16.0 serway physics

8/9/2019 16.0 Serway Physics

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Introduction

Electric Potential Energy (PE)

• A conservative force potential energy (PE)

conservative force= - gradient of potential energy

conservativeF PE = −∇

• Gravity force near the Earth surface:gF mg=

• Gravity potential energy near the Earth surface :

gPE mgh=

h mg

A ball is exerted by the

gravity force and has

the potential energy ofmgh.

ground


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Introduction

Electric Potential Energy (Cont.)

• The Coulomb force is a conservative force.

• Work done by a conservative force is equal to thenegative of the change in potential energy.

work= - change of potential energy

ABW PE = −∆

• It is possible to define an electrical potential energyfunction with this force. The concept of electric potential energy is useful in the study of electricity

eF PE ⇔


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Work and Potential Energy (Cont.)

• The change of the electric potential energy is

independent of the path but the distance

ΔPE (path 1)= ΔPE (path 2)

Section 16.1

A B

Path 1

Path 2

• The Electric potential energy is a scalar quantity(magnitude), not a vector (magnitude and direction).


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Comparison between electric and gradational potential energy

0

0

AB e

e

e

W F d qEd

PE

PE

= =

= −∆ >

∆ <

0

0

AB g

g

g

W F d mgd

PE

PE

= =

= −∆ >

∆ <

Section 16.1

Electric potential energy differs significantly from gradationalpotential energy

—There are two kinds of electrical charges, positive and negative

— mass has no negative sign.


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Work and Potential Energy -Example

• There is a uniform field

between the two plates.

• As the charge moves from A to B, work is done by

electric force on it.

W AB = F x Δ x =q E x ( x f – x i)

=-ΔPE

Section 16.1


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Work and Potential Energy -Example (Cont.)

•

Note: E x and q can be positive ornegative

• The work-energy theorem―change in electric potential

energy (SI unit: Joule (J))

Section 16.1

0 0, 0,

0 0, 0,

0 0, 0,

0 0, 0,

x

x

AB x

x

x

if q E

if q E W qE x

if q E

if q E

> > >> <

ABPE W ∆ = −


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Potential Energy Differences in an Electric Field

Section 16.1

• A proton is released from rest. The

change in the electric potentialenergy associated with the proton:

ΔPE=-WAB=-F eΔy=-(eE Δy) < 0

Proton is released

E

A

B

∆y

+e

y

e

F eE

=

• An electron is fired in the samedirection. The change in the electricpotential energy associated with the

electron:ΔPE=-WAB=- (F eΔy)=-((-eE)Δy) > 0

An electron is fired in

the same direction

E

A

B

∆y

-e

y

eF eE = −


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• When an electron is released fromrest in a constant electric field, thechange in the electric potentialenergy associated with the electron

become more negative with time.

―True or false?

Section 16.1

ΔPE = - WAB = -((-e)(-E))∆

y < 0 (True)

An electron is released

E

A

B

∆y

-e

y

Potential Energy Differences in an Electric Field

eF eE = −


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• When an electron is fired in a uniformelectric field, find the initial speed v0 if its final speed has fallen by half.

• Apply conservation of energy,

Section 16.1

An electron is fired.

E

A

B

∆y

-e

y

Dynamics of Charged Particles

( ) ( )

2 2

2

2

2

1 10 0

2 2

1 1 10

2 2 2

8 ( ) 83 8

8 3 3 3

e f e i

e i e i

e i i

e e e

KE PE m v m v PE

m v m v PE

e E y eE yPE m v PE v

m m m

∆ + ∆ = → − + ∆ =

− + ∆ =

− − ∆ ∆∆

− = −∆ → = = =

eF eE = −


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Electric Potential and Potential Difference

• A field the field potential

―independent of the object in the field

• Gravitational field Gravitational potential

―independent of the object with mass m in the field

• Electric field Electric potential

―independent of the object with charge q in the field

Section 16.1


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Electric Potential Difference

• The electric potential difference ΔV between points A

and B is defined as the change in the potential energy(final value minus initial value) of a charge q moved from

A to B divided by the size of the charge.

ΔV = VB – V

A = ΔPE / q

SI unit: J/C or V (Volt)

Section 16.1

• Alternately, the electric potential difference is the work

per unit charge that would have to be done by some

force to move a charge from point A to point B in theelectric field.

• Potential difference is not the same as potential energy.


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Potential Difference (Cont.)

• Another way to relate the potential energy differenceand the potential difference:

ΔPE = q ΔV

• Both electric potential energy and potential

difference are scalar quantities.

• A special case occurs when there is a uniform electric field .

∆V =ΔPE/q=-F e

Δx/q= -qEx

Δx/q= -E x ∆x

– Gives more information about units: N/C = V/m

since F e/q=E x=ΔV/Δx

Section 16.1


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Potential Energy Compared to Potential

• Electric potential is characteristic of the field only.

– Independent of any test charge that may be

placed in the field

• Electric potential energy is characteristic of thecharge-field system.

– Due to an interaction between the field and the

charge placed in the field

Section 16.1


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Electric Potential and Charge Movements

• When released from rest, positive charges acceleratespontaneously from regions of high potential to low potential.

ΔV =-E Δx=VB-VA0Δx


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Electric Potential of a Point Charge

• The point of zero electric potential is taken tobe at an infinite distance from the charge.

• The potential created by a point charge q at

any distance r from the charge is

Section 16.2

e

qV k

r =

0,

B A

B B

A

V V V

V when r

V V

∆ = −

→ → ∞

= −∆

VA VB=0

Ehigh low

r →∞


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Electric Field and Electric Potential Depend on

Distance

• The electric field is a vector

and is proportional to 1/r2

• The electric potential is a

scalar and is proportional

to 1/r

Section 16.2

2

0

ˆe

F q E k r

q r

≡ =

e

qV k

r =


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Electric Potential of Multiple Point Charges

• Superposition principle applies

• The total electric potential at some point P due to

several point charges is the algebraic sum of the

electric potentials due to the individual charges.

– The algebraic sum is used because potentials arescalar quantities.

Section 16.2

1 2 31 2 3

1 2 3

1 2 3

; ; ; ...

...

e e eP P P

P P P

P P P P

k q k q k qV V V

r r r

V V V V

= = =

= + + +

• P

q1

q2

q3

r1P r2Pr3P


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Dipole Example

• Two charges have equalmagnitudes and oppositecharges.

• Example of superposition

• Discussion:

Section 16.2

1 2

1 2

2 11 2

1 2

;e eP P

P P

P PP P P e

P P

k q k qV V

r r

r r V V V k q

r r

= = −

−= + =

q1=q

q2=-q

Pr1Pr2P

Potential of a dipole plotted on thevertical axis (in arbitrary units).

1

2

1 2

1 2

0;

0;

0 ;

0

P P

P P

P P P

P P P

V if r

V if r

V if r r

V if r and r

→ ∞ →

→ −∞ →

→ =

→ → ∞


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• The work required to bring q2

from infinity to P withoutacceleration (i.e., F ex=F e) is

Electrical Potential Energy of Two Charges

• V1 is the electric potential due to

q1 at some point P

Section 16.2

2 2 1 2 1( )exW PE q V q V V q V ∞∆ = ∆ = ∆ = − =

1 2

2 1 ; 0

e

q qPE q V k PE

r ∞= = =

1

1 e

qV k

r =

• This work is equal to the potentialenergy of the two particle system


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Electric Potential Energy of Two Charges (Cont.)

• If the charges have the same sign, PE is positive.

– Positive work must be done to force the twocharges near one another.

– The like charges would repel.

• If the charges have opposite signs, PE is negative.

– The electric force would be attractive. – Work must be done to hold back the unlike

charges from accelerating as they are broughtclose together.

Section 16.2

1 21 2

1 2

0 00 0

e for q qq qPE k for q qr

> >= <


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Electric Potential and Potential Energy (Ch 16.1)

ΔPE = q ΔV WAB=-ΔPE

For q>0: ΔPE 0: ΔPE >0 (PE A >PE B)

(gains energy)

For q


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Section 16.2

Brief Summary Of Potential and Potential Energy

(cont.)

O

● q>0

• If a positive charge is released in

the electric field, it will move

towards to lower electric potential.

q


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Problem Solving with Electric Potential

(Point Charges)

• Draw a diagram of all charges.

– Note the point of interest.

• Calculate the distance from each charge to the pointof interest.

