ch21

1

ElectrostaticsElectrostatics

Physics for Scientists & Engineers 2, Chapter 21 1

ElectrostaticsElectrostatics Physics 184Physics 184


Physics 184Physics 18421.1 Electromagnetism21.1 Electromagnetism

Electricity and MagnetismElectricity and Magnetism

Electricity and magnetism have been known for thousands of years

• The ancient Greeks knew that a piece of amber rubbed with fur would attract small, light objects

• The word for electron and electricity is derived from the Greek word for amber


from the Greek word for amber

• Naturally occurring magnetic materials called lodestones were used as early as 300 BC to construct compasses

The relationship between electricity and magnetism was not known until the middle of the 19th century

Fundamental Forces of NatureFundamental Forces of Nature


The Four ForcesThe Four Forces

In our model of the world, the four fundamental forces work by exchanging elementary particles• Gravity - graviton

(has not been observed yet)

• Electromagnetic - photon

• Weak - W and Z bosons


(observed in 1983)

• Strong – gluons(observed in 1979)

Thus forces can act a distance without touching• The Sun can attract the Earth from 93 million miles away

• Magnet can attract metal

Gravitational and Electric ForcesGravitational and Electric Forces

For gravity we defined a gravitational force

and a gravitational potential

F(r)=G

m1m2

r2


We will do the same for the electric force and the electric potential

We will introduce the concept of an electric field to help us understand the electromagnetic force

U(r)=-G

m1m2

r

2

Physics 184Physics 184


Physics 184Physics 18421.2a Electric Charge21.2a Electric Charge

Electric ChargeElectric Charge Everyday example: When walking on a carpet on a dry

winter’s day and then touching a door knob, one often experiences a spark• This process is called charging

• Charging: negatively charged electrons move from the atoms and molecules of the carpet to the soles of our shoes, to the body

• A spark occurs when the built-up charge discharges through


A spark occurs when the built up charge discharges through the metal of the door knob.

Similar phenomenon involving wind, rain and ice produces lightning

Electric ChargeElectric Charge

Normally objects around us do not seem to carry a net charge

They have equal amounts of positive and negative charge and thus are electrically neutral• Negative charge means an excess of electrons• Positive charge means a deficit of electrons

If a plastic rod is rubbed with fur, the rod will become


p ,charged• If two charged plastic rods together, they will repel each

other If a glass rod is rubbed with silk, the rod will become

charged• If we bring together a charged plastic rod and a charged

glass rod, they will attract each other

This result leads to the Law of Electric Charges

The unit of charge is the coulomb, abbreviated C• named after Charles-Augustin de Coulomb (1736 – 1806)

The coulomb is defined in terms of the SI unit for

Law of Electric ChargesLaw of Electric Charges

Like charges repel and opposite charges attract.Like charges repel and opposite charges attract.Like charges repel and opposite charges attract.

electric current, the ampere, abbreviated A• named after Andre-Marie Ampere (1775 – 1836)

The ampere is a basic SI unit like the meter, the second, and the kilogram.

The unit of charge is defined as


1 C = 1 A s1 C = 1 A s

Charge of an ElectronCharge of an Electron We can define the unit of charge in terms of the charge

of one electron

An electron is an elementary particle with charge q = -e where

• e = 1.602·10-19 C

• A proton is a particle with q = +e

A coulomb is a large amount of charge

Typically we deal with smaller amounts of charge


Typically we deal with smaller amounts of charge

• 1 microcoulomb = 1 μC = 1.0·10-6 C

• 1 nanocoulomb = 1 nC = 1.0·10-9 C

• 1 picocoulomb = 1 pC = 1.0·10-12 C

The number of electrons required to make a coulomb is

• Ne = 1 C/(1.602·10-19 C per electron) = 6.242·1018 electrons

Charge ConservationCharge Conservation Benjamin Franklin (1706 - 1790) introduced the idea of

positive and negative charge (amber or plastic is negative)

Franklin also proposed that electric charge is conserved

When a plastic rod is charged by rubbing it with a fur, charge is neither created nor destroyed, but instead electrons are transferred to the rod leaving a net positive charge on the fur


transferred to the rod leaving a net positive charge on the fur

Law of Charge Conservation

This law adds to our list of conservation laws• Conservation of energy• Conservation of momentum• Conservation of angular momentum

The total charge is constant.The total charge is constant.

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Physics 184Physics 184Example: Muscle TwitchExample: Muscle Twitch

Muscle TwitchMuscle Twitch

PROBLEM

A current of 5.0 mA is enough to make your muscles twitch. How many electrons flow through your skin if you are exposed to such a current for 10.0 s?

