Lecture 7

TOPIC 6

ELECTRIC POTENTIAL AND CAPACITOR

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Outline• Electric Potential Energy and Potential Difference

• Relation between Electric Potential and Electric Field

• The Electron Volt, a Unit of Energy

• Electric Potential Due to Point Charges

• Equipotential Surface

• Capacitance

• Dielectrics

• Storage of Electric Energy

• The Electric Battery

• Electric Current 2

Analogy between gravitational and electrical potential energy:

Electrostatic Potential Energy and Potential Difference

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BABAAB mghmghW PEPE BAABW PEPE

BABAAB mghmghW GPEGPE BAABW EPEEPE

Uniform gravitational field

Uniform electric field

AS

What is potential energy, PE ?

It is the energy due to position.

For the ball, because ball at A is at a higher position, it possess higher gravitational PE.

For the charge, because at A, nearer to the positive plate, it possess more electrical PE

o

B

o

A

o

AB

qqq

W PEPE

The potential energy per unit charge is called the electric potential.

The electric potential at a given point is the electric potential energy of a small test charge divided by the charge itself:

oqV

PE

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SI Unit of Electric Potential:

joule/coulomb = volt (V)

o

AB

o

A

o

BAB q

W

qqVV

PEPE

o

AB

o q

W

qV

PE

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Example 1

The work done by the electric force as the test charge (+2.0 x 10–6 C) moves from A to B is +5.0 x 10–5 J.

(a) Find the difference in PE between point B and point A. (-5.0x10-5J)

(b) Determine the potential difference between these points.(-25V)

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9

Solution

Relation between Electric Potential and Electric Field

Work is charge multiplied by potential:

Work is also force multiplied by distance:

)(PEPE ABABAB VVqW

10

Solving for the field,

If the field is not uniform, it can be calculated at multiple points:

d

VE BA

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The Electron Volt, a Unit of Energy

One electron volt (eV) is the energy gained by an electron moving through a potential difference of one volt.

12

JVCKE

VqW

q

WV

AB

o

ABAB

1919

0

106.11)106.1(

Electric Potential Due to Point Charges

The electric potential V at a point in an electric field is the work done to bring a unit positive charge from infinity to that point. The electric potential at infinity is considered zero.

The electric potential due to a point charge can be derived using calculus.

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There is an electric field around a point charge +Q. We now want to derive an expression for the electric potential V at a point P distance r from a point charge +Q.

Consider a charge +q at the point A distance x from the point charge +Q.

+Qr

P

+x

dx

F’

B

F

+qA

The force on the charge q is:

22 4

1

x

Qq

x

QqkF

14

To bring the charge q from A to B through a small displacement –dx:

Work done, dW = –F dx

From the definition of electric potential as the work done per unit charge to bring a positive charge from infinity to that point, the electric potential V at a point P distance r from the point charge +Q is:

15

112

229

2

1085.8

100.9 where, 4

1

1

1

Fm

mNCkr

Qr

Qk

x

Qk

dxx

Qqk

q

Fdxq

dWq

V

r

rx

rx

rx

16

These plots show the potential due to

(a) positive and

(b) negative charge.

Using potentials instead of fields can make solving problems much easier – potential is a scalar quantity, whereas the field is a vector. 17

The electric potential energy U of a point charge q which is at a distance of r from the point charge Q is:

r

qQr

qQk

qVU

4

When calculating the electric potential energy U of a charge in an electric field, the sign of charge must be consider

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Example 2

Three point charge of +q, +2q and -3q are arranged as shown in the figure. Find the electric potential energy of the system of three charges.(-7.59kq2/a)

+q

-3q +2q

a

aSolution:

Distance between the point charges:

+q and +2q r12 = √2a

+q and -3q r13 = a

+2q and -3q r23 = a 19

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Solution

Equipotential Surfaces

Equipotentials for:

(a) uniform field, (b) point charge, (c) dipole.

An equipotential surface is a surface on which all points have the same electric potential.

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The electric field created by any charge or group of charges is everywhere perpendicular to the associated equipotential surfaces and points in the direction of decreasing potential.

The net electric force does no work on a charge as it moves on an equipotential surface. 22

From the figures,

CBA VVV

then the work done to bring a test charge from B to A is given by

VVqVqW BAABBA

EA

BC

0W BA

No work is done in moving a charge along an equipotential surface. 23

Capacitance

A capacitor consists of two conductors that are close but not touching. A capacitor has the ability to store electric charge.

24

Parallel-plate capacitor connected to battery. Diagram (b) is a circuit diagram.

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When a capacitor is connected to a battery, the charge on its plates is proportional to the voltage:

Where: Q = charge

C = capacitance

V = potential difference.V

QC

CVQ

VQ

The quantity C is called the capacitance.

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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)

The capacitance does not depend on the voltage.

Unit of capacitance: farad (F)

1 F = 1 CV–1

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For a parallel-plate capacitor: ++

+

+

+

--

-

-

-

V

d

εo= permittivity of free space

= 8.85 x 10–12 Fm–1

The electric field strength between the plates is then:

d

VE

The capacitance of a parallel-plate capacitor if the space between the plates is free space is:

Plate area A

d

AC

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Example 3

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Ans: 0.44pF, 8.84x1010 C

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Solution

Capacitors in Series and in Parallel Circuit

Capacitors in series:

C1C2 C3

-Q-Q -Q +Q+Q +Q

V1 V2V3

VV

Cequ

-Q+Q

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• The charges on all the three capacitors are the same, Q.• The potential difference across the capacitors C1, C2, and

C3 are:

• The total potential difference is:

• So

• If C is the equivalent capacitance, then

3

3

2

2

1

1 , ,C

QV

C

QV

C

QV

321321 C

Q

C

Q

C

QVVVV

321

111

CCCQ

V

321

1111 and

CCCCV

QC

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Capacitors in parallel:

C1

C2

C3

Q 1

Q 2

Q 3

V

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• The charge on each is:

Q1 = C1V, Q2 = C2V, Q3 = C3V

• The total charge is

Q = Q1 + Q2 + Q3

= C1V + C2V + C3V

= V (C1 + C2 + C3)

If the equivalent capacitance is C, then

C = Q / V and so C = C1 + C2 + C3

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Example 4:Figure 3 shows five capacitors in a circuit. Calculate the equivalent capacitance between A and B terminal.

A

B

3 µF

4 µF

2 µF

1 µF 3 µF

Figure 3

Solution A

B

3 µF 2 µF

4 µF

1 µF3 µF

F1.2

3

1

2

11

1

TC

A

B

3 µF

4 µF

1 µF1.2 µF

A

B

3 µF

4 µF

2.2µF

Figure 3

F2.2

2.112

TC

FC

C

T

T

96.0

4

1

2.2

1

3

11

36

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Capacitors with Dielectrics

A dielectric is an insulator, and is characterized by a dielectric constant εr.

Capacitance Co of a parallel-plate capacitor with area of each plate A, and the plates separated in free space by a distance d is given by:

If the space between the plates of capacitor is filled with an insulator, also known as a dielectric, the capacitance C of the capacitor is given by:

d

AC

Cd

AC r

r 38

Dielectric strength is the maximum field a dielectric can experience without breaking down.

εr > 1 , the effect of having an insulator is to increase its capacitance.

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The molecules in a dielectric tend to become oriented in a way that reduces the external field.

This means that the electric field within the dielectric is less than it would be in air, allowing more charge to be stored for the same potential.

No dielectric Dielectric inserted

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Example 5:

A parallel-plate capacitor consists of two metal plates each of area 0.40 m2 and separated by a distance of 0.20 cm.

a) Find the capacitance of the capacitor if the space between the plates is

i) vacuum

ii) an insulator of dielectric constant 2.3

(Ans : (i) (1.77x10-9F)(ii) (4.07x10-9F)

b) With the insulator between the plates, the insulation breaks down when the electric field intensity exceeds 1.8 × 105 Vcm–1. What is the maximum potential difference that can be apply across the capacitor? (Ans:3.6x104V)

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Solutions:

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Storage of Electric Energy

A charged capacitor stores electric energy; the energy stored is equal to the work done to charge the capacitor.

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C

QCVQVPE

22

2

1

2

1

2

1

The sudden discharge of electric energy can be

harmful or fatal.

Capacitors can retain their charge indefinitely

even when disconnected from a voltage source

Be careful!

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Exercise 1:

As shown in diagram, find the total energy stored in C1, C2 and C3 when they are fully charged.

[Answer: 3.6 × 10–4 J]

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2112 CCC 55 μF 10 (2 marks)

312123

111

CCC

10

1

10

1

5

1

μF 5123 C (2 marks)

2

2

1CVW

26 121052

1

J 106.3 4 (2 marks)

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Ex 1Two point charges are arranged along the x axis as shown in the figure. At which of the following values of x is the electric potential equal to zero? Note: At infinity, the electric potential is zero.

 Ans: +0.29m

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Ex 2Three point charges –Q, –Q, and +3Q are arranged along a line as shown in the sketch.What is the electric potential at the point P?Ans: +1.6kQ/R

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Ex 3If the work required to move a +0.25 C charge from point A to point B is +175 J, what is the potential difference between the two points?Ans: 700 V

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Ex 4When two capacitors are connected in series, the equivalent capacitance of the combination is 120 µF. When the two are connected in parallel, however, the equivalent capacitance is 480 µF. What are the capacitances of the individual capacitors?Ans: 240 µF and 240 µF

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Ex 5How much energy is stored in the combination of capacitors shown?

Ans: 0.030 J

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