• Use the basic equation V = k eq/r

– Include the sign

– The potential is positive if the charge is positive

and negative if the charge is negative.

Section 16.2

• P

q1

q2

q3

r1P r2P

r3P


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Problem Solving with Electric Potential (Cont.)

• Use the superposition principle when you have

multiple charges.

– Take the algebraic sum

• Remember that potential is a scalar quantity.

– So no components to worry about

Section 16.2

1 2 3 ...P P P PV V V V = + + + • P

q1

q2

q3

r1P r2Pr3P


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Conductors in Equilibrium

• All of the charge resides at the

surface.• E = 0 inside the conductor.

• The electric field just outside theconductor is perpendicular to

the surface.

Section 16.3

0 0

inside einside

inside

W F lPE V

q q q

qE l lq

∆ ⋅ ∆∆∆ = = − = −

⋅ ∆= − = − ⋅ ∆ =

• V inside=constant 0 E =

l∆

A

•

•

B

• The potential everywhere inside the conductor isconstant and equal to its value at the surface (why?).


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Potentials and Charged Conductors

• All points on the surface of a chargedconductor in electrostatic equilibriumare at the same potential.

Section 16.3

cos90 0

0

surface e surface surface

surface

W F l qE l E l

PE W V

q q

∆ = ⋅ ∆ = ⋅ ∆ = ∆ =

∆∆ = = − =

V surface=constant l∆

•The electric potential is constant everywhere on thesurface of a charged conductor in electrostaticequilibrium and equal to its value at the inside.

V inside=V surface


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More Information About the Unit for Electric

Potential Energy: The Electron Volt (eV)

• The electron volt (eV) is defined as the kineticenergy that an electron gains when accelerated through a potential difference of 1 V.

— PE = eΔV =1 eV = 1.6 x 10-19 J

—Electrons in normal atoms have energies of 10’sof eV.

—Excited electrons have energies of 1000’

s of eV.—High energy gamma rays have energies of millionsof eV.

Section 16.3


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Equipotential Surfaces

•

An equipotential surface is a surface on whichall points are at the same potential.

Section 16.4

0 B AV V V ∆ = − =

—No work is required to move acharge at a constant speed on anequipotential surface.

—The electric field at every pointon an equipotential surface isperpendicular to the surface.


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Equipotential surface

E

Equipotentials and Electric Fields Lines –

Positive Charge

• The equipotentials for a

point charge are a family of

spheres centered on the

point charge.

– In blue

• The field lines are

perpendicular to theelectric potential at all

points.

– In orangeSection 16.4


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Equipotentials and Electric Fields Lines – Dipole

• Equipotential lines are

shown in blue.

• Electric field lines are

shown in orange.• The field lines are

perpendicular to the

equipotential lines at all

points.

Section 16.4


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d

Equipotentials in the Two Parallel Plates

Section 16.4

y

•

A

•

B

WAB=qE Δxcos90˚=0

Δx

ΔV=VA-VB=0

• Equipotential lines are shown in blue.

• Electric field lines are shown in orange.

• The field lines are perpendicular to the

equipotential lines at all points.


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Application – Electrostatic Precipitator

• It is used to remove

particulate matterfrom combustion gases

• Reduces air pollution

•Can eliminateapproximately 90% by

mass of the ash and

dust from smoke

• Recovers metal oxides

from the stack

Section 16.5


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Application – Electrostatic Precipitator (Cont.)

• The wire is maintained at a

negative electric potential withrespect to the outer wall, so the

electric field is directed toward

the wire.

• The electric field near the wire

reaches a high enough value to

cause a discharge around the wire

and the formation of positiveions, electrons, and negative ions.

Section 16.5

E

+q/-q


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Application – Electrostatic Precipitator (Cont.)

•These negative charge particlesaccelerate and the dirt particles

become charged negatively by

collision and ion capture.

• These negatively charged dirt

particles are drawn to the outer

wall.

•

When the duct is shaken, theparticles fall loose and are

collected at the bottom

E

+q/-q


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Application – Electrostatic Air Cleaner

• Used in homes to reduce the discomfort of allergy

sufferers.

• It uses many of the same principles as the

electrostatic precipitator.

Section 16.5


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Application – Xerographic Copiers

• The process of xerography is used for making

photocopies.

• Uses photoconductive materials

– A photoconductive material is a poor conductorof electricity in the dark but becomes a good

electric conductor when exposed to light.

Section 16.5


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h h ( )


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The Xerographic Process (Cont.)

•The negatively charged powder, called toner, is dusted

onto the photoconductor and adheres only to the

areas that contain the positively charged image (but

not neutralizes the positively charge image).

• It is then transferred to the surface of a sheet of

positively charged paper

• Finally, the toner is ‘fixed’ to the surface of the paper

by heat, resulting in a permanent copy of the original.

Section 16.5


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Application – Laser Printer

• The steps for producing a document on a laser printer

is similar to the steps in the xerographic process. – Steps a, c, and d are the same.

– The major difference is the way the image formson the selenium-coated drum.

• A rotating mirror inside the printer causes thebeam of the laser to sweep across the selenium-coated drum.

• The electrical signals form the desired letter in

positive charges on the selenium-coated drum.• Toner is applied and the process continues as in

the xerographic process.

Section 16.5


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Capacitance

Section 16.6

• Two parallel plates: ΔV=Ed=Qd/ ( A0ε0)

d

Q=σA0

• The capacitance, C, of a capacitor is defined as the

ratio of the magnitude of the charge on either

conductor (plate) to the magnitude of the potential

difference between the conductors (plates).

SI Units: Farad (F) – 1 F = 1 C / V

– A Farad is very large

– Often will see µF or pF

• Q/ ΔV=ε0 A0/d : characterize the

conductor and is independent of

Q and ΔV, called capacitance.

ΔV =Ed


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Parallel-Plate Capacitor

• The capacitor consists of twoparallel plates.

• Each has area A.

• They are separated by a distance d .

• The plates carry equal andopposite charges.

• When connected to the battery,

charge is pulled off from one plateand transferred to the other plate.

• The transfer stops when ∆Vcap =∆Vbattery

Section 16.7

• A capacitor is a device used in a variety of electric circuits.

-e


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Parallel-Plate Capacitor (Cont.)

• The capacitance of a device depends on the

geometric arrangement of the conductors.• For a parallel-plate capacitor whose plates are

separated by air:

Section 16.7

( ) 00/Q A A A

C V Ed d d

σ σ

ε σ ε = = = =∆

• The larger area can store more charge, therefore

enhances the capacitance.• The small plate separation can decrease the potential

difference and hence enhances the capacitance.


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Electric Field in a Realistic Parallel-Plate Capacitor

• The electric field between the plates is uniform.

– Near the center

– Nonuniform near the edges

•The field may be taken as constant throughout theregion between the plates.

Section 16.7


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Capacitors in Parallel

• When capacitors are first connected in the circuit,electrons are transferred from the left plates throughthe battery to the right plate, leaving the left platepositively charged and the right plate negativelycharged.

Section 16.8

-e -e

-e-e

+Q 1 -Q 1

+Q 2 -Q 2

• The capacitors reach their maximumcharge when the flow of chargeceases (completely charged).

• The flow of charges ceases whenthe voltage across the capacitorsequals that of the battery.

ΔV1

ΔV2

ΔV1=ΔV2 =ΔV


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Capacitors in Parallel (Cont.)

•

The potential differenceacross the capacitors is thesame:

– and each is equal to the

voltage of the batteryΔV1=ΔV2 =ΔV

Section 16.8

ΔV1=ΔV2

• The total charge, Q, isequal to the sum of thecharges on the capacitors.

Q = Q 1 + Q 2

ΔV1

ΔV2


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•

The capacitors can bereplaced with one capacitor

with a capacitance of Ceq

Section 16.8

1 2 1 1 2 2Q Q Q C V C V = + = ∆ + ∆

1 2( ) eqQ C C V C V = + ∆ = ∆

1 2eqC C C = +

–

The equivalent capacitormust have exactly the

same external effect on

the circuit as the original

capacitors.


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• Ceq = C1 + C2 + …

• The equivalent capacitance of a parallel

combination of capacitors is greater than any of

the individual capacitors.

Section 16.8

• Q = Q 1 + Q 2+…

• ΔV1=ΔV2 =…=ΔV


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Capacitors in Series

Section 16.8

• When a battery is connected to the circuit, electrons

are transferred from the left plate of C1 to the right plate of C2 through the battery.

-e-e

• The magnitude of the

charge must be the sameon the left plate of C1 and

the right plate of C2.

• As this negative charge accumulates on the right plate of C2, an equivalent amount of negative

charge is removed from the left plate of C1, leavingit with an excess positive charge.


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Capacitors in Series (Cont.)

Section 16.8

• All of the right plates gain charges of –Q and allthe left plates have charges of +Q.

• left plate of C2 are notconnected to the battery but

connected together and areelectrically neutral. Theircharge are induced by the left plate of C1 and the right plate

of C2.

-e

• How about the charge on the right plate of C1 and

the left plate of C2?

??

induced


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• An equivalent capacitor can befound that performs the samefunction as the seriescombination.

Section 16.8

QΔV

1 2

1 2

1 21 2

1 2

;

1 1 1

eq

eq

Q QV V

C C

Q Q Q

V V V C C C

C C C

∆ = ∆ =

∆ = ∆ + ∆ = + =

= +

•

The potential differences add up to the battery voltage.


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•

• The equivalent capacitance of a series combination is

always less than any individual capacitor in the

combination.

Section 16.8

• Q=Q1=Q2=…

P bl S l i St t


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Problem-Solving Strategy

• Combine capacitors following

the formulas. – When two or more capacitors

are connected in parallel , thepotential differences across

them are the same V1=V2. – The charge on each capacitor

is proportional to itscapacitance: Qi =C i ΔV .

– The capacitors add directly togive the equivalentcapacitance.

Section 16.8

Ceq = C1 + C2 + …

Q = Q 1 + Q 2

V1=V2

Parallel

Problem-Solving Strategy (Cont.)


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g gy ( )

• Be careful with the choice of units.

• Combine capacitors following theformulas.

– When two or more unequalcapacitors are connected in

series, they carry the sameQ 1=Q 2 charge, but the potentialdifferences across them are notthe same.

–

The capacitances add asreciprocals and the equivalentcapacitance is always less thanthe smallest individual capacitor.

Section 16.8

Q 1=Q 2

Series


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Problem-Solving Strategy (Cont.)

• A complicated circuit can often be reduced to oneequivalent capacitor.

—Replace capacitors in series or parallel withtheir equivalent.

—Redraw the circuit and continue.

Section 16.8

•To find the charge (Q) on, or the potential difference(ΔV ) across, one of the capacitors, start with yourfinal equivalent capacitor C eq and work back throughthe circuit reductions.

• Repeat the process until there is only one singleequivalent capacitor.

P bl S l i S E i S


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Problem-Solving Strategy, Equation Summary

• Use the following equations when working through

the circuit diagrams:

– Capacitance equation: C = Q / ∆V; Q=CΔV; ΔV=Q/C

– Capacitors in parallel: Ceq = C1 + C2 + …

–

Capacitors in parallel all have the same voltagedifferences as does the equivalent capacitance:

– Capacitors in series: 1/Ceq = 1/C1 + 1/C2 + …

– Capacitors in series all have the same charge, Q, asdoes their equivalent capacitance:

Section 16.8

V1=V2=V3=…

Q 1=Q 2=Q 3=…

d


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Energy Stored in a Capacitor

• A capacitor can store energy

Section 16.9

— How to calculate the stored energy?

― The battery transfers charges

from one plate to another when

they are connected to a battery.

-e

-e

-e

— The stored energy is the same as the work required to movecharge onto the plates done by

the battery.battery storeW E ∆ =

E St d i C it (C t )


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Energy Stored in a Capacitor (Cont.)

Section 16.9

;i i i iW PE V Q∆ = ∆ = ∆ ∆

• The work ∆W i required to transfer

charge ∆Qi from one plate to theother must be done against thepotential ∆Vi of the capacitor:

i

ii

QV

C

∆∆ =

ΔVi1

2iiW W Q V ∆ = ∆ = ∆∑

• The total work done by the battery :

∆V

∆Vi - ∆Qi ∆Qi


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• Energy stored = ΔW=½ QΔV

Section 16.9

• The stored energy is the same as the work required

to move charge onto the plates.

Q

ΔV QC

V

=

∆ Q=CΔV

ΔV=Q/C

• From the definition of capacitance, this can be

rewritten in different forms.