SOLUTION

3 C


3 C5.0 mA 5.0 10

s-= ⋅

( ) 3 C10.0 s 5.0 10 0.050 C

s-æ ö÷ç ⋅ =÷ç ÷çè ø

1719

1 electron0.050 C 3.1 10 electrons

1.602 10 C- = ⋅⋅



Physics 184Physics 18421.2b Elementary Charge21.2b Elementary Charge

Elementary ChargeElementary Charge

Electric charge is quantized

The smallest charge observable is the charge of anelectron

Established by Robert Millikan (1868 - 1953) in his famous


(1868 - 1953) in his famousoil drop experiment

In everyday life, we don’tnotice that charge is quantized because most electrical phenomena involve a large number of electrons

Structure of AtomsStructure of Atoms

Atoms are electrically neutral

Atoms are composed of a positively charged atomic nucleus surrounded by negative electrons


electrons The atomic nucleus is

composed of positively charge protons and electrically neutral neutrons

The number of protons is the same as the number of electrons

Description of AtomsDescription of Atoms

Atomic number = ZMass number = A# electrons = Z (charge = -Ze)# protons = Z (charge = +Ze)# neutrons = N = A – ZFor example, 12C has


6 protons6 neutrons6 electrons

Atomic mass = ZMp + NMn

ZMe – binding energy/c2

Atomic mass ≈ AMp

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ParticlesParticles

The electron is an elementary particle

The proton is composed of chargedparticles called quarks held together byuncharged particles call gluons• Quarks have charge ±(1/3)e and ±(2/3)e

• Quarks have never been observed directlyQuarks have never been observed directly

The proton is composed of two up quarkseach with charge +(2/3)e and one downquark with charge –(1/3)e

The neutron is composed of one up quark with charge +(2/3)e and twodown quarks with charge –(1/3)e




Physics 184Physics 184Example: Net ChargeExample: Net Charge

Net ChargeNet Charge

PROBLEM

Suppose we want to create a positive charge of 10.0 μC on a block of copper metal with mass 2.00 kg. What fraction of the electrons in the copper block would we remove?

SOLUTION


The atomic weight of copper is 63.55 grams per mole

The number of copper atoms is

Natom =2.00 kg( ) 6.02⋅1023 atoms/mole( )

0.06355 kg/mole=1.89⋅1025

Net ChargeNet Charge

eq eN=

The atomic number of copper is 29, which means

there are 29 electrons per atom

Ne = 29⋅Natom = 29⋅1.89⋅1025 =5.49⋅1026

The number of electrons in 10 C is thus


13e -11

26e

Percentage removed is

6.24 10% 100 100 1.14 10 %

5.49 10

N

N ⋅

= ⋅ = ⋅ = ⋅⋅

613

e -19

10 10 C6.24 10

1.602 10 C

qN

e

-⋅= = = ⋅

⋅



Physics 184Physics 18421.3 Insulators and Conductors21.3 Insulators and Conductors

Insulators and ConductorsInsulators and Conductors

The electronic structure of materials determines their ability to conduct electricity• “Conducting electricity” means the transport of electrons

Materials that conduct electricity well are called conductors• Electrons can move freely (some of the electrons)

M t l


• Metals• Water with dissolved materials

Materials that conduct electricity poorly are called insulators• Electrons cannot move freely

• Glass• Plastic• Cloth• Pure water

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SemiconductorsSemiconductors Semiconductors are materials that can be switched

between being an insulator and being a conductor

Semiconductors are the backbone of modern electronics and computers


Replica of first transistor in 1947

Modern computer chip with millions of transistors

SuperconductorsSuperconductors

Superconductors are materials that have no resistance to the conduction of electricity as opposed to normal conductors that conduct electricity well but with some losses

A typical superconductor is a niobium-titanium alloy that must be kept near the temperature of liquid


that must be kept near the temperature of liquid helium (4.2 K)

During the last 20 years, high temperature superconductors have been developed that operated at the temperature of liquid nitrogen (77.3 K)

Material that are superconductors at room temperature would be very useful

Applications of SuperconductorsApplications of Superconductors

The main application of superconductors is to produce electromagnets made with superconducting wire

Examples include• Magnetic resonance imaging (MRI) machines• Particle accelerators

• MSU’s K1200 Superconducting Cyclotron


• MSU’s upcoming superconducting LINAC (linear accelerator) for FRIB

• Tevatron at Fermilab near Chicago, Illinois• RHIC at Brookhaven National Laboratory on Long Island,

NY• LHC at CERN near Geneva, Switzerland

Magnetic Resonance Imaging - MRIMagnetic Resonance Imaging - MRI

MRI stands for nuclear magnetic resonance imaging

MRI produces high quality images of living tissue without causing any damage

The quality of an MRI image (signal to noise) is proportional to

Magnetic Field = 1.5 T

Yue Cao, Stephen Whalen, Jie Huang, Kevin L. Berger, and Mark C. DeLanHuman Brain Mapping 20:82–90(2003). (MSU Radiology)


(signal-to-noise) is proportional to the the magnitude of the magnetic field• High field means high quality images

Superconducting magnets can produce up to four times the magnetic field of a room-temperature magnet

Magnetic Field = 3.0 T



Physics 184Physics 18421.4 21.4 Electrostatic ChargingElectrostatic Charging

Electrostatic ChargingElectrostatic Charging

Giving a static charge to an object is calledelectrostatic charging

We will approach our study of electrostatic charging through a series of simple experiments

A power supply or a battery can provide positive and negative charge


negative charge

Insulating paddles can be positively or negatively charged

We will make a conducting connection to the Earth• We call this connection grounding

• The Earth is a nearly infinite reservoir of charge and neutralizes electrically charged objects connected to it

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ElectroscopeElectroscope

An electroscope gives an observable response when it is charged

This electroscope has two conductors that are initially vertical and touching when thevertical and touching when the electroscope is uncharged

When the electroscope is charged, the hinged conductor will move away from the fixed conductor

Note that we cannot tell the sign of the charge


Inducing a Negative ChargeInducing a Negative Charge

If we bring a negatively charged paddle near an electroscope, the electrons are repelled from the ball, inducing a negative charge on the conductors


Charging by ContactCharging by Contact

If the negatively charged paddle touches the electroscope, electrons will flow from the paddle to the conductors, producing a net negative charge


Positive Charging by InductionPositive Charging by Induction

We can create a positive charge• Starting with an uncharged electroscope

• Bringing a negatively charged paddle close to the electroscope

• Connecting a ground to the electroscope

• Removing the ground

• Taking the paddle away


QuestionQuestion

Two lightweight metal spheres are suspended near each other from insulating threads. One sphere has a net charge; the other sphere has no net charge. The spheres will

A. attract each other.

B exert no net electrostatic force on each otherB. exert no net electrostatic force on each other.

C. repel each other.

D. do any of these things depending on the sign of the charge on the one sphere.




Physics 184Physics 18421.5a Electrostatic Force 21.5a Electrostatic Force –– Coulomb’s LawCoulomb’s Law

7

Electrostatic Force – Coulomb’s LawElectrostatic Force – Coulomb’s Law The law of electric charges is evidence of a force

between any two charges at rest

Experiments show that for the electrostatic force exerted by charge 2 q2 on charge 1 q1, the force on q1

points toward q2 if the charges have opposite signs and away from q2 if the charges have like signs


Electrostatic Force – Coulomb’s LawElectrostatic Force – Coulomb’s Law

Coulomb’s Law gives the magnitude of this force as

k is Coulomb’s constant given by

1 2

2

1 2

1 2

and are electric charges

is the distance between the charges

q qF k

rq q

r r r

=

= -

We can relate Coulomb’s constant to theelectric permittivity of free space ε0


k = 8.99⋅109

N m2

C2

k =

1

40

0 = 8.85⋅10-12 C2

N m2

Electrostatic Force VectorElectrostatic Force Vector

Coulomb’s Law can be written in vector form as

( )1 22 1 2 13

q qF k r r

r

= - - 1 2

212ˆ

q qk r

r= -


Superposition PrincipleSuperposition Principle

Consider the force exerted on charge q3 by two other charges q1 and q2


( )1 3

1

1 3 23

ˆ q q

k xx x

F

= -- ( )

22 3

32

3 2

ˆ q q

k xx x

F

= --

1 3net 23 3F FF

= +



Physics 184Physics 184Example: Force between Two ChargesExample: Force between Two Charges

Force between Two Charges Force between Two Charges

You place two charges a distance r apart. Then you double each charge and double the distance between the charges. How does the force between the two charges change?

A. The new force is twice as large.

B The new force is half as largeB. The new force is half as large.

C. The new force is four times as large.

D. The new force is four times smaller.

E. The new force is the same.


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Force between Two Charges (2)Force between Two Charges (2)

PROBLEM

What is the magnitude of the force between two 1.00 C charges 1.00 m apart?

SOLUTION

1 2

2

q qF k

r=


This force is approximately the same as 450 loaded Space Shuttles!

( )( )

229

22

9

1.00 CN m8.99 10

C 1.00 m

8.99 10 N

F

F

æ ö÷ç ÷= ⋅ç ÷ç ÷çè ø

= ⋅

r Physics 184Physics 184


Physics 184Physics 184Example: Electrostatic Force inside the AtomExample: Electrostatic Force inside the Atom

Electrostatic Force inside the AtomElectrostatic Force inside the Atom PROBLEM 1

What is the magnitude of the electrostatic force between the two protons in the helium nucleus?

SOLUTION 1

The two neutrons and protons inside the helium nucleus are held together by the strong forcenucleus are held together by the strong force

The two protons each have charge +e and are separated by a distance of 2.0·10-15 m

The electrostatic force of repulsion between the two protons is


F= kq1q2

r2= 8.99⋅109

N m2

C2

æ

èççç

ö

ø÷÷÷÷

1.60⋅10-19 C( )2

2.0⋅10-15 m( )2 =58 N

Electrostatic Force inside the Atom(2)Electrostatic Force inside the Atom(2) PROBLEM 2

What is the magnitude of the electrostatic force between a gold nucleus and an electron of the gold atom in an orbit with radius 4.88·10–12 m?

SOLUTION 2

The attractive force between the negative electron and gthe positive gold nucleus is


F= kq1q2

r2= k

e Ze( )r2

= kZe2

r2

F= 8.99⋅109 N m2

C2

æ

èççç

ö

ø÷÷÷÷

79( ) 1.60⋅10-19 C( )2

4.88⋅10-12 m( )2 = 7.63⋅10-4 N



Physics 184Physics 184Example: Equilibrium PositionExample: Equilibrium Position

Equilibrium PositionEquilibrium Position

PROBLEM

Two charged particles are placed as shown in the figure:q1 = 0.15 μC is located at the origin, and q2 = 0.35 μC is located on the positive x-axis at x2 = 0.40 m. Where should a third charged particle, q3, be placed to be at an equilibrium point (the forces on it sum to zero)?


9


SOLUTION

Let’s figure out where not to put the third charge

If the charge is placed anywhere off the x-axis, there will be a component of the force toward or away from the x-axis• Third charge must lie on the x-axis

Look at three regionsx < 0, 0 < x < x2, x > x2



If x < 0 or x > x2 then the force on the third charge from the the other two charges would be in the same direction regardless of the sign of the third charge• No equilibrium possible

If 0 < x < x2 then the force on charge three from the other two charges would be in the opposite direction regardless of the sign of the third charge• Equilibrium possible



rF1 3 =

rF2 3 k

q1q32 = k

q2q32


F13 F23 kx3 -x1( )2 k

x2 -x3( )2

q1

x3 -x1( )2 =q2

x3 -x2( )2 q1 x2 -x3( )2 = q2 x3 -x1( )2

q1 x2 -x3( )= q2 x3 -x1( ) q1 x2 - q1 x3 = q2 x3 - q2 x1

q1 x3 + q2 x3 = q1 x2 + q2 x1


q x + q x


x3 =q1 x2 + q2 x1

q1 + q2

x3 =0.15⋅10-6 C 0.40 m( )

0.15⋅10-6 C + 0.35⋅10-6 Cx3 = 0.16 m



Physics 184Physics 184Solved Problem 21.1Solved Problem 21.1

Charged BallsCharged Balls


PROBLEM Two identical charged balls

hang from the ceiling by insulated ropes of equal length, = 1.50 mA h 25 0 C i li d A charge q = 25.0 μC is applied to each ball

Then the two balls hang at rest, and each supporting rope has an angle of 25.0° with respect to the vertical

What is the mass of each ball?


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SOLUTIONTHINK

Each ball has three forces acting on it• Force of gravity• Repulsive electrostatic force• Tension in the supporting rope

The forces must sum to zero


e esin 0 sinT F T F - = =

The forces must sum to zeroSKETCH

A free body diagram of one of theballs is shownRESEARCH

The sum of the x-components of the forces gives us


The sum of the y-components of the forces gives us

The electrostatic force is given by

The force of gravity is given by

g gcos 0 cosT F T F - = =

2

e 2

qF k

d=

F mg=

The distance between the balls is given by

The electrostatic force between the balls is


gF mg=

/ 2sin

2

d d

= =

( )

2 2

e 2 2 24 sin2 sin

q qF k k

= =


SIMPLIFY We divide two force component equations to get

Substitute in our equations for Fe and Fg

e e

g g

sin tan =

cos

T F F

T F F

=

2q

CALCULATE


22 2

2 24 sintan =

4 sin tan

qk kq

mmg g

=

( )

( )( )

