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MyChapter 17

LectureOutline

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Chapter 17: Electric Potential

•Electric Potential Energy

•Electric Potential

•How are the E-field and Electric Potential related?

•Motion of Point Charges in an E-field

•Capacitors

•Dielectrics

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§17.1 Electric Potential Energy

Electric potential energy (Ue) is energy stored in the electric field.

•Ue depends only on the location, not upon the path taken to get there (conservative force). Not a vector.

•Ue = 0 at some reference point.

•For two point particles take Ue = 0 at r = .

•For the electric force r

qkqU e

21=

4

Example: A proton and an electron, initially separated by a distance r, are brought closer together.

Bringing the charges closer together decreases r:.

For these two chargesr

keU e

2

−=

0<−=Δ eiefe UUU

This is like a mass falling near the surface of the Earth; positive work is done by the field.

(a) How does the potential energy of this system of charges charge?

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When q1 and q2 have the same algebraic sign then ΔUe > 0.

This means that work must be done by an external agent to bring the charges closer together.

Example continued:

(b) How will the electric potential energy change if both particles have positive (or negative) charges?

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Example: What is the potential energy of three point charges arranged as a right triangle? (See text Example 17.2)

12r

2q

1q 3q13r

23r

0=eU12

21

r

qkq+

23

32

13

31

r

qkq

r

qkq++

12r

2q

1q 3q13r

23r

0=eU12

21

r

qkq+

23

32

13

31

r

qkq

r

qkq++

Are these the same?

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§17.2 Electric Potential

Electric potential is the electric potential energy per unit charge.

testq

UV e=

Electric potential (or just potential) is a measurable scalar quantity. Its unit is the volt (1 V = 1 J/C).

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For a point charge of charge Q:r

kQ

q

UV e ==

test

When a charge q moves through a potential difference of ΔV, its potential energy change is ΔUe = qΔV.

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Q

b

a

c

e

d

g

f

Example: A charge Q = +1 nC is placed somewhere in space far from other charges. Take ra = 1.0 m and re = 2.0 m.

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Place a charge of +0.50 nC at point e. What will the change in potential (ΔV) be if this charge is moved to point a?

( )( )

( )( )Volts 0.9

m 1

nC 0.1/CNm 100.9

Volts 5.4m 2

nC 0.1/CNm 100.9

229

229

+=×

==

+=×

==

aa

ee

r

kQV

r

kQV

ΔV = Vf Vi = Va Ve = +4.5 Volts

Example continued:

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ΔUe = qΔV = (+0.50 nC)(+4.5 Volts)= +2.3 nJ

What is the change in potential energy (ΔU) of the +0.50 nC charge ?

Example continued:

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§17.3 The Relationship between E and V

Q

b

a

c

e

d

g

f

+9 V

+4.5 V

The circles are called equipotentials (surfaces of equal potential).

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The electric field will point in the direction of maximum potential decrease and will also be perpendicular to the equipotential surfaces.

Q

b

a

c

e

d

g

f

+9 V +4.5 V

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Equipotentials and field lines for a dipole.

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Uniform E-field

E

Equipotential surfaces

V1 V2 V3 V4

Edq

UV e −=

Δ=Δ Where d is the distance

over which ΔV occurs.

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§17.4 Moving Charges

When only electric forces act on a charge, its total mechanical energy will be conserved.

fi EE =

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Example (text problem 17.40): Point P is at a potential of 500.0 kV and point S is at a potential of 200.0 kV. The space between these points is evacuated. When a charge of +2e moves from P to S, by how much does its kinetic energy change?

ffii

fi

UKUK

EE

+=+

=

( )( )

( )( )J 106.9

kV0.5000.200214−×+=

−+−=

−−=Δ−=Δ−=

−−=−=−

e

VVqVqU

UUUUKK

ps

iffiif

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Example (text problem 17.41): An electron is accelerated from rest through a potential difference. If the electron reaches a speed of 7.26106 m/s, what is the potential difference?

ffii

fi

UKUK

EE

+=+

=

0

( )( )( )

Volts 150

C 1060.12

m/s1026.7kg 1011.9

2

2

1

19

26312

2

+=

×−

××−=−=Δ

Δ−=

Δ−=Δ−=

−

−

q

mvV

Vqmv

VqUK

f

f

f

Note: the electron moves from low V to high V.

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§17.5 Capacitors

A capacitor is a device that stores electric potential energy by storing separated positive and negative charges. Work must be done to separate the charges.

Parallel plate capacitor

+ + + ++ + +

– ––––––

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VQ

VE

QE

Δ∝∴Δ∝

∝

Written as an equality: Q = CΔV, where the proportionality constant C is called the capacitance.

For a parallel plate capacitor:

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. where 0

0

00

d

AC

VCVd

AQ

dA

QdEdV

ε

εεε

σ

=

Δ=Δ=∴

===Δ

Note: C depends only on constants and geometrical factors. The unit of capacitance is the farad (F). 1 F = 1 C2/J = 1 C/V

What is the capacitance for a parallel plate capacitor?

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Example (text problem 17.56): A parallel plate capacitor has a capacitance of 1.20 nF. There is a charge of magnitude 0.800 C on each plate.

(a) What is the potential difference between the plates?

Volts 667nF 20.1

C 800.0===Δ

Δ=

CQ

V

VCQ

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(b) If the plate separation is doubled, while the charge is kept constant, what will happen to the potential difference?

dV

A

Qd

C

QV

∝Δ

==Δ0ε

If d is doubled so is the potential difference.

Example continued:

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Example (text problem 17.100): A parallel plate capacitor has a charge of 0.020 C on each plate with a potential difference of 240 volts. The parallel plates are separated by 0.40 mm of air.

(a) What is the capacitance of this capacitor?

pF 83F 103.8Volts 240

020.0 11 =×==Δ

= −CVQ

C

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(b) What is the area of a single plate?

( )( )

22

22120

0

cm 38m 0038.0

/NmC 1085.8

mm 40.0pF 83

==

×==

=

−ε

ε

CdA

d

AC

Example continued:

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§17.6 Dielectrics

As more and more charge is placed on capacitor plates, there will come a point when the E-field becomes strong enough to begin to break down the material (medium) between the capacitor plates.

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To increase the capacitance, a dielectric can be placed between the capacitor plates.

d

AC 0

0

0

where

C C

εκ

=

=

and κ is the dielectric constant.

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Example (text problem 17.71): A capacitor can be made from two sheets of aluminum foil separated by a sheet of waxed paper. If the sheets of aluminum are 0.3 m by 0.4 m and the waxed paper, of slightly larger dimensions, is of thickness 0.030 mm and has κ = 2.5, what is the capacitance of this capacitor?

( )( )

( )( ) F. 1085.8F 1054.35.2C C and

F 1054.3

m10030.0

m30.0*40.0/CNm1085.8

880

8

3

22212

00

−−

−

−

−

×=×==

×=

×

×=

=

κ

ε

d

AC

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§17.7 Energy Stored in a Capacitor

A capacitor will store energy equivalent to the amount of work that it takes to separate the charges.

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These are found by using Q = CΔV and the first relationship.

( )

C

Q

VC

VQU

2

2

12

1

2

2

=

Δ=

Δ=

The energy stored in the electric field between the plates is:

}

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Example (text problem 17.79): A parallel plate capacitor is composed of two square plates, 10.0 cm on a side, separated by an air gap of 0.75 mm.

(a) What is the charge on this capacitor when the potential difference is 150 volts?

C 1077.1 80 −×=Δ=Δ= VdA

VCQε

(b) What energy is stored in this capacitor?

J 1033.12

1 6−×=Δ= VQU

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Summary

•Electric Potential Energy

•Electric Potential

•The Relationship Between E and V

•Motion of Point Charges (conservation of energy)

•Parallel Plate Capacitors (capacitance, dielectrics, energy storage)