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Applying Physics (Maximum Energy Design)

•

How should three capacitors and two batteries beconnected so that the capacitors will store the

maximum possible energy?

Section 16.9

• To have maximum capacitance, the three capacitors

must be parallel. Why?

( )221 1

2 2

eq eq Max Max Max E C V E C V = ∆ → = ∆

1 2 3 Max iC C C C C = = + +∑• To have maximum the potential difference, the two

batteries must be series. Why?

1 2 Max iV V V V ∆ = ∆ = ∆ + ∆∑


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Section 16.9

•

Maximum energy stored/maximum operatingvoltage—before discharging the capacitor

• In general, capacitors act as energy reservoirs that

can be slowly charged and then discharged quickly toprovide large amounts of energy in a short pulse.

+Q

-Q

( )2

max max

1

2 E C V = ∆

max

max

QQC

V V = =∆ ∆

Discharge


61/71

Capacitors with Dielectrics

Section 16.10

0 0

AC

d ε =

0 0

AC C

d κε κ = =

κ ≥ 1

2

0

1

2

E C V = ∆

2

0 0 012

E C V = ∆0

Capacitors with Dielectrics (Cont )


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Capacitors with Dielectrics (Cont.)

• A dielectric is an insulating material that, when placed

between the plates of a capacitor, increases thecapacitance.

Section 16.10

– Dielectrics include rubber, plastic, orwaxed paper.

• C = κ C o = κ εo( A/d ) ≥C 0 where κ ≥ 1

– The capacitance is multiplied by the

factor κ when the dielectric completelyfills the region between the plates.

– κ is called the dielectric constant .


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Dielectric Strength

• For any given plate separation, there is a

maximum electric field E max that can be

produced in the dielectric before it breaks

down and begins to conduct.

• This maximum electric field is called the

dielectric strength.

Section 16.10


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Section 16.10

Capacitors with Dielectrics (Examples)


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Capacitors with Dielectrics (Examples)

Section 16.10

C1

C2

A/2

A/2

Two capacitors in series:

1 2

1 1 1

eqC C C = +

1 1

1 1 0 0 0

/ 2;

2 2

A AC C

d d

κ κ

κ ε ε = = =

2 22 2 0 0 0

/ 2

2 2

A AC C

d d

κ κ

κ ε ε = = =

1 2eqC C C = +

Two capacitors in parallel:

1 1 0 1 0 1 02 2 ;

/ 2

A AC C

d d κ ε κ ε κ = = =

2 2 0 2 0 2 02 2 ;

/ 2

A AC C

d d

κ ε κ ε κ = = =

d/2 d/2

C1 C2

0 0

AC

d ε =

An Atomic Description of Dielectrics


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An Atomic Description of Dielectrics

• Polarization occurs when there is a separation

between the average positions of its negative chargeand its positive charge.

Section 16.10

• In a capacitor, the dielectric becomes polarized because it is in an electric field that exists between

the plates.

More Atomic Description


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Section 16.10

More Atomic Description

• The field generated by the two plates produces an induced polarization in the dielectric material.


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More Atomic Description (Cont.)

Section 16.10

•

The electric field from the polarizeddielectric will partially cancel the

electric field from the charge on the

capacitor plates.

0net ind E E E = +

0 0

0

;net

E E σ σ

ε ε

ε ε

= < = >

This decreases the net field inside the

capacitor, and decreases the potential

difference across the capacitor.

0net net V E d V ∆ = < ∆

More Atomic Description (cont )


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More Atomic Description (cont.)

Section 16.10

• To return the capacitor to its original

potential difference, more charge isneeded.

Q (with dielectric)>Q (vacuum)

• The net effect of the dielectric is to

increase the amount of charge stored on

a capacitor, and then increase the

capacitance.0

0

QQ

C C V V = > =∆ ∆

Q

0net V V ∆ < ∆

0

0

C C C

C κ κ = ⇒ =

net V V ∆ = ∆

Application – Computers


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Application – Computers

• Computers use capacitors in

many ways.

– Some keyboards usecapacitors at the bases of thekeys.

– When the key is pressed, thecapacitor spacing decreases and the capacitanceincreases.

– The key is recognized by thechange in capacitance.

Section 16.10

Commercial Capacitor Designs


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Commercial Capacitor Designs

16.0 serway physics

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