229 6

2

22 2

N m8.99 10 25.0 10 C

C0.764116 kg

4 9.81 m/s 1.50 m sin 25.0 tan 25.0m

-æ ö÷ç ÷⋅ ⋅ç ÷ç ÷çè ø

= =


ROUND

DOUBLE-CHECK To double-check our results, let’s make the small angle

approximation that sinθ ≈ tanθ ≈ θ and cosθ ≈ 1

0.764 kgm =

2 2


( )

( )

( ) ( )

2 2

e 22

229 6

22

2 32 3

sin2

N m8.99 10 25.0 10 C

C0.768 kg

4 4 1.50 m 0.436 rad

q qT mg F k k

d

kqm

g g

-

» = = »

æ ö÷ç ÷⋅ ⋅ç ÷ç ÷çè ø= = =



Physics 184Physics 184Solved Solved Problem 21.2Problem 21.2

Four ChargesFour Charges

Four ChargesFour ChargesPROBLEM

Consider four charges placed at the corners of a square with sides of length 1.25 m as shown on the right

What is the magnitude of the electric force on q4 resulting from the electric force from the remaining three charges?


charges?

SOLUTIONTHINK

The electrostatic force on q4 is the vector sum of the forces resulting from its interactions with the other three charges

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We need to determine the individual force components in each spatial direction and add those to find the components of the net force vector

SKETCH



RESEARCH

The net force on q4 is the vector sum of the forces

The x-component of these forces is1 4 2 4 3 4, ,F F F

Fx kq1q4

2 kq2q4

2cos45

kq42 q1

q2 cos45


The y-component of these forces is

The magnitude and angle of the net force are

Fy kq2q4

2d 2sin 45 k

q3q4

d 2 kq4

d 2

q2

2sin 45q3

Fx kd 2 k

2d 2cos45

d 2 q1 2cos45

F Fx2 Fy

2 tan Fy / Fx


SIMPLIFY

Putting in our expressions for the components of the force

Fkq4

d2 q1 q2

2cos45

2

kq4

d 2

q2

2sin 45q3

2

2

2


For the angle of the force we have

tan1Fy

Fx

tan1

kq4

d 2

q2

2sin 45q3

kq4

d 2 q1 q2

2cos45

tan1

q2

2sin 45q3

q1 q2

2cos45

F

kq4

d 2 q1 q2

2cos45

2

q2

2sin 45q3

2


CALCULATE

Putting in our numerical values, we get

2 2 2.50 Csin 45 cos 45 0.8839 C

2 2 2 2

q q

9 2 2

2 2

2

8.99 10 N m /C 4.50 C1.50 C 0.8839 C 0.8839 C 3.50 C

1.25 mF


ROUNDF 0.0916 N

47.7

0.09164 NF

1 0.8839 C 3.50 Ctan 47.66

1.50 C 0.8839 C


DOUBLE-CHECK

To double-check our result, we calculate the magnitude of the three forces acting on q4

F14 kq1q4

r142 8.99109 N m2 /C2 1.50 C 4.50 C

1.25 m 2 0.0388 N

F kq2q4 8 99 109 N 2 /C2 2.50 C 4.50 C

0 0324 N


The angle looks reasonable because the force points downward and to the right

F24 kq2q4

r242 8.99109 N m2 /C2

2 1.25 m 2 0.0324 N

F34 kq3q4

r342 8.99109 N m2 /C2 3.50 C 4.50 C

1.25 m 2 0.0906 N



Physics 184Physics 18421.5b Electrostatic Applications21.5b Electrostatic Applications

12

Electrostatic PrecipitatorElectrostatic Precipitator

An application of electrostatic charging and electrostatic forces is the cleaning of emissions from coal-fired power plants

An electrostatic precipitatorremoves ash and particulates


Laser PrinterLaser Printer

A laser printer uses electrostatic forces to print


Laser PrinterLaser Printer

Any spot the laser strikes onthe drum is discharged, allowing toner to produce an image


Completely charged drum



Physics 184Physics 18421.6 Coulomb’s Law and Newton’s21.6 Coulomb’s Law and Newton’s

Law of GravitationLaw of Gravitation

Forces between ElectronsForces between Electrons What is relative strength of the electric force

compared with the force of gravity for two electrons?

Fe

Fg

kqe

2

Gme2

(8.99109 N m2 / C2 )(1.6021019 C)2

(6.67 10-11 N m2 /kg2 )(9.10910-31 kg)2 4.21042

Fe = kqe

2

r2

Fg =Gme

2

r2


Gravity is irrelevant for atomic and subatomic processes – the electric force is much muchstronger

But sometimes gravity is most important; e.g, the motion of the planets

r