Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill...

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson

Copyright © 2008 – The McGraw-Hill Companies s.r.l.

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).

•Ue = 0 at some reference point.

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

•For the electric force r

qkqU e

21

3



Example: A proton and an electron, initially separated by a distance r, are brought closer together. How does the potential energy of this system of charges charge?

Bringing the charges closer together decreases r:.

For these two chargesr

keU e

2

0 eiefe UUU

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

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

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

Example continued

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

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What is the potential energy of a system (arrangement) of point charges? To calculate:

Begin by placing the first charge at a place in space far from any other charges. No work is required to do this.

Next, bring in the remaining charges one at a time until the desired configuration is finished.

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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

0eU12

21

r

qkq

23

32

13

31

r

qkq

r

qkq

12r

2q

1q 3q13r

23r

0eU12

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.

test

e

q

UV

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

test

e

When a charge q moves through a potential difference of V, its potential energy change is Ue = q V.

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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 = rb = rc = rd = 1.0 m and re = rf = rg = 2.0 m.

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(a) Compare the potential at points d and g.

Example continued:

Since Q>0 the potential at point d is greater than at point g, it is closer to the charge Q.

(b) Compare the potential at points a and b.

The potential at point a is the same as at point b; both are at the same distance from the charge Q.

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(c) Place a charge of +0.50 nC at point e. What will the change in potential (V) 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

V = Vf – Vi = Va-Ve = +4.5 Volts

Example continued:

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Ue =qV = (+0.50 nC)(+4.5 Volts)= +2.3 nJ

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

Example continued:

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(e) How would the results of the previous questions change if instead of a +1.0 nC charge there is a -1.0 nC charge in its place?

Example continued:

(a)The potential at point d is less than the potential at point g.

(b) Unchanged

(c) -4.5 V

(d) -2.3 nJ

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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

Where d is the distance

over which V occurs.

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If the electric field inside a conductor is zero, what is the value of the potential?

If E=0, then V=0. The potential is constant!

What is the value of V inside the conductor? It will be the value of V on the surface of the conductor.

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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 17.31): 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

e

VVqVqU

UUUUKK

ps

iffiif

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Example (text problem 17.32): 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

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

Written as an equality: Q = CV, 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

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 17.42): 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

C

QV

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

0

If d is doubled so is the potential difference.

Example continued:

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Example (text problem 17.86): 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

C

V

QC

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

22

22120

0

cm 38m 0038.0

Nm/C 1085.8

mm 40.0pF 83

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

and is the dielectric constant.

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Example (text problem 17.55): 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 = 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

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=CV and the first relationship.

C

Q

VC

VQU

2

2

12

1

2

2

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

}

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Example (text problem 17.63): 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 Vd

AVCQ

(b) What energy is stored in this capacitor?

J 1033.12

1 6 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)

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill...

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Transcript of Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill...