater

transferring heat into a warm place from a cooler one (heating mode like a window unit

that heats a house by extracting thermal energy from the cold outdoors) The coefficient

of performance of a refrigerator or of a heat pump in cooling mode is defined to be

COP cooling modecQ

W

10

For a heat pump in heating mode the coefficient of performance is

COP heating modehQ

W

Note that we donrsquot have to use the engineering work here because in these systems

work is done on the system and W is naturally positive

A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without

doing any work giving an infinite coefficient of performance Energy conservation alone

allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump

in heating mode would transfer Qh into the house without doing any work and so would

also have an infinite coefficient of performance The second law forbids this too A good

coefficient of performance for a real device would be around 5 or 6

Carnot Cycle

The most efficient of all possible engines is one that uses the Carnot cycle This cycle

employs an ideal gas has no friction and operates very slowly so that the gas can be in

thermal equilibrium at all parts of the cycle This means of course that it canrsquot

possibly be built and even if it could be built it would not run fast enough to be useful

Nevertheless this cycle is very important because it gives an upper bound on the

efficiency of real engines There cannot possibly be an engine that is more efficient

than one based on the Carnot cycle This cycle consists of the following four steps

1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its

volume The reason that heat is absorbed is that expansion tends to cool the gas but

thermal contact with the environment at Th keeps the temperature high by heat

conduction into the ideal gas

2 The ideal gas further increases its volume by an adiabatic expansion This expansion

causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc

3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in

volume The reason that heat is exhausted is that compression tends to heat the gas

but thermal contact with the environment at Tc keeps the temperature low by heat

conduction out of the ideal gas

4 The gas is adiabatically compressed back to its original volume (the volume it started

with in step 1) This compression heats the gas from Tc up to Th

11

The efficiency of a Carnot engine is given by the very simple formula

1 cC

h

Te

T

where the temperatures must in be Kelvin No real engine can be more efficient than

this

The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode

is given by

COP cooling mode cC

h c

T

T T

and no real refrigerator can have a coefficient of performance greater than this

The coefficient of performance of a Carnot heat pump in heating mode is

COP heating mode hC

h c

T

T T

So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency

To make the reversible steps in the cycle really reversible they would have to occur

infinitely slowly So the price you pay for making a perfect engine is that it takes

forever to get it to do any work

Entropy

The entropy of a system is defined in terms of its molecular makeup and measures

roughly the disorder of the system If the system is packed into a very small volume

then it is quite ordered and the entropy will be low If it occupies a large volume the

entropy is high (To see what this has to do with disorder note that socks in a drawer

occupy a small volume while socks on the bed in the corner by the door and

hanging from the chandelier occupy a large volume) If the system is very cold then

the molecules hardly move and may even reach out to each other and form a crystal

This is a highly ordered state and therefore has low entropy If the system is very hot

with rapidly speeding molecules crashing into the container walls and bouncing off

each other things are disordered and the entropy is high

It is possible to calculate the entropy of a system in terms of its macroscopic thermody-

namic properties ie pressure volume temperature number of moles etc The key to this

calculation is the concept of a reversible process A reversible process is one that is

12

carried out without friction and so slowly that the process can be reversed at any stage by

making an infinitesimal change in the environment of the system The slow expansion of the

gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment

is reversible If we turned around and began slowly to compress the gas would just slowly

exhaust heat to the environment in the exact reverse way that it absorbed it during expansion

Most processes however are irreversible For example if a gas-filled box were suddenly

increased in size so that the particles were free to wander into the void created by the sudden

expansion then the gas would eventually fill the new volume uniformly at the same

temperature as before the expansion (The temperature is unchanged in this imaginary

process because the kinetic energy of the molecules would be unaffected by such an

instantaneous expansion of the container walls) This imaginary but highly thought-

stimulating process is called a free expansion and it is impossible to reverse it During the

expansion we didnrsquot push on any of the molecules so reversing this process would mean

making them go back into their original volume without pushing on them they simply will

not cooperate to this extent Another way to see that just pushing them back where they came

from does not reverse the free expansion is to think about what would happen if we just

compressed either adiabatically or isothermally An adiabatic compression back to the

original volume would heat the gas above its original temperature and an isothermal

compression would require that heat be exhausted to the environment But the free expansion

involved neither temperature changes nor heat exchanges so neither of these two processes

is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in

a way that takes us back to the initial state

It is possible to calculate the change of entropy for both reversible and irreversible processes

Letrsquos consider a reversible process first In a reversible process the entropy change is given

by the formula

dQS

T

where dQ is the amount of heat added to the system during a small step of the process

The total energy change during the process may then simply be calculated by integration

f f

f i i i

dQS S S dS

T

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11

2

heating process in just the right way its pressure could remain fixed In isobaric

processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)

Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is

then zero and so we have ΔEint = Q

Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint

so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in

an isothermal process The work done on the gas is then given by

lnf

i

Vi

Vf

VW PdV nRT

V

Heat Conduction

The quantity P is defined to be the rate at which heat flows through an object and is a

power having units of watts It is analogous to electric current which is the rate at

which charge flows through an object If the flow of heat through a slab of length L

and cross-sectional area A is steady in time then P is given by the equation

h cT TdQkA

dt L

P =

where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The

heat flows of course because of this temperature difference The quantity k is called

the thermal conductivity and is a constant that is characteristic of the material It is

analogous to the electrical conductivity h cT TL is sometimes called the temperature

gradient and is written dTd dTdx

R-Values

It is common to have the heat-conducting properties of materials described by their R-

values especially for insulating materials like fiberglass batting The connection

between k and R is R = Lk where L is the material thickness In this country R-values

always have units of 2ft F hourBtu

Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11

(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)

The heat flow rate through a slab of area A is given by

3

h cT TA

R

P

in units of Btuhour Note that A must be in square feet and the temperatures must be

in degrees Fahrenheit

Convection

Convection is the transfer of thermal energy by flow of material For instance a home

furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct

throughout the house instead it quickly pumps warm air to all of the rooms

Generally convection is a much faster way to transfer heat than conduction

Radiation

Electromagnetic radiation can also transfer heat When you warm yourself near a

campfire which has burned itself down into a bed of glowing embers you are

receiving radiant heat from the infrared portion of the electromagnetic spectrum The

rate at which an object emits radiant heat is given by Stefanrsquos law

4AeTP

where P is the radiated power in watts σ is a constant

8 2 45696 10 W m K

A is the surface area of the object in m2 and T is the temperature in kelvins The

constant e is called the emissivity and it varies from substance to substance A perfect

absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =

0 Hence black objects radiate very well while shiny ones do not Also an object that

is hotter than its surroundings radiates more energy than it absorbs whereas an object

that is cooler than its surroundings absorbs more energy than it radiates

Terminology

Transfer variables vs state variables

Energy transfer by heat as well as work done depends on the initial final and

intermediate states of the system They are transfer variables But their sum (Q + W =

4

Eint) is a state variable

Figure 205

5

Serway Chapter 21

Kinetic Theory

The ideal gas law works for all atoms and molecules at low pressure It is rather

amazing that it does Kinetic theory explains why The properties of an ideal gas can

be understood by thinking of it as N rapidly moving particles of mass m As these

particles collide with the container walls momentum is imparted to the walls which

we call the force of gas pressure In this picture the pressure is related to the average

of the square of the particle velocity 2v by

22 1( )

3 2

NP mv

V

Using the ideal gas law we obtain the average translational kinetic energy per

molecule

21 3

2 2 Bmv k T

The rms speed is then given by

2rms

3 3Bk T RTv v

m M

where M is the molecular mass in kgmol

Degrees of Freedom

Roughly speaking a degree of freedom is a way in which a molecule can store energy

For instance since there are three different directions in space along which a molecule

can move there are three degrees of freedom for the translational kinetic energy

There are also three different axes of rotation about which a polyatomic molecule can

spin so we say there are three degrees of freedom for the rotational kinetic energy

There are even degrees of freedom associated with the various ways in which a

molecule can vibrate and with the different energy levels in which the electrons of

the molecule can exist

Internal Energy and Degrees of Freedom The internal energy of an ideal gas made

up of molecules with J degrees of freedom is given by

int 2 2 B

J JE nRT Nk T

6

Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of

molar heat capacities CV and CP These are the heat capacities per mole and the

subscript V on CV means that the volume is being held constant while for CP the

pressure is held constant For example to raise the temperature of n moles of a gas

whose pressure is held constant by 10 K we would have to supply an amount of heat Q

= nCP (10) K

Molar Specific Heat of an Ideal Gas at Constant Volume

VQ nC T

3monatomic

2VC R

5diatomic

2VC R

5polyatomic

2VC

Real gases deviate from these formulas because in addition to the translational and ro-

tational degrees of freedom they also have vibrational and electronic degrees of

freedom These are unimportant at low temperatures due to quantum mechanical

effects but become increasingly important at higher temperatures The rough rule is

No of degrees of freedom

2VC R

Molar Specific Heat of an Ideal Gas at Constant Pressure

PQ nC T

P VC C R

The internal energy of an ideal gas depends only on the temperature

int VE nC T

Adiabatic Processes in an Ideal Gas

7

An adiabatic process is one in which no heat is exchanged between the system and the

environment When an ideal gas expands or contracts adiabatically not only does its

pressure change as expected from the ideal gas law but its temperature changes as

well Under these conditions the final pressure Pf can be computed from the initial

pressure Pi and from the final and initial volumes Vf and Vi by

or constantf f i iP V PV PV

where γ = CPCV The quantity γ is called the adiabatic exponent Note that this

doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be

combined with the adiabatic law for pressure given above to obtain the adiabatic law

for temperatures

1 constantTV

Compressions in sound waves are adiabatic because they happen too rapidly for any

appreciable amount of heat to flow This is why the adiabatic exponent appears in the

formula for the speed of sound in an ideal gas

RTv

M

Note that v depends only on T and not on P Because it depends only on the

temperature the speed of sound is the same in Provo as at sea level in spite of the

lower pressure here due to the difference in elevation

Equipartition of Energy

Every kind of molecule has a certain number of degrees of freedom which are

independent ways in which it can store energy Each such degree of freedom has

associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1

2 RT per mole)

(Note since a molecule has so many possible degrees of freedom it would seem that

there should be a lot of 12 sBk T to spread around But because energy is quantized

some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high

enough that 12 Bk T is as big as the lowest quantum of energy

8

Serway Chapter 22

Second Law of Thermodynamics

There are several equivalent forms of this important law

Kelvin It is not possible to change heat completely into work with no other change

taking place Or in other words there are no perfect heat engines

Clausius It is not possible for heat to flow from one body to another body at a higher

temperature with no other change taking place Or in other words there are no

perfect refrigerators

Entropy In any thermodynamic process that proceeds from one equilibrium state to

another the entropy of the system + environment either remains unchanged or

increases The total entropy never decreases This law is a bit of an oddity among the

laws of physics because it is not absolute Things are forbidden by the second law not

because it is impossible for them to happen but because it is extremely unlikely for

them to happen (See below for more information about entropy)

Reversible and Irreversible Processes

A reversible process is one which occurs so slowly that it is in thermal equilibrium (or

very nearly so) at all times A hallmark of such processes is that a motion picture of

them looks perfectly normal whether run forward or backward Imagine for instance

the slow expansion of a gas at constant temperature in a cylinder whose volume is being

increased by a slowly moving piston Run the movie backwards and what do you see

You see the slow compression of a gas at constant temperature which looks perfectly

normal

An irreversible process is one which occurs in such a way that thermal equilibrium is

not maintained throughout the process The mark of this kind of process is that a motion

picture of it looks very odd when run backward Imagine the sudden expansion of a gas

into a previously evacuated chamber because a hole was punched in the wall between a

pressurized chamber and the evacuated one Run the movie backward and what do you

see You see the gas in the soon-to-be-evacuated chamber gather itself together and

stream through a tiny hole into a chamber in which there is already plenty of gas If you

have ever seen this happen get in touch with the support group for those who have

witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW

9

Heat Engines

Because of the vexing difference in sign between work done on and system and work done

by a system we will invent a new work variable Weng Heat engines do work and so the net

W for these engines is negative But in engineering applications hidden minus signs are

regarded as evil so for heat engines we donrsquot talk about W instead we talk about its

magnitude engW W So for heat engines the first law is

int engE Q W

But for heat pumps and refrigerators work is done on the system so we use the usual work

W when we talk about these systems

A heat engine is a machine that absorbs heat converts part of it to work and exhausts the

rest The heat must be absorbed at high temperature and exhausted at low temperature If the

absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then

eng h cW Q Q

and the efficiency of the engine is defined to be

eng

h

We

Q

A perfect engine would convert the heat hQ completely into work Weng giving an effi-

ciency of e = 1 Energy conservation alone allows a perfect engine but the second law

requires e lt 1

Refrigerators and Heat Pumps

A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high

temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine

by some source of power A heat pump is a machine that either works like a refrigerator

keeping a place cold by transferring heat from this cold place to a higher temperature

environment (cooling mode like an air conditioner) or it functions as a heater

transferring heat into a warm place from a cooler one (heating mode like a window unit

that heats a house by extracting thermal energy from the cold outdoors) The coefficient

of performance of a refrigerator or of a heat pump in cooling mode is defined to be

COP cooling modecQ

W

10

For a heat pump in heating mode the coefficient of performance is

COP heating modehQ

W

Note that we donrsquot have to use the engineering work here because in these systems

work is done on the system and W is naturally positive

A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without

doing any work giving an infinite coefficient of performance Energy conservation alone

allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump

in heating mode would transfer Qh into the house without doing any work and so would

also have an infinite coefficient of performance The second law forbids this too A good

coefficient of performance for a real device would be around 5 or 6

Carnot Cycle

The most efficient of all possible engines is one that uses the Carnot cycle This cycle

employs an ideal gas has no friction and operates very slowly so that the gas can be in

thermal equilibrium at all parts of the cycle This means of course that it canrsquot

possibly be built and even if it could be built it would not run fast enough to be useful

Nevertheless this cycle is very important because it gives an upper bound on the

efficiency of real engines There cannot possibly be an engine that is more efficient

than one based on the Carnot cycle This cycle consists of the following four steps

1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its

volume The reason that heat is absorbed is that expansion tends to cool the gas but

thermal contact with the environment at Th keeps the temperature high by heat

conduction into the ideal gas

2 The ideal gas further increases its volume by an adiabatic expansion This expansion

causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc

3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in

volume The reason that heat is exhausted is that compression tends to heat the gas

but thermal contact with the environment at Tc keeps the temperature low by heat

conduction out of the ideal gas

4 The gas is adiabatically compressed back to its original volume (the volume it started

with in step 1) This compression heats the gas from Tc up to Th

11

The efficiency of a Carnot engine is given by the very simple formula

1 cC

h

Te

T

where the temperatures must in be Kelvin No real engine can be more efficient than

this

The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode

is given by

COP cooling mode cC

h c

T

T T

and no real refrigerator can have a coefficient of performance greater than this

The coefficient of performance of a Carnot heat pump in heating mode is

COP heating mode hC

h c

T

T T

So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency

To make the reversible steps in the cycle really reversible they would have to occur

infinitely slowly So the price you pay for making a perfect engine is that it takes

forever to get it to do any work

Entropy

The entropy of a system is defined in terms of its molecular makeup and measures

roughly the disorder of the system If the system is packed into a very small volume

then it is quite ordered and the entropy will be low If it occupies a large volume the

entropy is high (To see what this has to do with disorder note that socks in a drawer

occupy a small volume while socks on the bed in the corner by the door and

hanging from the chandelier occupy a large volume) If the system is very cold then

the molecules hardly move and may even reach out to each other and form a crystal

This is a highly ordered state and therefore has low entropy If the system is very hot

with rapidly speeding molecules crashing into the container walls and bouncing off

each other things are disordered and the entropy is high

It is possible to calculate the entropy of a system in terms of its macroscopic thermody-

namic properties ie pressure volume temperature number of moles etc The key to this

calculation is the concept of a reversible process A reversible process is one that is

12

carried out without friction and so slowly that the process can be reversed at any stage by

making an infinitesimal change in the environment of the system The slow expansion of the

gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment

is reversible If we turned around and began slowly to compress the gas would just slowly

exhaust heat to the environment in the exact reverse way that it absorbed it during expansion

Most processes however are irreversible For example if a gas-filled box were suddenly

increased in size so that the particles were free to wander into the void created by the sudden

expansion then the gas would eventually fill the new volume uniformly at the same

temperature as before the expansion (The temperature is unchanged in this imaginary

process because the kinetic energy of the molecules would be unaffected by such an

instantaneous expansion of the container walls) This imaginary but highly thought-

stimulating process is called a free expansion and it is impossible to reverse it During the

expansion we didnrsquot push on any of the molecules so reversing this process would mean

making them go back into their original volume without pushing on them they simply will

not cooperate to this extent Another way to see that just pushing them back where they came

from does not reverse the free expansion is to think about what would happen if we just

compressed either adiabatically or isothermally An adiabatic compression back to the

original volume would heat the gas above its original temperature and an isothermal

compression would require that heat be exhausted to the environment But the free expansion

involved neither temperature changes nor heat exchanges so neither of these two processes

is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in

a way that takes us back to the initial state

It is possible to calculate the change of entropy for both reversible and irreversible processes

Letrsquos consider a reversible process first In a reversible process the entropy change is given

by the formula

dQS

T

where dQ is the amount of heat added to the system during a small step of the process

The total energy change during the process may then simply be calculated by integration

f f

f i i i

dQS S S dS

T

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11

3

h cT TA

R

P

in units of Btuhour Note that A must be in square feet and the temperatures must be

in degrees Fahrenheit

Convection

Convection is the transfer of thermal energy by flow of material For instance a home

furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct

throughout the house instead it quickly pumps warm air to all of the rooms

Generally convection is a much faster way to transfer heat than conduction

Radiation

Electromagnetic radiation can also transfer heat When you warm yourself near a

campfire which has burned itself down into a bed of glowing embers you are

receiving radiant heat from the infrared portion of the electromagnetic spectrum The

rate at which an object emits radiant heat is given by Stefanrsquos law

4AeTP

where P is the radiated power in watts σ is a constant

8 2 45696 10 W m K

A is the surface area of the object in m2 and T is the temperature in kelvins The

constant e is called the emissivity and it varies from substance to substance A perfect

absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =

0 Hence black objects radiate very well while shiny ones do not Also an object that

is hotter than its surroundings radiates more energy than it absorbs whereas an object

that is cooler than its surroundings absorbs more energy than it radiates

Terminology

Transfer variables vs state variables

Energy transfer by heat as well as work done depends on the initial final and

intermediate states of the system They are transfer variables But their sum (Q + W =

4

Eint) is a state variable

Figure 205

5

Serway Chapter 21

Kinetic Theory

The ideal gas law works for all atoms and molecules at low pressure It is rather

amazing that it does Kinetic theory explains why The properties of an ideal gas can

be understood by thinking of it as N rapidly moving particles of mass m As these

particles collide with the container walls momentum is imparted to the walls which

we call the force of gas pressure In this picture the pressure is related to the average

of the square of the particle velocity 2v by

22 1( )

3 2

NP mv

V

Using the ideal gas law we obtain the average translational kinetic energy per

molecule

21 3

2 2 Bmv k T

The rms speed is then given by

2rms

3 3Bk T RTv v

m M

where M is the molecular mass in kgmol

Degrees of Freedom

Roughly speaking a degree of freedom is a way in which a molecule can store energy

For instance since there are three different directions in space along which a molecule

can move there are three degrees of freedom for the translational kinetic energy

There are also three different axes of rotation about which a polyatomic molecule can

spin so we say there are three degrees of freedom for the rotational kinetic energy

There are even degrees of freedom associated with the various ways in which a

molecule can vibrate and with the different energy levels in which the electrons of

the molecule can exist

Internal Energy and Degrees of Freedom The internal energy of an ideal gas made

up of molecules with J degrees of freedom is given by

int 2 2 B

J JE nRT Nk T

6

Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of

molar heat capacities CV and CP These are the heat capacities per mole and the

subscript V on CV means that the volume is being held constant while for CP the

pressure is held constant For example to raise the temperature of n moles of a gas

whose pressure is held constant by 10 K we would have to supply an amount of heat Q

= nCP (10) K

Molar Specific Heat of an Ideal Gas at Constant Volume

VQ nC T

3monatomic

2VC R

5diatomic

2VC R

5polyatomic

2VC

Real gases deviate from these formulas because in addition to the translational and ro-

tational degrees of freedom they also have vibrational and electronic degrees of

freedom These are unimportant at low temperatures due to quantum mechanical

effects but become increasingly important at higher temperatures The rough rule is

No of degrees of freedom

2VC R

Molar Specific Heat of an Ideal Gas at Constant Pressure

PQ nC T

P VC C R

The internal energy of an ideal gas depends only on the temperature

int VE nC T

Adiabatic Processes in an Ideal Gas

7

An adiabatic process is one in which no heat is exchanged between the system and the

environment When an ideal gas expands or contracts adiabatically not only does its

pressure change as expected from the ideal gas law but its temperature changes as

well Under these conditions the final pressure Pf can be computed from the initial

pressure Pi and from the final and initial volumes Vf and Vi by

or constantf f i iP V PV PV

where γ = CPCV The quantity γ is called the adiabatic exponent Note that this

doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be

combined with the adiabatic law for pressure given above to obtain the adiabatic law

for temperatures

1 constantTV

Compressions in sound waves are adiabatic because they happen too rapidly for any

appreciable amount of heat to flow This is why the adiabatic exponent appears in the

formula for the speed of sound in an ideal gas

RTv

M

Note that v depends only on T and not on P Because it depends only on the

temperature the speed of sound is the same in Provo as at sea level in spite of the

lower pressure here due to the difference in elevation

Equipartition of Energy

Every kind of molecule has a certain number of degrees of freedom which are

independent ways in which it can store energy Each such degree of freedom has

associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1

2 RT per mole)

(Note since a molecule has so many possible degrees of freedom it would seem that

there should be a lot of 12 sBk T to spread around But because energy is quantized

some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high

enough that 12 Bk T is as big as the lowest quantum of energy

8

Serway Chapter 22

Second Law of Thermodynamics

There are several equivalent forms of this important law

Kelvin It is not possible to change heat completely into work with no other change

taking place Or in other words there are no perfect heat engines

Clausius It is not possible for heat to flow from one body to another body at a higher

temperature with no other change taking place Or in other words there are no

perfect refrigerators

Entropy In any thermodynamic process that proceeds from one equilibrium state to

another the entropy of the system + environment either remains unchanged or

increases The total entropy never decreases This law is a bit of an oddity among the

laws of physics because it is not absolute Things are forbidden by the second law not

because it is impossible for them to happen but because it is extremely unlikely for

them to happen (See below for more information about entropy)

Reversible and Irreversible Processes

A reversible process is one which occurs so slowly that it is in thermal equilibrium (or

very nearly so) at all times A hallmark of such processes is that a motion picture of

them looks perfectly normal whether run forward or backward Imagine for instance

the slow expansion of a gas at constant temperature in a cylinder whose volume is being

increased by a slowly moving piston Run the movie backwards and what do you see

You see the slow compression of a gas at constant temperature which looks perfectly

normal

An irreversible process is one which occurs in such a way that thermal equilibrium is

not maintained throughout the process The mark of this kind of process is that a motion

picture of it looks very odd when run backward Imagine the sudden expansion of a gas

into a previously evacuated chamber because a hole was punched in the wall between a

pressurized chamber and the evacuated one Run the movie backward and what do you

see You see the gas in the soon-to-be-evacuated chamber gather itself together and

stream through a tiny hole into a chamber in which there is already plenty of gas If you

have ever seen this happen get in touch with the support group for those who have

witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW

9

Heat Engines

Because of the vexing difference in sign between work done on and system and work done

by a system we will invent a new work variable Weng Heat engines do work and so the net

W for these engines is negative But in engineering applications hidden minus signs are

regarded as evil so for heat engines we donrsquot talk about W instead we talk about its

magnitude engW W So for heat engines the first law is

int engE Q W

But for heat pumps and refrigerators work is done on the system so we use the usual work

W when we talk about these systems

A heat engine is a machine that absorbs heat converts part of it to work and exhausts the

rest The heat must be absorbed at high temperature and exhausted at low temperature If the

absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then

eng h cW Q Q

and the efficiency of the engine is defined to be

eng

h

We

Q

A perfect engine would convert the heat hQ completely into work Weng giving an effi-

ciency of e = 1 Energy conservation alone allows a perfect engine but the second law

requires e lt 1

Refrigerators and Heat Pumps

A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high

temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine

by some source of power A heat pump is a machine that either works like a refrigerator

keeping a place cold by transferring heat from this cold place to a higher temperature

environment (cooling mode like an air conditioner) or it functions as a heater

transferring heat into a warm place from a cooler one (heating mode like a window unit

that heats a house by extracting thermal energy from the cold outdoors) The coefficient

of performance of a refrigerator or of a heat pump in cooling mode is defined to be

COP cooling modecQ

W

10

For a heat pump in heating mode the coefficient of performance is

COP heating modehQ

W

Note that we donrsquot have to use the engineering work here because in these systems

work is done on the system and W is naturally positive

A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without

doing any work giving an infinite coefficient of performance Energy conservation alone

allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump

in heating mode would transfer Qh into the house without doing any work and so would

also have an infinite coefficient of performance The second law forbids this too A good

coefficient of performance for a real device would be around 5 or 6

Carnot Cycle

The most efficient of all possible engines is one that uses the Carnot cycle This cycle

employs an ideal gas has no friction and operates very slowly so that the gas can be in

thermal equilibrium at all parts of the cycle This means of course that it canrsquot

possibly be built and even if it could be built it would not run fast enough to be useful

Nevertheless this cycle is very important because it gives an upper bound on the

efficiency of real engines There cannot possibly be an engine that is more efficient

than one based on the Carnot cycle This cycle consists of the following four steps

1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its

volume The reason that heat is absorbed is that expansion tends to cool the gas but

thermal contact with the environment at Th keeps the temperature high by heat

conduction into the ideal gas

2 The ideal gas further increases its volume by an adiabatic expansion This expansion

causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc

3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in

volume The reason that heat is exhausted is that compression tends to heat the gas

but thermal contact with the environment at Tc keeps the temperature low by heat

conduction out of the ideal gas

4 The gas is adiabatically compressed back to its original volume (the volume it started

with in step 1) This compression heats the gas from Tc up to Th

11

The efficiency of a Carnot engine is given by the very simple formula

1 cC

h

Te

T

where the temperatures must in be Kelvin No real engine can be more efficient than

this

The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode

is given by

COP cooling mode cC

h c

T

T T

and no real refrigerator can have a coefficient of performance greater than this

The coefficient of performance of a Carnot heat pump in heating mode is

COP heating mode hC

h c

T

T T

So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency

To make the reversible steps in the cycle really reversible they would have to occur

infinitely slowly So the price you pay for making a perfect engine is that it takes

forever to get it to do any work

Entropy

The entropy of a system is defined in terms of its molecular makeup and measures

roughly the disorder of the system If the system is packed into a very small volume

then it is quite ordered and the entropy will be low If it occupies a large volume the

entropy is high (To see what this has to do with disorder note that socks in a drawer

occupy a small volume while socks on the bed in the corner by the door and

hanging from the chandelier occupy a large volume) If the system is very cold then

the molecules hardly move and may even reach out to each other and form a crystal

This is a highly ordered state and therefore has low entropy If the system is very hot

with rapidly speeding molecules crashing into the container walls and bouncing off

each other things are disordered and the entropy is high

It is possible to calculate the entropy of a system in terms of its macroscopic thermody-

namic properties ie pressure volume temperature number of moles etc The key to this

calculation is the concept of a reversible process A reversible process is one that is

12

carried out without friction and so slowly that the process can be reversed at any stage by

making an infinitesimal change in the environment of the system The slow expansion of the

gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment

is reversible If we turned around and began slowly to compress the gas would just slowly

exhaust heat to the environment in the exact reverse way that it absorbed it during expansion

Most processes however are irreversible For example if a gas-filled box were suddenly

increased in size so that the particles were free to wander into the void created by the sudden

expansion then the gas would eventually fill the new volume uniformly at the same

temperature as before the expansion (The temperature is unchanged in this imaginary

process because the kinetic energy of the molecules would be unaffected by such an

instantaneous expansion of the container walls) This imaginary but highly thought-

stimulating process is called a free expansion and it is impossible to reverse it During the

expansion we didnrsquot push on any of the molecules so reversing this process would mean

making them go back into their original volume without pushing on them they simply will

not cooperate to this extent Another way to see that just pushing them back where they came

from does not reverse the free expansion is to think about what would happen if we just

compressed either adiabatically or isothermally An adiabatic compression back to the

original volume would heat the gas above its original temperature and an isothermal

compression would require that heat be exhausted to the environment But the free expansion

involved neither temperature changes nor heat exchanges so neither of these two processes

is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in

a way that takes us back to the initial state

It is possible to calculate the change of entropy for both reversible and irreversible processes

Letrsquos consider a reversible process first In a reversible process the entropy change is given

by the formula

dQS

T

where dQ is the amount of heat added to the system during a small step of the process

The total energy change during the process may then simply be calculated by integration

f f

f i i i

dQS S S dS

T

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11

4

Eint) is a state variable

Figure 205

5

Serway Chapter 21

Kinetic Theory

The ideal gas law works for all atoms and molecules at low pressure It is rather

amazing that it does Kinetic theory explains why The properties of an ideal gas can

be understood by thinking of it as N rapidly moving particles of mass m As these

particles collide with the container walls momentum is imparted to the walls which

we call the force of gas pressure In this picture the pressure is related to the average

of the square of the particle velocity 2v by

22 1( )

3 2

NP mv

V

Using the ideal gas law we obtain the average translational kinetic energy per

molecule

21 3

2 2 Bmv k T

The rms speed is then given by

2rms

3 3Bk T RTv v

m M

where M is the molecular mass in kgmol

Degrees of Freedom

Roughly speaking a degree of freedom is a way in which a molecule can store energy

For instance since there are three different directions in space along which a molecule

can move there are three degrees of freedom for the translational kinetic energy

There are also three different axes of rotation about which a polyatomic molecule can

spin so we say there are three degrees of freedom for the rotational kinetic energy

There are even degrees of freedom associated with the various ways in which a

molecule can vibrate and with the different energy levels in which the electrons of

the molecule can exist

Internal Energy and Degrees of Freedom The internal energy of an ideal gas made

up of molecules with J degrees of freedom is given by

int 2 2 B

J JE nRT Nk T

6

Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of

molar heat capacities CV and CP These are the heat capacities per mole and the

subscript V on CV means that the volume is being held constant while for CP the

pressure is held constant For example to raise the temperature of n moles of a gas

whose pressure is held constant by 10 K we would have to supply an amount of heat Q

= nCP (10) K

Molar Specific Heat of an Ideal Gas at Constant Volume

VQ nC T

3monatomic

2VC R

5diatomic

2VC R

5polyatomic

2VC

Real gases deviate from these formulas because in addition to the translational and ro-

tational degrees of freedom they also have vibrational and electronic degrees of

freedom These are unimportant at low temperatures due to quantum mechanical

effects but become increasingly important at higher temperatures The rough rule is

No of degrees of freedom

2VC R

Molar Specific Heat of an Ideal Gas at Constant Pressure

PQ nC T

P VC C R

The internal energy of an ideal gas depends only on the temperature

int VE nC T

Adiabatic Processes in an Ideal Gas

7

An adiabatic process is one in which no heat is exchanged between the system and the

environment When an ideal gas expands or contracts adiabatically not only does its

pressure change as expected from the ideal gas law but its temperature changes as

well Under these conditions the final pressure Pf can be computed from the initial

pressure Pi and from the final and initial volumes Vf and Vi by

or constantf f i iP V PV PV

where γ = CPCV The quantity γ is called the adiabatic exponent Note that this

doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be

combined with the adiabatic law for pressure given above to obtain the adiabatic law

for temperatures

1 constantTV

Compressions in sound waves are adiabatic because they happen too rapidly for any

appreciable amount of heat to flow This is why the adiabatic exponent appears in the

formula for the speed of sound in an ideal gas

RTv

M

Note that v depends only on T and not on P Because it depends only on the

temperature the speed of sound is the same in Provo as at sea level in spite of the

lower pressure here due to the difference in elevation

Equipartition of Energy

Every kind of molecule has a certain number of degrees of freedom which are

independent ways in which it can store energy Each such degree of freedom has

associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1

2 RT per mole)

(Note since a molecule has so many possible degrees of freedom it would seem that

there should be a lot of 12 sBk T to spread around But because energy is quantized

some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high

enough that 12 Bk T is as big as the lowest quantum of energy

8

Serway Chapter 22

Second Law of Thermodynamics

There are several equivalent forms of this important law

Kelvin It is not possible to change heat completely into work with no other change

taking place Or in other words there are no perfect heat engines

Clausius It is not possible for heat to flow from one body to another body at a higher

temperature with no other change taking place Or in other words there are no

perfect refrigerators

Entropy In any thermodynamic process that proceeds from one equilibrium state to

another the entropy of the system + environment either remains unchanged or

increases The total entropy never decreases This law is a bit of an oddity among the

laws of physics because it is not absolute Things are forbidden by the second law not

because it is impossible for them to happen but because it is extremely unlikely for

them to happen (See below for more information about entropy)

Reversible and Irreversible Processes

A reversible process is one which occurs so slowly that it is in thermal equilibrium (or

very nearly so) at all times A hallmark of such processes is that a motion picture of

them looks perfectly normal whether run forward or backward Imagine for instance

the slow expansion of a gas at constant temperature in a cylinder whose volume is being

increased by a slowly moving piston Run the movie backwards and what do you see

You see the slow compression of a gas at constant temperature which looks perfectly

normal

An irreversible process is one which occurs in such a way that thermal equilibrium is

not maintained throughout the process The mark of this kind of process is that a motion

picture of it looks very odd when run backward Imagine the sudden expansion of a gas

into a previously evacuated chamber because a hole was punched in the wall between a

pressurized chamber and the evacuated one Run the movie backward and what do you

see You see the gas in the soon-to-be-evacuated chamber gather itself together and

stream through a tiny hole into a chamber in which there is already plenty of gas If you

have ever seen this happen get in touch with the support group for those who have

witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW

9

Heat Engines

Because of the vexing difference in sign between work done on and system and work done

by a system we will invent a new work variable Weng Heat engines do work and so the net

W for these engines is negative But in engineering applications hidden minus signs are

regarded as evil so for heat engines we donrsquot talk about W instead we talk about its

magnitude engW W So for heat engines the first law is

int engE Q W

But for heat pumps and refrigerators work is done on the system so we use the usual work

W when we talk about these systems

A heat engine is a machine that absorbs heat converts part of it to work and exhausts the

rest The heat must be absorbed at high temperature and exhausted at low temperature If the

absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then

eng h cW Q Q

and the efficiency of the engine is defined to be

eng

h

We

Q

A perfect engine would convert the heat hQ completely into work Weng giving an effi-

ciency of e = 1 Energy conservation alone allows a perfect engine but the second law

requires e lt 1

Refrigerators and Heat Pumps

A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high

temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine

by some source of power A heat pump is a machine that either works like a refrigerator

keeping a place cold by transferring heat from this cold place to a higher temperature

environment (cooling mode like an air conditioner) or it functions as a heater

transferring heat into a warm place from a cooler one (heating mode like a window unit

that heats a house by extracting thermal energy from the cold outdoors) The coefficient

of performance of a refrigerator or of a heat pump in cooling mode is defined to be

COP cooling modecQ

W

10

For a heat pump in heating mode the coefficient of performance is

COP heating modehQ

W

Note that we donrsquot have to use the engineering work here because in these systems

work is done on the system and W is naturally positive

A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without

doing any work giving an infinite coefficient of performance Energy conservation alone

allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump

in heating mode would transfer Qh into the house without doing any work and so would

also have an infinite coefficient of performance The second law forbids this too A good

coefficient of performance for a real device would be around 5 or 6

Carnot Cycle

The most efficient of all possible engines is one that uses the Carnot cycle This cycle

employs an ideal gas has no friction and operates very slowly so that the gas can be in

thermal equilibrium at all parts of the cycle This means of course that it canrsquot

possibly be built and even if it could be built it would not run fast enough to be useful

Nevertheless this cycle is very important because it gives an upper bound on the

efficiency of real engines There cannot possibly be an engine that is more efficient

than one based on the Carnot cycle This cycle consists of the following four steps

1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its

volume The reason that heat is absorbed is that expansion tends to cool the gas but

thermal contact with the environment at Th keeps the temperature high by heat

conduction into the ideal gas

2 The ideal gas further increases its volume by an adiabatic expansion This expansion

causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc

3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in

volume The reason that heat is exhausted is that compression tends to heat the gas

but thermal contact with the environment at Tc keeps the temperature low by heat

conduction out of the ideal gas

4 The gas is adiabatically compressed back to its original volume (the volume it started

with in step 1) This compression heats the gas from Tc up to Th

11

The efficiency of a Carnot engine is given by the very simple formula

1 cC

h

Te

T

where the temperatures must in be Kelvin No real engine can be more efficient than

this

The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode

is given by

COP cooling mode cC

h c

T

T T

and no real refrigerator can have a coefficient of performance greater than this

The coefficient of performance of a Carnot heat pump in heating mode is

COP heating mode hC

h c

T

T T

So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency

To make the reversible steps in the cycle really reversible they would have to occur

infinitely slowly So the price you pay for making a perfect engine is that it takes

forever to get it to do any work

Entropy

The entropy of a system is defined in terms of its molecular makeup and measures

roughly the disorder of the system If the system is packed into a very small volume

then it is quite ordered and the entropy will be low If it occupies a large volume the

entropy is high (To see what this has to do with disorder note that socks in a drawer

occupy a small volume while socks on the bed in the corner by the door and

hanging from the chandelier occupy a large volume) If the system is very cold then

the molecules hardly move and may even reach out to each other and form a crystal

This is a highly ordered state and therefore has low entropy If the system is very hot

with rapidly speeding molecules crashing into the container walls and bouncing off

each other things are disordered and the entropy is high

It is possible to calculate the entropy of a system in terms of its macroscopic thermody-

namic properties ie pressure volume temperature number of moles etc The key to this

calculation is the concept of a reversible process A reversible process is one that is

12

carried out without friction and so slowly that the process can be reversed at any stage by

making an infinitesimal change in the environment of the system The slow expansion of the

gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment

is reversible If we turned around and began slowly to compress the gas would just slowly

exhaust heat to the environment in the exact reverse way that it absorbed it during expansion

Most processes however are irreversible For example if a gas-filled box were suddenly

increased in size so that the particles were free to wander into the void created by the sudden

expansion then the gas would eventually fill the new volume uniformly at the same

temperature as before the expansion (The temperature is unchanged in this imaginary

process because the kinetic energy of the molecules would be unaffected by such an

instantaneous expansion of the container walls) This imaginary but highly thought-

stimulating process is called a free expansion and it is impossible to reverse it During the

expansion we didnrsquot push on any of the molecules so reversing this process would mean

making them go back into their original volume without pushing on them they simply will

not cooperate to this extent Another way to see that just pushing them back where they came

from does not reverse the free expansion is to think about what would happen if we just

compressed either adiabatically or isothermally An adiabatic compression back to the

original volume would heat the gas above its original temperature and an isothermal

compression would require that heat be exhausted to the environment But the free expansion

involved neither temperature changes nor heat exchanges so neither of these two processes

is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in

a way that takes us back to the initial state

It is possible to calculate the change of entropy for both reversible and irreversible processes

Letrsquos consider a reversible process first In a reversible process the entropy change is given

by the formula

dQS

T

where dQ is the amount of heat added to the system during a small step of the process

The total energy change during the process may then simply be calculated by integration

f f

f i i i

dQS S S dS

T

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11

5

Serway Chapter 21

Kinetic Theory

The ideal gas law works for all atoms and molecules at low pressure It is rather

amazing that it does Kinetic theory explains why The properties of an ideal gas can

be understood by thinking of it as N rapidly moving particles of mass m As these

particles collide with the container walls momentum is imparted to the walls which

we call the force of gas pressure In this picture the pressure is related to the average

of the square of the particle velocity 2v by

22 1( )

3 2

NP mv

V

Using the ideal gas law we obtain the average translational kinetic energy per

molecule

21 3

2 2 Bmv k T

The rms speed is then given by

2rms

3 3Bk T RTv v

m M

where M is the molecular mass in kgmol

Degrees of Freedom

Roughly speaking a degree of freedom is a way in which a molecule can store energy

For instance since there are three different directions in space along which a molecule

can move there are three degrees of freedom for the translational kinetic energy

There are also three different axes of rotation about which a polyatomic molecule can

spin so we say there are three degrees of freedom for the rotational kinetic energy

There are even degrees of freedom associated with the various ways in which a

molecule can vibrate and with the different energy levels in which the electrons of

the molecule can exist

Internal Energy and Degrees of Freedom The internal energy of an ideal gas made

up of molecules with J degrees of freedom is given by

int 2 2 B

J JE nRT Nk T

6

Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of

molar heat capacities CV and CP These are the heat capacities per mole and the

subscript V on CV means that the volume is being held constant while for CP the

pressure is held constant For example to raise the temperature of n moles of a gas

whose pressure is held constant by 10 K we would have to supply an amount of heat Q

= nCP (10) K

Molar Specific Heat of an Ideal Gas at Constant Volume

VQ nC T

3monatomic

2VC R

5diatomic

2VC R

5polyatomic

2VC

Real gases deviate from these formulas because in addition to the translational and ro-

tational degrees of freedom they also have vibrational and electronic degrees of

freedom These are unimportant at low temperatures due to quantum mechanical

effects but become increasingly important at higher temperatures The rough rule is

No of degrees of freedom

2VC R

Molar Specific Heat of an Ideal Gas at Constant Pressure

PQ nC T

P VC C R

The internal energy of an ideal gas depends only on the temperature

int VE nC T

Adiabatic Processes in an Ideal Gas

7

An adiabatic process is one in which no heat is exchanged between the system and the

environment When an ideal gas expands or contracts adiabatically not only does its

pressure change as expected from the ideal gas law but its temperature changes as

well Under these conditions the final pressure Pf can be computed from the initial

pressure Pi and from the final and initial volumes Vf and Vi by

or constantf f i iP V PV PV

where γ = CPCV The quantity γ is called the adiabatic exponent Note that this

doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be

combined with the adiabatic law for pressure given above to obtain the adiabatic law

for temperatures

1 constantTV

Compressions in sound waves are adiabatic because they happen too rapidly for any

appreciable amount of heat to flow This is why the adiabatic exponent appears in the

formula for the speed of sound in an ideal gas

RTv

M

Note that v depends only on T and not on P Because it depends only on the

temperature the speed of sound is the same in Provo as at sea level in spite of the

lower pressure here due to the difference in elevation

Equipartition of Energy

Every kind of molecule has a certain number of degrees of freedom which are

independent ways in which it can store energy Each such degree of freedom has

associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1

2 RT per mole)

(Note since a molecule has so many possible degrees of freedom it would seem that

there should be a lot of 12 sBk T to spread around But because energy is quantized

some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high

enough that 12 Bk T is as big as the lowest quantum of energy

8

Serway Chapter 22

Second Law of Thermodynamics

There are several equivalent forms of this important law

Kelvin It is not possible to change heat completely into work with no other change

taking place Or in other words there are no perfect heat engines

Clausius It is not possible for heat to flow from one body to another body at a higher

temperature with no other change taking place Or in other words there are no

perfect refrigerators

Entropy In any thermodynamic process that proceeds from one equilibrium state to

another the entropy of the system + environment either remains unchanged or

increases The total entropy never decreases This law is a bit of an oddity among the

laws of physics because it is not absolute Things are forbidden by the second law not

because it is impossible for them to happen but because it is extremely unlikely for

them to happen (See below for more information about entropy)

Reversible and Irreversible Processes

A reversible process is one which occurs so slowly that it is in thermal equilibrium (or

very nearly so) at all times A hallmark of such processes is that a motion picture of

them looks perfectly normal whether run forward or backward Imagine for instance

the slow expansion of a gas at constant temperature in a cylinder whose volume is being

increased by a slowly moving piston Run the movie backwards and what do you see

You see the slow compression of a gas at constant temperature which looks perfectly

normal

An irreversible process is one which occurs in such a way that thermal equilibrium is

not maintained throughout the process The mark of this kind of process is that a motion

picture of it looks very odd when run backward Imagine the sudden expansion of a gas

into a previously evacuated chamber because a hole was punched in the wall between a

pressurized chamber and the evacuated one Run the movie backward and what do you

see You see the gas in the soon-to-be-evacuated chamber gather itself together and

stream through a tiny hole into a chamber in which there is already plenty of gas If you

have ever seen this happen get in touch with the support group for those who have

witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW

9

Heat Engines

Because of the vexing difference in sign between work done on and system and work done

by a system we will invent a new work variable Weng Heat engines do work and so the net

W for these engines is negative But in engineering applications hidden minus signs are

regarded as evil so for heat engines we donrsquot talk about W instead we talk about its

magnitude engW W So for heat engines the first law is

int engE Q W

But for heat pumps and refrigerators work is done on the system so we use the usual work

W when we talk about these systems

A heat engine is a machine that absorbs heat converts part of it to work and exhausts the

rest The heat must be absorbed at high temperature and exhausted at low temperature If the

absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then

eng h cW Q Q

and the efficiency of the engine is defined to be

eng

h

We

Q

A perfect engine would convert the heat hQ completely into work Weng giving an effi-

ciency of e = 1 Energy conservation alone allows a perfect engine but the second law

requires e lt 1

Refrigerators and Heat Pumps

A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high

temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine

by some source of power A heat pump is a machine that either works like a refrigerator

keeping a place cold by transferring heat from this cold place to a higher temperature

environment (cooling mode like an air conditioner) or it functions as a heater

transferring heat into a warm place from a cooler one (heating mode like a window unit

that heats a house by extracting thermal energy from the cold outdoors) The coefficient

of performance of a refrigerator or of a heat pump in cooling mode is defined to be

