Slide 1

Slide 2 Fig 26-CO, p.795

Consider two conductors carrying charges of equal magnitude but of

opposite sign, Such a combination of two conductors is called a capacitor.

The capacitance C of a capacitor is the ratio of

the magnitude of the charge on either conductor

to the magnitude of the potential difference

between them:

V

QC

The SI unit of capacitance is the farad (F) = coulombs per volt,

Volt

ColumFarad 1

Slide 3

26.2 CALCULATING CAPACITANCE

1. Parallel-Plate Capacitors

A parallel-plate capacitor consists of two parallel

conducting plates, each of area A, separated by a

distance d. When the capacitor is charged by

connecting the plates to the terminals of a battery,

the plates carry equal amounts of charge. One

plate carries positive charge +Q, and the other

carries negative charge -Q.

0 0

QE

A

The value of the electric field between the plates is

Slide 4 Fig 26-3a, p.799

0

QV Ed d

A

0

QC

VQQd

A

The capacitance of a parallel-plate capacitor is proportional to the area of its plates and inversely proportional to the plate separation

0

AC

d

Slide 5 Fig 26-3b, p.799

0

AC

d

C = 8.85 x 10-12 (C2/N.m2) . 2x 10-4(m2)/ 1x 10-3 (m)

1.77 x 10-12 F = 1.77 pF

Example A parallel-plate capacitor has an area A = 2.00 x 104 m2 and a plate separation d = 1.00 mm. Find its capacitance.

Slide 6 Fig 26-6, p.801

2 ln( )e

Cb

ka

2. The Cylindrical Capacitor

A cylindrical capacitor consists of

a solid cylindrical conductor of

radius a and length surrounded

by a coaxial cylindrical

shell of radius b.

Slide 7

3.The Spherical Capacitor

A spherical capacitor consists of an inner sphere ofradius a surrounded by a concentric spherical shell of radius b.

( )e

abC

k b a

Slide 8

Other different capacitors shape

Slide 9

Slide 10

26.3 COMBINATIONS OF CAPACITORS

Parallel Combination

Let us call the maximum charges on the two capacitors Q 1 and Q 2 . The total charge Q stored by the two capacitors is

Q= Q1+Q2

Q1= C1 V Q2= C2 V

The equivalent capacitor Q= Ceq V

Ceq V= C1 V+ C2 V

Ceq = C1 + C2

Slide 11 Fig 26-9b, p.803

Slide 12 Fig 26-10, p.804

Series Combination

21

21

21

111

CCC

C

Q

C

Q

C

Q

VVV

eq

eq

eq

Slide 13 Fig 26-11, p.806

Example : Find the equivalent capacitance between a and b for the combination of capacitors shown in Figure

Slide 14

26.4 Energy stored in a charged capacitor

Suppose that q is the charge on the

capacitor at some instant during the

charging process. At the same instant, the

potential difference across the capacitor is

V = q/C.

We know that the work necessary to transfer

an increment of charge dq from the plate

carrying charge -q to the plate carrying

charge +q (which is at the higher electric

potential) is

dW dqV

Slide 15

0

0

Q

Q

W dqV

qW dq

c

21

2

QW

C

21

2

QU

C

221 1 1

2 2 2

QU QV CV

C

This result applies to any capacitor, regardless of its geometry. We

see that for a given capacitance, the stored energy increases as the

charge increases and as the potential difference increases

Slide 16

0

AC

d

2 20

20

1 1( ) ( )

2 21

( )2

AU CV Ed

d

Ad E

Energy stored in a parallel-plate capacitor

For a parallel-plate capacitor, the potential difference is related to the

electric field through the relationship V = Ed.. The capacitance is given by

By substituting

The energy per unit volume known as the energy density, is

The energy density in any electric field is proportional to the square of the magnitude of the electric field at a given point

202

1 EAd

UUE

Slide 17

26.5. Capacitors with Dielectrics

Dielectric is a non-conducting material, such as rubber, glass, or

waxed paper. When a dielectric is inserted between the plates of a

capacitor, the capacitance increases. If the dielectric completely fills

the space between the plates, the capacitance increases by a

dimensionless factor k , which is called the dielectric constant.

0

0

o

o

V ECkC V E

Slide 18

For a parallel-plate capacitor, we can express

the capacitance when the capacitor is filled

with a dielectric as

0 0&A A A

C k kd d d

We see that a dielectric provides the following

advantages:

• Increase in capacitance

• Increase in maximum operating voltage

• Possible mechanical support between the

plates, which allows the plates to be

close together without touching, thereby

decreasing d and increasing C

Slide 19

Slide 20 Table 26-1, p.812

Slide 21

Slide 22

Slide 23