COP cooling modecQ

W

10

For a heat pump in heating mode the coefficient of performance is

COP heating modehQ

W

Note that we donrsquot have to use the engineering work here because in these systems

work is done on the system and W is naturally positive

A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without

doing any work giving an infinite coefficient of performance Energy conservation alone

allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump

in heating mode would transfer Qh into the house without doing any work and so would

also have an infinite coefficient of performance The second law forbids this too A good

coefficient of performance for a real device would be around 5 or 6

Carnot Cycle

The most efficient of all possible engines is one that uses the Carnot cycle This cycle

employs an ideal gas has no friction and operates very slowly so that the gas can be in

thermal equilibrium at all parts of the cycle This means of course that it canrsquot

possibly be built and even if it could be built it would not run fast enough to be useful

Nevertheless this cycle is very important because it gives an upper bound on the

efficiency of real engines There cannot possibly be an engine that is more efficient

than one based on the Carnot cycle This cycle consists of the following four steps

1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its

volume The reason that heat is absorbed is that expansion tends to cool the gas but

thermal contact with the environment at Th keeps the temperature high by heat

conduction into the ideal gas

2 The ideal gas further increases its volume by an adiabatic expansion This expansion

causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc

3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in

volume The reason that heat is exhausted is that compression tends to heat the gas

but thermal contact with the environment at Tc keeps the temperature low by heat

conduction out of the ideal gas

4 The gas is adiabatically compressed back to its original volume (the volume it started

with in step 1) This compression heats the gas from Tc up to Th

11

The efficiency of a Carnot engine is given by the very simple formula

1 cC

h

Te

T

where the temperatures must in be Kelvin No real engine can be more efficient than

this

The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode

is given by

COP cooling mode cC

h c

T

T T

and no real refrigerator can have a coefficient of performance greater than this

The coefficient of performance of a Carnot heat pump in heating mode is

COP heating mode hC

h c

T

T T

So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency

To make the reversible steps in the cycle really reversible they would have to occur

infinitely slowly So the price you pay for making a perfect engine is that it takes

forever to get it to do any work

Entropy

The entropy of a system is defined in terms of its molecular makeup and measures

roughly the disorder of the system If the system is packed into a very small volume

then it is quite ordered and the entropy will be low If it occupies a large volume the

entropy is high (To see what this has to do with disorder note that socks in a drawer

occupy a small volume while socks on the bed in the corner by the door and

hanging from the chandelier occupy a large volume) If the system is very cold then

the molecules hardly move and may even reach out to each other and form a crystal

This is a highly ordered state and therefore has low entropy If the system is very hot

with rapidly speeding molecules crashing into the container walls and bouncing off

each other things are disordered and the entropy is high

It is possible to calculate the entropy of a system in terms of its macroscopic thermody-

namic properties ie pressure volume temperature number of moles etc The key to this

calculation is the concept of a reversible process A reversible process is one that is

12

carried out without friction and so slowly that the process can be reversed at any stage by

making an infinitesimal change in the environment of the system The slow expansion of the

gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment

is reversible If we turned around and began slowly to compress the gas would just slowly

exhaust heat to the environment in the exact reverse way that it absorbed it during expansion

Most processes however are irreversible For example if a gas-filled box were suddenly

increased in size so that the particles were free to wander into the void created by the sudden

expansion then the gas would eventually fill the new volume uniformly at the same

temperature as before the expansion (The temperature is unchanged in this imaginary

process because the kinetic energy of the molecules would be unaffected by such an

instantaneous expansion of the container walls) This imaginary but highly thought-

stimulating process is called a free expansion and it is impossible to reverse it During the

expansion we didnrsquot push on any of the molecules so reversing this process would mean

making them go back into their original volume without pushing on them they simply will

not cooperate to this extent Another way to see that just pushing them back where they came

from does not reverse the free expansion is to think about what would happen if we just

compressed either adiabatically or isothermally An adiabatic compression back to the

original volume would heat the gas above its original temperature and an isothermal

compression would require that heat be exhausted to the environment But the free expansion

involved neither temperature changes nor heat exchanges so neither of these two processes

is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in

a way that takes us back to the initial state

It is possible to calculate the change of entropy for both reversible and irreversible processes

Letrsquos consider a reversible process first In a reversible process the entropy change is given

by the formula

dQS

T

where dQ is the amount of heat added to the system during a small step of the process

The total energy change during the process may then simply be calculated by integration

f f

f i i i

dQS S S dS

T

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11

6

Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of

molar heat capacities CV and CP These are the heat capacities per mole and the

subscript V on CV means that the volume is being held constant while for CP the

pressure is held constant For example to raise the temperature of n moles of a gas

whose pressure is held constant by 10 K we would have to supply an amount of heat Q

= nCP (10) K

Molar Specific Heat of an Ideal Gas at Constant Volume

VQ nC T

3monatomic

2VC R

5diatomic

2VC R

5polyatomic

2VC

Real gases deviate from these formulas because in addition to the translational and ro-

tational degrees of freedom they also have vibrational and electronic degrees of

freedom These are unimportant at low temperatures due to quantum mechanical

effects but become increasingly important at higher temperatures The rough rule is

No of degrees of freedom

2VC R

Molar Specific Heat of an Ideal Gas at Constant Pressure

PQ nC T

P VC C R

The internal energy of an ideal gas depends only on the temperature

int VE nC T

Adiabatic Processes in an Ideal Gas

7

An adiabatic process is one in which no heat is exchanged between the system and the

environment When an ideal gas expands or contracts adiabatically not only does its

pressure change as expected from the ideal gas law but its temperature changes as

well Under these conditions the final pressure Pf can be computed from the initial

pressure Pi and from the final and initial volumes Vf and Vi by

or constantf f i iP V PV PV

where γ = CPCV The quantity γ is called the adiabatic exponent Note that this

doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be

combined with the adiabatic law for pressure given above to obtain the adiabatic law

for temperatures

1 constantTV

Compressions in sound waves are adiabatic because they happen too rapidly for any

appreciable amount of heat to flow This is why the adiabatic exponent appears in the

formula for the speed of sound in an ideal gas

RTv

M

Note that v depends only on T and not on P Because it depends only on the

temperature the speed of sound is the same in Provo as at sea level in spite of the

lower pressure here due to the difference in elevation

Equipartition of Energy

Every kind of molecule has a certain number of degrees of freedom which are

independent ways in which it can store energy Each such degree of freedom has

associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1

2 RT per mole)

(Note since a molecule has so many possible degrees of freedom it would seem that

there should be a lot of 12 sBk T to spread around But because energy is quantized

some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high

enough that 12 Bk T is as big as the lowest quantum of energy

8

Serway Chapter 22

Second Law of Thermodynamics

There are several equivalent forms of this important law

Kelvin It is not possible to change heat completely into work with no other change

taking place Or in other words there are no perfect heat engines

Clausius It is not possible for heat to flow from one body to another body at a higher

temperature with no other change taking place Or in other words there are no

perfect refrigerators

Entropy In any thermodynamic process that proceeds from one equilibrium state to

another the entropy of the system + environment either remains unchanged or

increases The total entropy never decreases This law is a bit of an oddity among the

laws of physics because it is not absolute Things are forbidden by the second law not

because it is impossible for them to happen but because it is extremely unlikely for

them to happen (See below for more information about entropy)

Reversible and Irreversible Processes

A reversible process is one which occurs so slowly that it is in thermal equilibrium (or

very nearly so) at all times A hallmark of such processes is that a motion picture of

them looks perfectly normal whether run forward or backward Imagine for instance

the slow expansion of a gas at constant temperature in a cylinder whose volume is being

increased by a slowly moving piston Run the movie backwards and what do you see

You see the slow compression of a gas at constant temperature which looks perfectly

normal

An irreversible process is one which occurs in such a way that thermal equilibrium is

not maintained throughout the process The mark of this kind of process is that a motion

picture of it looks very odd when run backward Imagine the sudden expansion of a gas

into a previously evacuated chamber because a hole was punched in the wall between a

pressurized chamber and the evacuated one Run the movie backward and what do you

see You see the gas in the soon-to-be-evacuated chamber gather itself together and

stream through a tiny hole into a chamber in which there is already plenty of gas If you

have ever seen this happen get in touch with the support group for those who have

witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW

9

Heat Engines

Because of the vexing difference in sign between work done on and system and work done

by a system we will invent a new work variable Weng Heat engines do work and so the net

W for these engines is negative But in engineering applications hidden minus signs are

regarded as evil so for heat engines we donrsquot talk about W instead we talk about its

magnitude engW W So for heat engines the first law is

int engE Q W

But for heat pumps and refrigerators work is done on the system so we use the usual work

W when we talk about these systems

A heat engine is a machine that absorbs heat converts part of it to work and exhausts the

rest The heat must be absorbed at high temperature and exhausted at low temperature If the

absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then

eng h cW Q Q

and the efficiency of the engine is defined to be

eng

h

We

Q

A perfect engine would convert the heat hQ completely into work Weng giving an effi-

ciency of e = 1 Energy conservation alone allows a perfect engine but the second law

requires e lt 1

Refrigerators and Heat Pumps

A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high

temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine

by some source of power A heat pump is a machine that either works like a refrigerator

keeping a place cold by transferring heat from this cold place to a higher temperature

environment (cooling mode like an air conditioner) or it functions as a heater

transferring heat into a warm place from a cooler one (heating mode like a window unit

that heats a house by extracting thermal energy from the cold outdoors) The coefficient

of performance of a refrigerator or of a heat pump in cooling mode is defined to be

COP cooling modecQ

W

10

For a heat pump in heating mode the coefficient of performance is

COP heating modehQ

W

Note that we donrsquot have to use the engineering work here because in these systems

work is done on the system and W is naturally positive

A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without

doing any work giving an infinite coefficient of performance Energy conservation alone

allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump

in heating mode would transfer Qh into the house without doing any work and so would

also have an infinite coefficient of performance The second law forbids this too A good

coefficient of performance for a real device would be around 5 or 6

Carnot Cycle

The most efficient of all possible engines is one that uses the Carnot cycle This cycle

employs an ideal gas has no friction and operates very slowly so that the gas can be in

thermal equilibrium at all parts of the cycle This means of course that it canrsquot

possibly be built and even if it could be built it would not run fast enough to be useful

Nevertheless this cycle is very important because it gives an upper bound on the

efficiency of real engines There cannot possibly be an engine that is more efficient

than one based on the Carnot cycle This cycle consists of the following four steps

1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its

volume The reason that heat is absorbed is that expansion tends to cool the gas but

thermal contact with the environment at Th keeps the temperature high by heat

conduction into the ideal gas

2 The ideal gas further increases its volume by an adiabatic expansion This expansion

causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc

3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in

volume The reason that heat is exhausted is that compression tends to heat the gas

but thermal contact with the environment at Tc keeps the temperature low by heat

conduction out of the ideal gas

4 The gas is adiabatically compressed back to its original volume (the volume it started

with in step 1) This compression heats the gas from Tc up to Th

11

The efficiency of a Carnot engine is given by the very simple formula

1 cC

h

Te

T

where the temperatures must in be Kelvin No real engine can be more efficient than

this

The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode

is given by

COP cooling mode cC

h c

T

T T

and no real refrigerator can have a coefficient of performance greater than this

The coefficient of performance of a Carnot heat pump in heating mode is

COP heating mode hC

h c

T

T T

So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency

To make the reversible steps in the cycle really reversible they would have to occur

infinitely slowly So the price you pay for making a perfect engine is that it takes

forever to get it to do any work

Entropy

The entropy of a system is defined in terms of its molecular makeup and measures

roughly the disorder of the system If the system is packed into a very small volume

then it is quite ordered and the entropy will be low If it occupies a large volume the

entropy is high (To see what this has to do with disorder note that socks in a drawer

occupy a small volume while socks on the bed in the corner by the door and

hanging from the chandelier occupy a large volume) If the system is very cold then

the molecules hardly move and may even reach out to each other and form a crystal

This is a highly ordered state and therefore has low entropy If the system is very hot

with rapidly speeding molecules crashing into the container walls and bouncing off

each other things are disordered and the entropy is high

It is possible to calculate the entropy of a system in terms of its macroscopic thermody-

namic properties ie pressure volume temperature number of moles etc The key to this

calculation is the concept of a reversible process A reversible process is one that is

12

carried out without friction and so slowly that the process can be reversed at any stage by

making an infinitesimal change in the environment of the system The slow expansion of the

gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment

is reversible If we turned around and began slowly to compress the gas would just slowly

exhaust heat to the environment in the exact reverse way that it absorbed it during expansion

Most processes however are irreversible For example if a gas-filled box were suddenly

increased in size so that the particles were free to wander into the void created by the sudden

expansion then the gas would eventually fill the new volume uniformly at the same

temperature as before the expansion (The temperature is unchanged in this imaginary

process because the kinetic energy of the molecules would be unaffected by such an

instantaneous expansion of the container walls) This imaginary but highly thought-

stimulating process is called a free expansion and it is impossible to reverse it During the

expansion we didnrsquot push on any of the molecules so reversing this process would mean

making them go back into their original volume without pushing on them they simply will

not cooperate to this extent Another way to see that just pushing them back where they came

from does not reverse the free expansion is to think about what would happen if we just

compressed either adiabatically or isothermally An adiabatic compression back to the

original volume would heat the gas above its original temperature and an isothermal

compression would require that heat be exhausted to the environment But the free expansion

involved neither temperature changes nor heat exchanges so neither of these two processes

is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in

a way that takes us back to the initial state

It is possible to calculate the change of entropy for both reversible and irreversible processes

Letrsquos consider a reversible process first In a reversible process the entropy change is given

by the formula

dQS

T

where dQ is the amount of heat added to the system during a small step of the process

The total energy change during the process may then simply be calculated by integration

f f

f i i i

dQS S S dS

T

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11

7

An adiabatic process is one in which no heat is exchanged between the system and the

environment When an ideal gas expands or contracts adiabatically not only does its

pressure change as expected from the ideal gas law but its temperature changes as

well Under these conditions the final pressure Pf can be computed from the initial

pressure Pi and from the final and initial volumes Vf and Vi by

or constantf f i iP V PV PV

where γ = CPCV The quantity γ is called the adiabatic exponent Note that this

doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be

combined with the adiabatic law for pressure given above to obtain the adiabatic law

for temperatures

1 constantTV

Compressions in sound waves are adiabatic because they happen too rapidly for any

appreciable amount of heat to flow This is why the adiabatic exponent appears in the

formula for the speed of sound in an ideal gas

RTv

M

Note that v depends only on T and not on P Because it depends only on the

temperature the speed of sound is the same in Provo as at sea level in spite of the

lower pressure here due to the difference in elevation

Equipartition of Energy

Every kind of molecule has a certain number of degrees of freedom which are

independent ways in which it can store energy Each such degree of freedom has

associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1

2 RT per mole)

(Note since a molecule has so many possible degrees of freedom it would seem that

there should be a lot of 12 sBk T to spread around But because energy is quantized

some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high

enough that 12 Bk T is as big as the lowest quantum of energy

8

Serway Chapter 22

Second Law of Thermodynamics

There are several equivalent forms of this important law

Kelvin It is not possible to change heat completely into work with no other change

taking place Or in other words there are no perfect heat engines

Clausius It is not possible for heat to flow from one body to another body at a higher

temperature with no other change taking place Or in other words there are no

perfect refrigerators

Entropy In any thermodynamic process that proceeds from one equilibrium state to

another the entropy of the system + environment either remains unchanged or

increases The total entropy never decreases This law is a bit of an oddity among the

laws of physics because it is not absolute Things are forbidden by the second law not

because it is impossible for them to happen but because it is extremely unlikely for

them to happen (See below for more information about entropy)

Reversible and Irreversible Processes

A reversible process is one which occurs so slowly that it is in thermal equilibrium (or

very nearly so) at all times A hallmark of such processes is that a motion picture of

them looks perfectly normal whether run forward or backward Imagine for instance

the slow expansion of a gas at constant temperature in a cylinder whose volume is being

increased by a slowly moving piston Run the movie backwards and what do you see

You see the slow compression of a gas at constant temperature which looks perfectly

normal

An irreversible process is one which occurs in such a way that thermal equilibrium is

not maintained throughout the process The mark of this kind of process is that a motion

picture of it looks very odd when run backward Imagine the sudden expansion of a gas

into a previously evacuated chamber because a hole was punched in the wall between a

pressurized chamber and the evacuated one Run the movie backward and what do you

see You see the gas in the soon-to-be-evacuated chamber gather itself together and

stream through a tiny hole into a chamber in which there is already plenty of gas If you

have ever seen this happen get in touch with the support group for those who have

witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW

9

Heat Engines

Because of the vexing difference in sign between work done on and system and work done

by a system we will invent a new work variable Weng Heat engines do work and so the net

W for these engines is negative But in engineering applications hidden minus signs are

regarded as evil so for heat engines we donrsquot talk about W instead we talk about its

magnitude engW W So for heat engines the first law is

int engE Q W

But for heat pumps and refrigerators work is done on the system so we use the usual work

W when we talk about these systems

A heat engine is a machine that absorbs heat converts part of it to work and exhausts the

rest The heat must be absorbed at high temperature and exhausted at low temperature If the

absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then

eng h cW Q Q

and the efficiency of the engine is defined to be

eng

h

We

Q

A perfect engine would convert the heat hQ completely into work Weng giving an effi-

ciency of e = 1 Energy conservation alone allows a perfect engine but the second law

requires e lt 1

Refrigerators and Heat Pumps

A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high

temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine

by some source of power A heat pump is a machine that either works like a refrigerator

keeping a place cold by transferring heat from this cold place to a higher temperature

environment (cooling mode like an air conditioner) or it functions as a heater

transferring heat into a warm place from a cooler one (heating mode like a window unit

that heats a house by extracting thermal energy from the cold outdoors) The coefficient

of performance of a refrigerator or of a heat pump in cooling mode is defined to be

COP cooling modecQ

W

10

For a heat pump in heating mode the coefficient of performance is

COP heating modehQ

W

Note that we donrsquot have to use the engineering work here because in these systems

work is done on the system and W is naturally positive

A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without

doing any work giving an infinite coefficient of performance Energy conservation alone

allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump

in heating mode would transfer Qh into the house without doing any work and so would

also have an infinite coefficient of performance The second law forbids this too A good

coefficient of performance for a real device would be around 5 or 6

Carnot Cycle

The most efficient of all possible engines is one that uses the Carnot cycle This cycle

employs an ideal gas has no friction and operates very slowly so that the gas can be in

thermal equilibrium at all parts of the cycle This means of course that it canrsquot

possibly be built and even if it could be built it would not run fast enough to be useful

Nevertheless this cycle is very important because it gives an upper bound on the

efficiency of real engines There cannot possibly be an engine that is more efficient

than one based on the Carnot cycle This cycle consists of the following four steps

1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its

volume The reason that heat is absorbed is that expansion tends to cool the gas but

thermal contact with the environment at Th keeps the temperature high by heat

conduction into the ideal gas

2 The ideal gas further increases its volume by an adiabatic expansion This expansion

causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc

3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in

volume The reason that heat is exhausted is that compression tends to heat the gas

but thermal contact with the environment at Tc keeps the temperature low by heat

conduction out of the ideal gas

4 The gas is adiabatically compressed back to its original volume (the volume it started

with in step 1) This compression heats the gas from Tc up to Th

11

The efficiency of a Carnot engine is given by the very simple formula

1 cC

h

Te

T

where the temperatures must in be Kelvin No real engine can be more efficient than

this

The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode

is given by

COP cooling mode cC

h c

T

T T

and no real refrigerator can have a coefficient of performance greater than this

The coefficient of performance of a Carnot heat pump in heating mode is

COP heating mode hC

h c

T

T T

So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency

To make the reversible steps in the cycle really reversible they would have to occur

infinitely slowly So the price you pay for making a perfect engine is that it takes

forever to get it to do any work

Entropy

The entropy of a system is defined in terms of its molecular makeup and measures

roughly the disorder of the system If the system is packed into a very small volume

then it is quite ordered and the entropy will be low If it occupies a large volume the

entropy is high (To see what this has to do with disorder note that socks in a drawer

occupy a small volume while socks on the bed in the corner by the door and

hanging from the chandelier occupy a large volume) If the system is very cold then

the molecules hardly move and may even reach out to each other and form a crystal

This is a highly ordered state and therefore has low entropy If the system is very hot

with rapidly speeding molecules crashing into the container walls and bouncing off

each other things are disordered and the entropy is high

It is possible to calculate the entropy of a system in terms of its macroscopic thermody-

namic properties ie pressure volume temperature number of moles etc The key to this

calculation is the concept of a reversible process A reversible process is one that is

12

carried out without friction and so slowly that the process can be reversed at any stage by

making an infinitesimal change in the environment of the system The slow expansion of the

gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment

is reversible If we turned around and began slowly to compress the gas would just slowly

exhaust heat to the environment in the exact reverse way that it absorbed it during expansion

Most processes however are irreversible For example if a gas-filled box were suddenly

increased in size so that the particles were free to wander into the void created by the sudden

expansion then the gas would eventually fill the new volume uniformly at the same

temperature as before the expansion (The temperature is unchanged in this imaginary

process because the kinetic energy of the molecules would be unaffected by such an

instantaneous expansion of the container walls) This imaginary but highly thought-

stimulating process is called a free expansion and it is impossible to reverse it During the

expansion we didnrsquot push on any of the molecules so reversing this process would mean

making them go back into their original volume without pushing on them they simply will

not cooperate to this extent Another way to see that just pushing them back where they came

from does not reverse the free expansion is to think about what would happen if we just

compressed either adiabatically or isothermally An adiabatic compression back to the

original volume would heat the gas above its original temperature and an isothermal

compression would require that heat be exhausted to the environment But the free expansion

involved neither temperature changes nor heat exchanges so neither of these two processes

is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in

a way that takes us back to the initial state

It is possible to calculate the change of entropy for both reversible and irreversible processes

Letrsquos consider a reversible process first In a reversible process the entropy change is given

by the formula

dQS

T

where dQ is the amount of heat added to the system during a small step of the process

The total energy change during the process may then simply be calculated by integration

f f

f i i i

dQS S S dS

T

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11

8

Serway Chapter 22

Second Law of Thermodynamics

There are several equivalent forms of this important law

Kelvin It is not possible to change heat completely into work with no other change

taking place Or in other words there are no perfect heat engines

Clausius It is not possible for heat to flow from one body to another body at a higher

temperature with no other change taking place Or in other words there are no

perfect refrigerators

Entropy In any thermodynamic process that proceeds from one equilibrium state to

another the entropy of the system + environment either remains unchanged or

increases The total entropy never decreases This law is a bit of an oddity among the

laws of physics because it is not absolute Things are forbidden by the second law not

because it is impossible for them to happen but because it is extremely unlikely for

them to happen (See below for more information about entropy)

Reversible and Irreversible Processes

A reversible process is one which occurs so slowly that it is in thermal equilibrium (or

very nearly so) at all times A hallmark of such processes is that a motion picture of

them looks perfectly normal whether run forward or backward Imagine for instance

the slow expansion of a gas at constant temperature in a cylinder whose volume is being

increased by a slowly moving piston Run the movie backwards and what do you see

You see the slow compression of a gas at constant temperature which looks perfectly

normal

An irreversible process is one which occurs in such a way that thermal equilibrium is

not maintained throughout the process The mark of this kind of process is that a motion

picture of it looks very odd when run backward Imagine the sudden expansion of a gas

into a previously evacuated chamber because a hole was punched in the wall between a

pressurized chamber and the evacuated one Run the movie backward and what do you

see You see the gas in the soon-to-be-evacuated chamber gather itself together and

stream through a tiny hole into a chamber in which there is already plenty of gas If you

have ever seen this happen get in touch with the support group for those who have

witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW

9

Heat Engines

Because of the vexing difference in sign between work done on and system and work done

by a system we will invent a new work variable Weng Heat engines do work and so the net

W for these engines is negative But in engineering applications hidden minus signs are

regarded as evil so for heat engines we donrsquot talk about W instead we talk about its

magnitude engW W So for heat engines the first law is

int engE Q W

But for heat pumps and refrigerators work is done on the system so we use the usual work

W when we talk about these systems

A heat engine is a machine that absorbs heat converts part of it to work and exhausts the

rest The heat must be absorbed at high temperature and exhausted at low temperature If the

absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then

eng h cW Q Q

and the efficiency of the engine is defined to be

eng

h

We

Q

A perfect engine would convert the heat hQ completely into work Weng giving an effi-

ciency of e = 1 Energy conservation alone allows a perfect engine but the second law

requires e lt 1

Refrigerators and Heat Pumps

A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high

temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine

by some source of power A heat pump is a machine that either works like a refrigerator

keeping a place cold by transferring heat from this cold place to a higher temperature

environment (cooling mode like an air conditioner) or it functions as a heater

transferring heat into a warm place from a cooler one (heating mode like a window unit

that heats a house by extracting thermal energy from the cold outdoors) The coefficient

of performance of a refrigerator or of a heat pump in cooling mode is defined to be

COP cooling modecQ

W

10

For a heat pump in heating mode the coefficient of performance is

COP heating modehQ

W

Note that we donrsquot have to use the engineering work here because in these systems

work is done on the system and W is naturally positive

A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without

doing any work giving an infinite coefficient of performance Energy conservation alone

allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump

in heating mode would transfer Qh into the house without doing any work and so would

also have an infinite coefficient of performance The second law forbids this too A good

coefficient of performance for a real device would be around 5 or 6

Carnot Cycle

The most efficient of all possible engines is one that uses the Carnot cycle This cycle

employs an ideal gas has no friction and operates very slowly so that the gas can be in

thermal equilibrium at all parts of the cycle This means of course that it canrsquot

possibly be built and even if it could be built it would not run fast enough to be useful

Nevertheless this cycle is very important because it gives an upper bound on the

efficiency of real engines There cannot possibly be an engine that is more efficient

than one based on the Carnot cycle This cycle consists of the following four steps

1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its

volume The reason that heat is absorbed is that expansion tends to cool the gas but

thermal contact with the environment at Th keeps the temperature high by heat

conduction into the ideal gas

2 The ideal gas further increases its volume by an adiabatic expansion This expansion

causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc

3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in

volume The reason that heat is exhausted is that compression tends to heat the gas

but thermal contact with the environment at Tc keeps the temperature low by heat

conduction out of the ideal gas

4 The gas is adiabatically compressed back to its original volume (the volume it started

with in step 1) This compression heats the gas from Tc up to Th

11

The efficiency of a Carnot engine is given by the very simple formula

1 cC

h

Te

T

where the temperatures must in be Kelvin No real engine can be more efficient than

this

The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode

is given by

COP cooling mode cC

h c

T

T T

and no real refrigerator can have a coefficient of performance greater than this

The coefficient of performance of a Carnot heat pump in heating mode is

COP heating mode hC

h c

T

T T

So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency

To make the reversible steps in the cycle really reversible they would have to occur

infinitely slowly So the price you pay for making a perfect engine is that it takes

forever to get it to do any work

Entropy

The entropy of a system is defined in terms of its molecular makeup and measures

roughly the disorder of the system If the system is packed into a very small volume

then it is quite ordered and the entropy will be low If it occupies a large volume the

entropy is high (To see what this has to do with disorder note that socks in a drawer

occupy a small volume while socks on the bed in the corner by the door and

hanging from the chandelier occupy a large volume) If the system is very cold then

the molecules hardly move and may even reach out to each other and form a crystal

This is a highly ordered state and therefore has low entropy If the system is very hot

with rapidly speeding molecules crashing into the container walls and bouncing off

each other things are disordered and the entropy is high

It is possible to calculate the entropy of a system in terms of its macroscopic thermody-

namic properties ie pressure volume temperature number of moles etc The key to this

calculation is the concept of a reversible process A reversible process is one that is

12

carried out without friction and so slowly that the process can be reversed at any stage by

making an infinitesimal change in the environment of the system The slow expansion of the

gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment

is reversible If we turned around and began slowly to compress the gas would just slowly

exhaust heat to the environment in the exact reverse way that it absorbed it during expansion

Most processes however are irreversible For example if a gas-filled box were suddenly

increased in size so that the particles were free to wander into the void created by the sudden

expansion then the gas would eventually fill the new volume uniformly at the same

temperature as before the expansion (The temperature is unchanged in this imaginary

process because the kinetic energy of the molecules would be unaffected by such an

instantaneous expansion of the container walls) This imaginary but highly thought-

stimulating process is called a free expansion and it is impossible to reverse it During the

expansion we didnrsquot push on any of the molecules so reversing this process would mean

making them go back into their original volume without pushing on them they simply will

not cooperate to this extent Another way to see that just pushing them back where they came

from does not reverse the free expansion is to think about what would happen if we just

compressed either adiabatically or isothermally An adiabatic compression back to the

original volume would heat the gas above its original temperature and an isothermal

compression would require that heat be exhausted to the environment But the free expansion

involved neither temperature changes nor heat exchanges so neither of these two processes

is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in

a way that takes us back to the initial state

It is possible to calculate the change of entropy for both reversible and irreversible processes

Letrsquos consider a reversible process first In a reversible process the entropy change is given

by the formula

dQS

T

where dQ is the amount of heat added to the system during a small step of the process

The total energy change during the process may then simply be calculated by integration

f f

f i i i

dQS S S dS

T

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11

9

Heat Engines

Because of the vexing difference in sign between work done on and system and work done

by a system we will invent a new work variable Weng Heat engines do work and so the net

W for these engines is negative But in engineering applications hidden minus signs are

regarded as evil so for heat engines we donrsquot talk about W instead we talk about its

magnitude engW W So for heat engines the first law is

int engE Q W

But for heat pumps and refrigerators work is done on the system so we use the usual work

W when we talk about these systems

A heat engine is a machine that absorbs heat converts part of it to work and exhausts the

rest The heat must be absorbed at high temperature and exhausted at low temperature If the

absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then

eng h cW Q Q

and the efficiency of the engine is defined to be

eng

h

We

Q

A perfect engine would convert the heat hQ completely into work Weng giving an effi-

ciency of e = 1 Energy conservation alone allows a perfect engine but the second law

requires e lt 1

Refrigerators and Heat Pumps

A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high

temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine

by some source of power A heat pump is a machine that either works like a refrigerator

keeping a place cold by transferring heat from this cold place to a higher temperature

environment (cooling mode like an air conditioner) or it functions as a heater

transferring heat into a warm place from a cooler one (heating mode like a window unit

that heats a house by extracting thermal energy from the cold outdoors) The coefficient

of performance of a refrigerator or of a heat pump in cooling mode is defined to be

COP cooling modecQ

W

10

For a heat pump in heating mode the coefficient of performance is

COP heating modehQ

W

Note that we donrsquot have to use the engineering work here because in these systems

work is done on the system and W is naturally positive

A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without

doing any work giving an infinite coefficient of performance Energy conservation alone

allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump

in heating mode would transfer Qh into the house without doing any work and so would

also have an infinite coefficient of performance The second law forbids this too A good

coefficient of performance for a real device would be around 5 or 6

Carnot Cycle

The most efficient of all possible engines is one that uses the Carnot cycle This cycle

employs an ideal gas has no friction and operates very slowly so that the gas can be in

thermal equilibrium at all parts of the cycle This means of course that it canrsquot

possibly be built and even if it could be built it would not run fast enough to be useful

Nevertheless this cycle is very important because it gives an upper bound on the

efficiency of real engines There cannot possibly be an engine that is more efficient

than one based on the Carnot cycle This cycle consists of the following four steps

1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its

volume The reason that heat is absorbed is that expansion tends to cool the gas but

thermal contact with the environment at Th keeps the temperature high by heat

conduction into the ideal gas

2 The ideal gas further increases its volume by an adiabatic expansion This expansion

causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc

3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in

volume The reason that heat is exhausted is that compression tends to heat the gas

but thermal contact with the environment at Tc keeps the temperature low by heat

conduction out of the ideal gas

4 The gas is adiabatically compressed back to its original volume (the volume it started

with in step 1) This compression heats the gas from Tc up to Th

11

The efficiency of a Carnot engine is given by the very simple formula

1 cC

h

Te

T

where the temperatures must in be Kelvin No real engine can be more efficient than

this

The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode

is given by

COP cooling mode cC

h c

T

T T

and no real refrigerator can have a coefficient of performance greater than this

The coefficient of performance of a Carnot heat pump in heating mode is

COP heating mode hC

h c

T

T T

So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency

To make the reversible steps in the cycle really reversible they would have to occur

infinitely slowly So the price you pay for making a perfect engine is that it takes

forever to get it to do any work

Entropy

The entropy of a system is defined in terms of its molecular makeup and measures

roughly the disorder of the system If the system is packed into a very small volume

then it is quite ordered and the entropy will be low If it occupies a large volume the

entropy is high (To see what this has to do with disorder note that socks in a drawer

occupy a small volume while socks on the bed in the corner by the door and

hanging from the chandelier occupy a large volume) If the system is very cold then

the molecules hardly move and may even reach out to each other and form a crystal

This is a highly ordered state and therefore has low entropy If the system is very hot

with rapidly speeding molecules crashing into the container walls and bouncing off

each other things are disordered and the entropy is high

It is possible to calculate the entropy of a system in terms of its macroscopic thermody-

namic properties ie pressure volume temperature number of moles etc The key to this

calculation is the concept of a reversible process A reversible process is one that is

12

carried out without friction and so slowly that the process can be reversed at any stage by

making an infinitesimal change in the environment of the system The slow expansion of the

gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment

is reversible If we turned around and began slowly to compress the gas would just slowly

exhaust heat to the environment in the exact reverse way that it absorbed it during expansion

Most processes however are irreversible For example if a gas-filled box were suddenly

increased in size so that the particles were free to wander into the void created by the sudden

expansion then the gas would eventually fill the new volume uniformly at the same

temperature as before the expansion (The temperature is unchanged in this imaginary

process because the kinetic energy of the molecules would be unaffected by such an

instantaneous expansion of the container walls) This imaginary but highly thought-

stimulating process is called a free expansion and it is impossible to reverse it During the

expansion we didnrsquot push on any of the molecules so reversing this process would mean

making them go back into their original volume without pushing on them they simply will

not cooperate to this extent Another way to see that just pushing them back where they came

from does not reverse the free expansion is to think about what would happen if we just

compressed either adiabatically or isothermally An adiabatic compression back to the

original volume would heat the gas above its original temperature and an isothermal

compression would require that heat be exhausted to the environment But the free expansion

involved neither temperature changes nor heat exchanges so neither of these two processes

is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in

a way that takes us back to the initial state

It is possible to calculate the change of entropy for both reversible and irreversible processes

Letrsquos consider a reversible process first In a reversible process the entropy change is given

by the formula

dQS

T

where dQ is the amount of heat added to the system during a small step of the process

The total energy change during the process may then simply be calculated by integration

f f

f i i i

dQS S S dS

T

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11

10

For a heat pump in heating mode the coefficient of performance is

COP heating modehQ

W

Note that we donrsquot have to use the engineering work here because in these systems

work is done on the system and W is naturally positive

A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without

doing any work giving an infinite coefficient of performance Energy conservation alone

allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump

in heating mode would transfer Qh into the house without doing any work and so would

also have an infinite coefficient of performance The second law forbids this too A good

coefficient of performance for a real device would be around 5 or 6

Carnot Cycle

The most efficient of all possible engines is one that uses the Carnot cycle This cycle

employs an ideal gas has no friction and operates very slowly so that the gas can be in

thermal equilibrium at all parts of the cycle This means of course that it canrsquot

possibly be built and even if it could be built it would not run fast enough to be useful

Nevertheless this cycle is very important because it gives an upper bound on the

efficiency of real engines There cannot possibly be an engine that is more efficient

than one based on the Carnot cycle This cycle consists of the following four steps

1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its

volume The reason that heat is absorbed is that expansion tends to cool the gas but

thermal contact with the environment at Th keeps the temperature high by heat

conduction into the ideal gas

2 The ideal gas further increases its volume by an adiabatic expansion This expansion

causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc

3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in

volume The reason that heat is exhausted is that compression tends to heat the gas

but thermal contact with the environment at Tc keeps the temperature low by heat

conduction out of the ideal gas

4 The gas is adiabatically compressed back to its original volume (the volume it started

with in step 1) This compression heats the gas from Tc up to Th

11

The efficiency of a Carnot engine is given by the very simple formula

1 cC

h

Te

T

where the temperatures must in be Kelvin No real engine can be more efficient than

this

The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode

is given by

COP cooling mode cC

h c

T

T T

and no real refrigerator can have a coefficient of performance greater than this

The coefficient of performance of a Carnot heat pump in heating mode is

COP heating mode hC

h c

T

T T

So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency

To make the reversible steps in the cycle really reversible they would have to occur

infinitely slowly So the price you pay for making a perfect engine is that it takes

forever to get it to do any work

Entropy

The entropy of a system is defined in terms of its molecular makeup and measures

roughly the disorder of the system If the system is packed into a very small volume

then it is quite ordered and the entropy will be low If it occupies a large volume the

entropy is high (To see what this has to do with disorder note that socks in a drawer

occupy a small volume while socks on the bed in the corner by the door and

hanging from the chandelier occupy a large volume) If the system is very cold then

the molecules hardly move and may even reach out to each other and form a crystal

This is a highly ordered state and therefore has low entropy If the system is very hot

with rapidly speeding molecules crashing into the container walls and bouncing off

each other things are disordered and the entropy is high

It is possible to calculate the entropy of a system in terms of its macroscopic thermody-

namic properties ie pressure volume temperature number of moles etc The key to this

calculation is the concept of a reversible process A reversible process is one that is

12

carried out without friction and so slowly that the process can be reversed at any stage by

making an infinitesimal change in the environment of the system The slow expansion of the

gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment

is reversible If we turned around and began slowly to compress the gas would just slowly

exhaust heat to the environment in the exact reverse way that it absorbed it during expansion

Most processes however are irreversible For example if a gas-filled box were suddenly

increased in size so that the particles were free to wander into the void created by the sudden

expansion then the gas would eventually fill the new volume uniformly at the same

temperature as before the expansion (The temperature is unchanged in this imaginary

process because the kinetic energy of the molecules would be unaffected by such an

instantaneous expansion of the container walls) This imaginary but highly thought-

stimulating process is called a free expansion and it is impossible to reverse it During the

expansion we didnrsquot push on any of the molecules so reversing this process would mean

making them go back into their original volume without pushing on them they simply will

not cooperate to this extent Another way to see that just pushing them back where they came

from does not reverse the free expansion is to think about what would happen if we just

compressed either adiabatically or isothermally An adiabatic compression back to the

original volume would heat the gas above its original temperature and an isothermal

compression would require that heat be exhausted to the environment But the free expansion

involved neither temperature changes nor heat exchanges so neither of these two processes

is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in

a way that takes us back to the initial state

It is possible to calculate the change of entropy for both reversible and irreversible processes

Letrsquos consider a reversible process first In a reversible process the entropy change is given

by the formula

dQS

T

where dQ is the amount of heat added to the system during a small step of the process

The total energy change during the process may then simply be calculated by integration

f f

f i i i

dQS S S dS

T

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11

11

The efficiency of a Carnot engine is given by the very simple formula

1 cC

h

Te

T

where the temperatures must in be Kelvin No real engine can be more efficient than

this

The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode

is given by

COP cooling mode cC

h c

T

T T

and no real refrigerator can have a coefficient of performance greater than this

The coefficient of performance of a Carnot heat pump in heating mode is

COP heating mode hC

h c

T

T T

So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency

To make the reversible steps in the cycle really reversible they would have to occur

infinitely slowly So the price you pay for making a perfect engine is that it takes

forever to get it to do any work

Entropy

The entropy of a system is defined in terms of its molecular makeup and measures

roughly the disorder of the system If the system is packed into a very small volume

then it is quite ordered and the entropy will be low If it occupies a large volume the

entropy is high (To see what this has to do with disorder note that socks in a drawer

occupy a small volume while socks on the bed in the corner by the door and

hanging from the chandelier occupy a large volume) If the system is very cold then

the molecules hardly move and may even reach out to each other and form a crystal

This is a highly ordered state and therefore has low entropy If the system is very hot

with rapidly speeding molecules crashing into the container walls and bouncing off

each other things are disordered and the entropy is high

It is possible to calculate the entropy of a system in terms of its macroscopic thermody-

namic properties ie pressure volume temperature number of moles etc The key to this

calculation is the concept of a reversible process A reversible process is one that is

12

carried out without friction and so slowly that the process can be reversed at any stage by

making an infinitesimal change in the environment of the system The slow expansion of the

gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment

is reversible If we turned around and began slowly to compress the gas would just slowly

exhaust heat to the environment in the exact reverse way that it absorbed it during expansion

Most processes however are irreversible For example if a gas-filled box were suddenly

increased in size so that the particles were free to wander into the void created by the sudden

expansion then the gas would eventually fill the new volume uniformly at the same

temperature as before the expansion (The temperature is unchanged in this imaginary

process because the kinetic energy of the molecules would be unaffected by such an

instantaneous expansion of the container walls) This imaginary but highly thought-

stimulating process is called a free expansion and it is impossible to reverse it During the

expansion we didnrsquot push on any of the molecules so reversing this process would mean

making them go back into their original volume without pushing on them they simply will

not cooperate to this extent Another way to see that just pushing them back where they came

from does not reverse the free expansion is to think about what would happen if we just

compressed either adiabatically or isothermally An adiabatic compression back to the

original volume would heat the gas above its original temperature and an isothermal

compression would require that heat be exhausted to the environment But the free expansion

involved neither temperature changes nor heat exchanges so neither of these two processes

is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in

a way that takes us back to the initial state

It is possible to calculate the change of entropy for both reversible and irreversible processes

Letrsquos consider a reversible process first In a reversible process the entropy change is given

by the formula

dQS

T

where dQ is the amount of heat added to the system during a small step of the process

The total energy change during the process may then simply be calculated by integration

f f

f i i i

dQS S S dS

T

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11

12

carried out without friction and so slowly that the process can be reversed at any stage by

making an infinitesimal change in the environment of the system The slow expansion of the

gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment

is reversible If we turned around and began slowly to compress the gas would just slowly

exhaust heat to the environment in the exact reverse way that it absorbed it during expansion

Most processes however are irreversible For example if a gas-filled box were suddenly

increased in size so that the particles were free to wander into the void created by the sudden

expansion then the gas would eventually fill the new volume uniformly at the same

temperature as before the expansion (The temperature is unchanged in this imaginary

process because the kinetic energy of the molecules would be unaffected by such an

instantaneous expansion of the container walls) This imaginary but highly thought-

stimulating process is called a free expansion and it is impossible to reverse it During the

expansion we didnrsquot push on any of the molecules so reversing this process would mean

making them go back into their original volume without pushing on them they simply will

not cooperate to this extent Another way to see that just pushing them back where they came

from does not reverse the free expansion is to think about what would happen if we just

compressed either adiabatically or isothermally An adiabatic compression back to the

original volume would heat the gas above its original temperature and an isothermal

compression would require that heat be exhausted to the environment But the free expansion

involved neither temperature changes nor heat exchanges so neither of these two processes

is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in

a way that takes us back to the initial state

It is possible to calculate the change of entropy for both reversible and irreversible processes

Letrsquos consider a reversible process first In a reversible process the entropy change is given

by the formula

dQS

T

where dQ is the amount of heat added to the system during a small step of the process

The total energy change during the process may then simply be calculated by integration

f f

f i i i

dQS S S dS

T

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11

13

(Just as in the case of energy we are mainly interested in differences rather than in

absolute magnitudes)

This integration method does not work for irreversible processes For instance in the

rapid free expansion discussed above no heat is added to the system but its disorder

obviously goes up We need to find some other way to calculate the entropy The key

is the fact that the entropy of a system depends only on its current state and not at all

on how it arrived there So to calculate the entropy change in an irreversible process

first find out what the initial and final conditions of the process are Then invent a

reversible process that takes the system from the initial state to the final state Since

the entropy depends only on the state of system and not on the process the entropy

change for the reversible process is the same as that for the irreversible process

Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2

then the pressure difference is P2 - Pl regardless of how the pressure change was

made This seems obvious for pressure but not for entropy only because you arent

comfortable with entropy yet

Entropy of an Ideal Gas

For n moles of an ideal gas the difference in entropy between a state with temperature

T and volume V and some standard state with temperature To and volume Vo is given

by

ln lno o Vo o

T VS T V S T V nC nR

T V

Entropy in a Phase Change

Calculating entropy change in a phase change like melting or boiling is easy It is

QT Where Q is in the latent heat for example the flows in to cause the melting and

T is the temperature of the phase change There is NO integral to do

14

Serway Chapter 35

Angle of Reflection

If plane waves are incident on a reflecting surface with the propagation direction of

the waves making angle θ1 with the normal direction to the surface then the reflected

angle θ1prime relative to the surface normal is simply

1 1

ie the incident angles and reflected angles are the same

Refraction

If plane waves traveling through medium 1 are incident on a plane interface between

medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and

the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law

2 2

1 1

sin

sin

v

v

where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of

incidence and the angle of refraction are both measured between the wave propagation

direction and the normal to the interface In terms of indices of refraction in the case

of light waves Snellrsquos law takes the more familiar form

1 1 2 2sin sinn n

where ni = cvi

Total Internal Reflection

If a wave is incident from a medium of low wave speed into a medium of high wave

speed the law of refraction requires that the angle of refraction be greater than the

angle of incidence If the angle of refraction is required to be greater than 90deg then no

refracted wave can exist and total internal reflection occurs The critical incident

angle θc beyond which total internal reflection occurs is given by

2

1

sin c

n

n

15

Dispersion of Light

In addition to the speed of light varying from material to material it also varies with

wavelength within each material This means that the index of refraction is generally a

function of wavelength

c

nv

Since the wave speed is not constant such a medium is dispersive meaning in this

context that refraction actually disperses white light into its various colors because

Snellrsquos law gives a different angle for each wavelength In most materials the

variation with wavelength is quite small but this small effect is responsible for some

of the most spectacular color effects we ever see including rainbows a flashing

crystal chandelier and the colored fire of a diamond solitaire by candlelight

16

Serway Chapter 36

Real and Virtual Images

When light rays are focused at a certain plane producing an image if a sheet of white

paper is placed there we call the image a real image The images produced by film

projectors and overhead projectors are examples of real images

When light rays appear to come from a certain location but no image is produced

when a screen is placed there we say that there is a virtual image at that location For

instance when you look in a mirror it appears that someone is behind the mirror but

a screen placed back there in the dark would show nothing Your image in the mirror

is a virtual image

Ray Tracing

There are lots of rules about how to find the images in optical systems but the best

way to keep things straight is to learn how to draw the principal rays for curved

mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615

The rays for converging and diverging thin lenses are shown in Fig 3627 You should

memorize the principal rays and know how to use them to locate images

Curved Mirrors

The focal length of a curved mirror with radius of curvature R is given by

2

Rf

The relation between object distance p image distance q and focal length f is

1 1 1

p q f

If the mirror is a diverging mirror f should be negative and if q should turn out to be

negative the image is virtual

Lateral Magnification

The lateral magnification in an optical system is defined by the ratio of the image size

to the object size

17

Image height

Object height

qM

p

for curved mirrors

As usual there are sign conventions here too but this formula just gives the

magnitude It is better to keep keep track of upright images versus inverted images by

means of ray diagrams rather than by memorizing sign conventions

Thin Lenses

The focal length of a thin lens is related to the radii of curvature of the two faces R1

and R2 of the lens by

1 2

1 1 11n

f R R

Note that this formula differs from Eq (3611) in the text by not having a minus sign

between the two R-terms We like this form better because for a simple converging

lens like a magnifying glass we just use positive values of R for both surfaces If one

of the faces is concave producing divergence use a negative value for R And if a

surface is flat use R = infin If the face is flat the radius is infinite

The relation between the image object and focal distances for a thin lens is the same

as that for a curved mirror

1 1 1

q p f

Use a negative focal length if the lens is diverging

The lateral magnification for a thin lens is the same as for a curved mirror

Image height

Object height

qM

p

Camera

The lens system in a camera projects a real image of an object onto the film (or CCD

array in a digital camera) The position of the image is adjusted to be on the film by

moving the lens into or out of the camera

18

Eye

The eye is like a camera in that a real image is formed on the retina Unlike a camera

the image position is adjusted by changing the focal length of the lens This is done by

the ciliary muscle which squeezes the lens changing its shape

Near Point The near point is the closest distance from the eye for which the lens can

focus an image on the retina It is usually 18-25 cm for young persons

Far Point The far point is the greatest distance from the eye for which the lens can

focus an image on the retina For a person with normal vision the far point is at

infinity

Nearsightedness A person is nearsighted if their far point is at some finite distance

less than infinity This condition can be corrected with a lens that takes an object at

infinity and produces a virtual image at the personrsquos far point

Farsightedness and Presbyopia A person is farsighted if their near point is too far

away for comfortable near work like reading or knitting This can be corrected by a

lens which takes an object at a normal near point distance of 18-25 cm and produces a

virtual image at the personrsquos natural near point Presbyopia involves a similar

problem which nearly all people experience as they age The ciliary muscle becomes

too weak and the lens becomes too stiff to allow the eye to provide for both near and

far vision The solution for this problem is either reading glasses or bifocal lenses

Reading classes are just weak magnifying glasses mounted on eyeglass frames

Bifocal lenses are split into upper and lower halves The lower half is a lens which

gives the proper correction for near work and the upper half is a different lens for

proper focusing at infinity

Angular Size

When an object is brought closer to the eye it appears to be larger because the image

on the retina is larger The size of this image is directly proportional to the objectrsquos

angular size which is the angle subtended by the object measured from the center of

the lens of the eye In optical instruments which are to be used with the eye the

angular size of the final image is whatrsquos important because it determines how large

the image will appear to the viewer

19

Simple Magnifier

A simple magnifier is a single converging lens or magnifying glass It takes an object

closer to the eye than a normal near point and produces a virtual image at or beyond

this near point The angular magnification is defined to be the ratio of the angular size

when viewed through the lens to the angular size of the object when viewed at the

normal near point (without aid of the lens)

Microscope

This instrument has two lenses (1) The objective is near the object being viewed and

produces a greatly magnified real image (2) The eyepiece is a simple magnifier which

the viewer uses to closely examine the image from (1)

Telescope

This instrument also has two lenses (1) The objective at the front of the telescope

takes light from a distant object and produces a real inverted image (which is rather

small) near its focal point (2) This small real image is then examined by the eyepiece

functioning as a simple magnifier to produce a virtual image with a larger angular

size

20

Serway Chapter 37

Two-Slit Interference

If light is incident on two closely spaced narrow slits a pattern of light and dark

stripes is produced beyond the slits The bright stripes or fringes are caused by

constructive interference of the two waves coming from the slits Constructive

interference occurs whenever two waves arrive at a location in phase with each other

This occurs when the distance x1 from slit 1 to a point P on the screen and the

distance x2 from slit 2 to point P differ by in integral number of wavelengths

1 2 where 0 1 2x x m m

where λ is the wavelength of the light When x1 and x2 are much larger than the slit

spacing d this condition reduces to

sind m

where θ is the angle between the direction of the incident light and the direction of the

light arriving at the screen

Thin Films

When light is partially reflected and partially transmitted by a thin film of transparent

material it is possible to have interference between the wave reflected from the front

of the film and light reflected from the back of the film (The colored reflections from

the thin film of oil on the water in a rain-soaked parking lot are an example of this

effect) It is difficult to write down formulas that will work in all cases so we will

just review the important principles here

1 If the two reflected waves are in phase with each other the film has enhanced

reflection (constructive interference) but if the two reflected waves are out of phase

with each other reflection is diminished (destructive interference) Phase shifts occur

due to reflection and due to the extra path length through the film of the wave

reflected from the back of the film

2 The phase change due to reflection is determined by the difference in index of

refraction between the two media involved in the reflection If the wave is incident

21

from a medium with a low index of refraction into a medium with a high index of

refraction a phase change of 180deg occurs and the reflected wave is inverted If

incident from high to low no phase shift occurs and the reflected wave is non-

inverted

3 The extra path length through the film of the wave reflected from the back of the

film is equal to 2t where t is the thickness of the film (The incident light is assumed

to be normal to the surface of the film) The number of wavelengths contained in the

extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the

film

4 Rules for reflection from thin films

If one ray is inverted and the other is not then we have

12 constructive

2 nt m

2 destructivent m

If either both rays are inverted or both are non-inverted we have

2 constructivent m

12 destructive

2 nt m

22

Serway Chapter 38

Diffraction Grating

A diffraction grating is simply a fancy version of two-slit interference with the two

slits replaced by thousands of slits Just as in the two-slit case the bright fringes

occur at angles given by

sin md

but in this case d the distance between neighboring slits is made to be very small

The effect of having many slits instead of two is to make each bright fringe highly

localized with wide dark regions between neighboring maxima

Single Slit Diffraction

When light passes through an opening in an opaque screen an interference pattern is

produced beyond the opening To understand why we may replace the single opening

by many small coherent sources of light These many sources interfere with each

other producing a pattern known as a diffraction pattern (Note that many authors do

not distinguish between interference and diffraction treating them as interchangeable

terms) If the opening is a slit of width a then the diffraction pattern far from the slit

will have a bright central maximum with a succession of minima and weaker maxima

on either side The angle between the incident direction and the minima is given by

sin ma

where m = plusmn1 plusmn2

If the opening is circular with diameter D the angle between the incident direction

and the first minimum is given by

sin 122D

Optical Resolution and Rayleighrsquos Criterion

Two point sources can just be resolved (distinguished from each other) if the peak of

the diffraction image of the first source overlies the first minimum of the diffraction

image of the second source For circular holes of the kind usually encountered in

23

optical devices this condition is approximately satisfied when the angular separation

between the two sources as viewed from the optical instrument is greater than or equal

to the critical angle

min 122D

where λ is the wavelength of the light and where D is the diameter of the aperture in

the instrument

Polarization

We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot

change direction in a random fashion The simplest kind of polarization is linear

polarization in which the electric field vector oscillates back and forth along the

same axis in space Polarized light can be produced from normal unpolarized light by

selective absorption (as in Polaroid sunglasses) by passing light through crystals that

have different indices of refraction for different polarizations (double refraction) by

scattering (the blue sky is polarized) and by reflection (glare) Polarization by

reflection occurs when light reflects from a shiny insulating (non-metallic) surface

The amount of polarization is greatest for reflection at Brewsterrsquos angle

2

1

tan P

n

n

where θP is the incidence angle of light from medium 1 onto medium 2 and where n1

and n2 are the indices of refraction for the media (In our everyday experience n1 = 1

since the light comes in through the air and n2 is the index of refraction of the shiny

insulating material producing the glare eg water glass plastic paint etc)

Malusrsquos Law

The intensity of transmitted polarized light through a perfect polarizer is related to the

incident intensity of polarized light by Malusrsquos law

2cosoI I

where Io is the intensity of the incident light and where θ is the angle between the

electric field vector in the incident wave and the transmission axis the polarizer

24

Serway Chapter 39

Principles of Relativity

All of the weirdness of relativity flows from two simple principles

(1) The laws of physics must be the same in all inertial (non-accelerating) reference

frames

(2) The speed of light in vacuum has the same value in all inertial reference frames

Note that (1) seems reasonable but (2) is very odd It says that if two space ships are

approaching each other at nearly the speed of light and a laser pulse is shot from ship

1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it

goes by it is moving at 3 times 108 ms the same as if the ships were stationary

Simultaneity

If observer 1 sees two events in her own frame as simultaneous at two different

locations a moving observer 2 will see these two events happening at different times

Relativistic gamma

The factor γ (gamma) appears regularly in the formulas of relativity

2

2

1

1 vc

where v is the relative speed between two inertial frames

Time Dilation

If observer 1 sees two events at the same location in space separated by time Δtp in his

own frame then observer 2 moving at speed v relative to observer 1 will see these

two events separated by a longer time Δt

pt t

Moving clocks run slow

Length Contraction

If observer 1 measures the length of an object along the x-direction at some instant of

time in her frame to be Lp then observer 2 moving in the x-direction at speed v will

measure the length L of the object to be shorter

pLL

25

Moving meter sticks are short

Velocity Addition

If a particle is moving at velocity vac relative to frame c and if frame c is moving at

speed vcb with respect to frame b (with both velocities directed along the same line)

what would be the velocity of the particle in frame b In classical physics the answer is

ab ac cbv v v

but if any of these velocities are an appreciable fraction of the speed of light the

answer changes

21ac cb

abac cb

v Vv

v v c

In using this equation make sure that vac has the same sign in the numerator and in the

denominator and that vbc also has the same sign in both places Note that this equation

is different (and in our opinion easier to understand) than the equations in the text

Momentum and Energy

The momentum and total energy of a particle moving at velocity v (its speed is v v )

are given by

22

2 2

22

1 1 vvc c

m mcm E mc

vp v

Rest Energy

When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It

includes the potential energy of the particle so that if we change the potential energy

of a particle we change its rest mass

Kinetic Energy The kinetic energy is the difference between the total energy and the

rest energy

2 2K mc mc

For v ltlt c this reduces to K = mv22

General Relativity

Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity

of a point a gravitational field is equivalent to an accelerated frame of reference in

26

the absence of gravitational effects For example there is no difference whatever in

the physics experienced by an astronaut on earth and one who is in interstellar space

far away from any stars or planets but who has just turned on her spaceshiprsquos engines

and is accelerating at 98 ms2

27

Serway Chapter 40

Photons

We now know that even though light behaves like a wave it is better described as

consisting of small packets of energy called photons The energy of a photon is related

to its frequency by

E hf

where h is Planckrsquos constant

346626 10 J sh

Photons also carry momentum given by

hf hp

c

Since the amount of energy in each photon is fixed the difference between dim light

and intense light (of the same wavelength) is that dim light consists of fewer photons

than intense light

Photoelectric Effect

In the photoelectric effect light shining onto a metal surface gives the electrons in the

metal enough energy to escape and be detected It requires a certain amount of energy

(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the

metal so the light must deliver at least this much energy to an electron to produce the

effect It is observed that red light no matter how intense never produces electrons

But ultraviolet light even if quite dim will eject electrons from the metal Since red

light consists of 2 eV photons and ultraviolet light has photons with energies around

4-6 eV the photon idea explains the behavior of the photoelectric effect The

maximum energy that an ejected electron can have is

maxK hf

where f is the photon frequency The minimum frequency that light can have and cause

electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0

cf h

28

Compton Effect

When high-frequency light interacts with free electrons the scattered light does not

have the same wavelength as the incident light contrary to what classical

electromagnetism would predict But the photon picture in which photons have

momentum and energy allows us to treat scattering as a collision between the photon

and the electron In this collision the electron and the scattered photon both have a

different momentum and energy than they did before And since p = hλ for a photon

if the momentum is different the wavelength will also be different

after before 1 cose

h

m c

where the angle θ is the angle between the incoming photon direction and the

direction of the scattered photon

Particles are Waves

Since photons behave like particles sometimes it is not surprising that elementary

particles can behave like waves sometimes The wavelength of a particle with

momentum p is given by

h

p

Wave-particle Duality

Both photons and elementary particles have a dual nature sometimes they behave like

particles and sometimes they behave like waves It is hard for us to comprehend the

nature of such an object by making mental pictures but experiments definitively show

that this is the case Since we have no direct experience with photons and elementary

particles (because their energies are so much smaller than the energies of the everyday

objects we encounter) it is perhaps not surprising that we have a hard time forming a

mental picture of how they behave

Electron Interference

Since an electron is both a particle and a wave just like a photon it should be able to

produce an interference pattern This is observed If an electron beam is shot at two

closely-spaced slits and if the electrons that pass through the slits are detected

downstream it is found that there are some locations where electrons are never detected

29

and others where lots of electrons are detected The pattern is exactly the same as the

one observed for light waves provided that we use the electron wavelength λ = hp in

place of the wavelength of light This pattern is observed even though each electron is

detected as a single dot on the screen Only after many such dots are collected does the

pattern emerge And if we try to understand how this effect could possibly work by

looking closely at each slit to see which one the electron came through the pattern

disappears the act of measurement destroys the interference This means that each

single electron somehow comes through both slits (as a wave would)

Uncertainty Principle

In classical physics we always imagine that the positions and momenta of moving

particles have definite values It might be hard to measure them but surely at each

instant of time a particle should be precisely located at some point in space and have a

similarly precise momentum This turns out not to be true Instead both position and

momentum are required to be uncertain with their uncertainties Δx and Δp satisfying

the Heisenberg uncertainty relation

2x p

So if the particle were known to be precisely at some particular location (so that Δx =

0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum

were exactly known we couldnrsquot know its position

There is a similar relation involving the particlersquos energy E and the time interval Δt

over which this energy is measured

2E t

This means that energy is not actually conserved if we are considering very small time

intervals and this brief non-conservation of energy has been observed

30

Serway Chapter 41

Wave Function ψ and Probability

Quantum mechanics does not predict exactly what an electron or a photon will do Instead

it specifies the wave function or probability amplitude ψ of an electron or a photon

This wave function is a complex-valued function of space and time whose squared

magnitude is the probability density P for finding a particle at a particular place in

space at a certain time

2P

where is the complex conjugate of ψ

For example the wave function of an electron with perfectly specified momentum p

would have a wavelength given by

h

p

and its probability amplitude would be proportional to

2i x ipxe e

The corresponding probability density would then be

21ipx ip ipxP e e e

which means that the electron is equally probable to be anywhere along the x axis

This is in accord with the uncertainty principle since we specified the momentum

precisely we canrsquot have any idea about the position of the electron

As another example you have probably seen ldquofuzzy ballrdquo drawings of electron

orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution

of the probability density 2

P in the orbital

Particle in a Box

A simple example in which we can calculate the wave function is the case of a particle of

mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this

case the general wave function is a linear superposition of wave functions ψn of the form

sin niE tn

n xx t A e

L

31

where A is a positive constant where n = 1 2 3 and where the energy associated

with each of the quantum states ψn is given by

22

28n

hE n

mL

This wave function is zero at x = 0 and x = L which means that the particle will never

be found at the walls of the box The wave function has maximum values in the

interior and at these places the particle is most likely to be found and it also has

places where it is zero and at these places the particle will also never be found as

expressed by the formula

2sinn x

PL

The particle in the box is interfering with itself producing a probability interference

pattern across the box just like the interference patterns we studied with light and

sound

32

Serway Chapter 42

Atoms

Many people picture an atom as a miniature solar system where electrons orbit around

a massive nucleus at the center This picture is misleading because in atoms the wave

nature of electrons dominates The electrons form 3-dimensional standing waves

(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the

answer is ψ the probability amplitude (see Chapter 41)

Standing waves on a string can occur only for certain frequencies (the fundamental

and higher harmonics) Similarly atomic orbitals occur only for certain energies For

the hydrogen atom the energies of the orbitals have the a particularly simple form

2

1136 eVnE

n

where n = 1 2 3 is called the principal quantum number For other atoms the

determination of the orbital energies requires numerical calculation by computers

Atomic Spectra

If an electron is somehow given extra extra energy (we say that it is excited) so that it

occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each

time an electron falls to a lower orbital it loses the difference in energy between the

two orbitals in the form of a photon Since the orbital energies are discrete so are the

energy differences and so are the wavelengths of the emitted light The entire set of

these discrete wavelengths is called the atomic spectrum and it is unique to each type

of atom For hydrogen the spectrum can be simply written as

H 2 2

1 1 1

f i

Rn n

where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as

a formula

Orbital Angular Momentum Electrons in atoms also have quantized values of

angular momentum The orbital quantum number ℓ specifies the value of this

quantized angular momentum through the formula

1L

33

If we want to know the value of the angular momentum along some direction in space

say the z direction the answer is not L but rather

zL m

where mℓ is another quantum number which runs from

1 1m

This quantum number is important when an atom sits in a magnetic field

Spin Angular Momentum It has been found experimentally that electrons and other

charged particles also carry internal angular momentum which we call spin

Electrons have an intrinsic spin angular momentum s along a specified axis that is

extremely quantized it can only take on 2 possible values

1

2zs s

We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-

cles Since s is an angular momentum it obeys the same rule as that for orbital angular

momentum namely that if its value along some axis is s then its total magnitude is

given by

31

2S s s

I know this seems weird but quantum mechanics is weird The only excuse for this

bizarre way of looking at the world is that it predicts what happens in experiments

Exclusion Principle The answer to the question of how many electrons (or any other

spin one-half particle) can be in one particular quantum state was discovered by

Wolfgang Pauli and is called the exclusion principle

ldquoNo two electrons can ever be in the same quantum state therefore no two electrons

in the same atom can have the same set of quantum numbersrdquo

This is the reason that we have atoms with different properties instead of every atom

simple having all of its electrons in the ground state All of the variety we see around

us in the world is the result of chemical differences and these differences would not

exist unless electrons obeyed this important principle The entire structure of the

periodic table (see pages 1377-1379 in Serway) is an expression of this principle

34

Serway Chapter 44

Nuclear Properties

The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of

neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit

-271 u=1660540 10 kg

The atomic number Z counts the number of protons in a nucleus while the neutron

number N counts the number of neutrons The mass number A is the sum of the two

A N Z

Protons and nuetrons have about the same mass and some times called baryons (heavy

ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon

number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron

has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei

with different numbers of neutrons These nuclear siblings are called isotopes

These numbers are used to label nuclei according to the pattern

5626 ie FeA

Z X

denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons

The nucleus is roughly spherical with a radius given approximately by

1 3 150 0where 12 10 mr r A r

Nuclear Stability

Because the positively charged protons electrically repel each other with an enormous

force at distances as small as 10-15 m there must be some really strong force that that

overcomes electrical repulsion to hold protons and neutrons together This force is

called with some lack of imagination the strong force It is a very short-range force

(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons

neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force

the coulomb repulsion of the protons is still present so anything that might keep the

protons from being right next to each other would help keep the nucleus from

35

exploding This role is played by the neutrons and for nuclei with Z le 20 the stable

nuclei roughly have N = Z

For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are

needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of

neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of

(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These

elements have one or two isotopes with half-lifes of billions of years so there are

substantial amounts of such elements on Earth This fact makes it possible to have

practical fission devices

Radioactive Decay

There are three types of radioactive decay

Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the

original nucleus is called X and the new nucleus is called Y then the decay would look

like this

4 42 2X Y+ HeA A

Z Z

Beta decay the nucleus either kicks out an electron (endash) or its positively-charged

antimatter twin the positron (e+) plus either an electron neutrino v or an electron

anti-neutrino v

1X Y+e electron decayA AZ Z v

1X Y+e positron decayA AZ Z v

A neutrino is a particle with no charge hardly any mass (much less than the electron

mass) and interacts so weakly with matter that most neutrinos upon encountering the

planet earth just pass right through it as if it werenrsquot there

Gamma decay the nucleons in the nucleus X are in an excited energy state X

(perhaps as a result of having undergone alpha or beta decay) and they drop down to a

lower energy state shedding the energy as a high frequency photon

X XA AZ Z

36

This process is exactly analogous to the way that the electrons in atoms emit photons

Decay Rate and Half Life

There is no way to predict exactly when an unstable or excited nucleus will decay but

there is an average rate at which this decay occurs called the decay constant λ The

meaning of this constant is that if there are a large number N of nuclei in a sample

then the number of decays per second that will be observed (called the decay rate R)

is R = λN In mathematical language

dMR N

dt

This simple differential equation has for its solution

0tN t N e

where N0 is the number of nuclei in the sample at time t = 0

The half-life is the time it takes for half of the nuclei in the sample to decay and is

related to the decay constant by

1 2

ln 2 0693T

Disintegration Energy

When a nucleus decays it is making a transition to an overall state of lower energy

which means according to Einsteinrsquos famous formula E = mc2 that the sum of the

masses after the decay must be less than the mass before with the lost mass appearing

as kinetic energy among the decay products For example in alpha decay this kinetic

energy called the disintegration energy Q is given by

2X YQ M M M c

37

Serway Chapter 45

Nuclear energy

There are two ways to extract energy from the nucleus fission and fusion For nuclei

with Z greater than 26 breaking the nucleus apart into pieces leads to a lower

2mc energy than the original nucleus so energy can be extracted by fission For

nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy

can be extracted by fusion

Fission

Since neutrons have no charge they are not repelled from nuclei as protons are for

this reason their behavior is the key to understanding how fission works

Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron

down This elastic energy loss is most effective if the other nuclei have low mass (like

hydrogen) and these materials are called moderators because of their ability to slow

down fast neutrons

The reason that slowing neutrons is important is that slow neutrons are much more

likely to be absorbed by a nucleus which then leads to nuclear reactions of various

kinds For a few very large nuclei like uranium-235 and some plutonium isotopes

absorption of a slow neutron causes the nucleus to split into two large fragments plus

2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron

can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow

can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is

allowed to proceed unchecked it produces a large explosion If a neutron absorbing

material is added to the mix (like the cadmium in reactor control rods) it is possible to

keep the reaction under control and to extract the released energy as heat to drive

steam turbines and produce electricity

Fusion

Fusion involves mashing two nuclei together and since they are both charged and repel

each other this reaction is much harder to make go The nuclei must have enough energy to

overcome the coulomb repulsion which is why this reaction requires a high temperature

(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to

helium via fusion) or like that in the center of the fission explosion that is used to detonate

38

a hydrogen bomb

This reaction is of interest for power production in spite of this difficult temperature

requirement because of the abundance of fusion fuel on the planet There are about

012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4

cents to extract it The fusion energy available from this minuscule amount of

deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so

abundant on earth that if fusion were to work we would have an essentially

inexhaustible source of energy

So why donrsquot we have fusion power plants Well the fuel is cheap but the match is

incredibly expensive The only way we know to control this difficult high-temperature

reaction is with large and expensive pieces of equipment involving either large

magnetic fields and complex high-power electromagnetic antennas or with gigantic

(football-fieldsized) laser facilities involving more than a hundred of the highest-

energy lasers ever built Power plants based on these current methods for controlling

fusion are unattractive to the fiscally-minded people who run the electric power

industry Hopefully better designs will be discovered as experiments continue

39

c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium

boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium

carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium

40

UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units

prefix symbol factor tera T 1012 giga G 109

mega M 106 kilo k 103

centi c 10-2 milli m 10-3

micro μ 10-6 nano n 10-9 pico p 10-12

femto f 10-15

length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m

mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg

force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N

pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa

time second (s) minute 1 min = 60 s hour 1 h =3600 s

frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz

energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J

power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W

charge coulomb (C = A bull s)

electric potential volt (V = kg bull m2 s3 bull A)

current ampere (A)

resistance ohm (Ω = kg bull m2s3 bull A2)

capacitance farad (F = s4 bull A2kg bull m2)

magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T

magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb

magnetic inductance henry (H = kg bull m2s2 bullA2)

temperature kelvin (K) degrees Celsius 0degC = 27315 K

angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad

41

Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg

931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton

2B e

em 9274 008 99 (37) x 10-24 JT

Bohr magneton μB 927 x 10-24 JT Bohr radius

0

2

2e em e k

a 5291 772 083 (19) x 10-11 m

Boltzmanns constant B A

RNk 1380 650 3 (24) x 10-23 JK

Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength

C eh

m c 2426 310 215 (18) x 10-12 m

Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u

electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg

5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2

electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg

1008 664 915 78 (55) u 939565 330 (38) MeVc2

neutron mass 1675 x 10-27 kg Nuclear magneton

2n p

em 5050 783 17 (20) x 10-27 JT

permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h

2h

6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js

Planckrsquos constant h ħ

6626 x 10-34 Js 1055 x 10-34 Js

Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2

proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a

These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits

NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table

42

INDEX

Absolute zero 9

Adiabatic 17

Adiabatic exponent 18

Adiabatic process 17

Alpha decay 46

Amplitude 4

Angle of reflection 25

Angular frequency 3

Angular magnification 30

Angular momentum atomic 43

Angular size 29

Archimedes Principle 1

Atomic spectra 43

Atoms 43

Avogadrorsquos number 10

Beats 8

Bernoullirsquos Equation 2

Beta decay 46

Bifocals 29

Brewsterrsquos angle 34

British Thermal Unit Btu 11

Buoyancy 2

calorie 11

Calorie 11

Camera 28

Carnot cycle 21

Carnot efficiency 22

Celsius scale 9

Ciliary muscle 29

Coefficient of performance 21

Compton effect 39

Constructive interference 7

Continuity equation of 2

Contraction length 35

Convection 14

Curved mirrors 27

Decay constant 46

Decay rate radioactive 46

Decibel scale 5

Degrees of freedom 16

Density 1

Destructive interference 7

Diffraction single slit 33

Diffraction grating 33

Dilation time 35

Disintegration energy 47

Dispersion of light 26

Doppler effect 6

Double slit interference 31

Efficiency 20

Electron interference 39

Emissivity 15

Energy nuclear decay 47

Energy relativistic 36

43

Engineering work 20

Entropy 22

Entropy ideal gas 24

Equipartition of energy 18

Equivalence principle 36

Exclusion principle 44

Expansion thermal 9

Expansion coefficient linear 9

Eye 29

Fahrenheit scale 9

Far point 29

Farsightedness 29

First Law of Thermodynamics 12

Fission 48

Flux volume 2

Free expansion 23

Fringes 31

Fusion 48

Fusion heat of 11

Gamma relativistic 35

Gamma decay 46

General relativity 36

Half-life 46

Heat 11

Heat capacity 11

Heat conduction 13

Heat engine 20

Heat of fusion 11

Heat of vaporization 11

Heat pump 20

Heats of transformation 11

Hydrostatics 1

Ideal Gas Law 10

Images real and virtual 27

Intensity sound 5

Interference 7

Interference two-slit 31

Internal energy 12

Internal energy degrees of freedom 16

Irreversible process 19 23

Isotopes 45

Joule 11

Kelvin scale 9

Kinetic energy relativity 36

Kinetic theory 16

Latent heat 11

Length contraction 35

Linear expansion coefficient 9

Linear polarization 34

Linear superposition 7

Longitudinal wave 3

Loudness 5

Magnification lateral 27

Magnifying glass 30

Malusrsquos law 34

Microscope 30

Momentum relativistic 36

Muscle ciliary 29

44

Musical instruments 8

Musical scale 8

Near point 29

Nearsightedness 29

Nonlinear 7

Nuclear energy 48

Nuclear properties 45

Nuclear stability 45

Octave 8

Optical resolution 33

Orbital quantum number ℓ 43

Orbitals 43

Particle in a Box 41

Particles are waves 39

Pascalrsquos Principle 1

Period 3

Photoelectric effect 38

Photons 38

Pitch 8

Polarization 34

Power sound 5

Presbyopia 29

Pressure 1

Principal quantum number n 43

Principle of equivalence 36

Principle of linear superposition 7

Probability amplitude ψ 41

Processes thermodynamic 13

R-value 14

Radiation thermal 14

Radioactive decay 46

Radioactive decay rate 46

Ray tracing 27

Rayleighrsquos criterion 33

Reading glasses 29

Real image 27

Refraction 25

Refrigerator 20

Relativistic gamma 35

Relativity principles 35

Resolved for light sources 33

Rest energy 36

Reversible process 19 22

Rope wave speed 4

Second Law of Thermodynamics 19

Shock waves 6

Simple magnifier 30

Simultaneity 35

Single slit diffraction 33

Snellrsquos law 25

Sound speed 5

Specific heat 11

Spin Angular momentum 44

Standing waves 7

Stefanrsquos law 14

Strong force 45

Telescope 30

Temperature 9

45

Temperature Scales 9

Thermal conductivity 14

Thermal energy 12

Thermal expansion 9

Thin film interference 31

Thin lenses 28

Time dilation 35

Tone musical 8

Total internal reflection 25

Transverse wave 3

Traveling Waves 3

Two-slit interference 31

Uncertainty principle 40

Vaporization heat of 11

Velocity addition relativity 36

Virtual image 27

Volume flux 2

Wave function 41

Wave Function ψ and Probability 41

Wave speed 3

Wave-particle duality 39

Wavelength 3

Wavenumber 3

Wien Displacement Law 16

Work 11