Chapter 25

Capacitors

25-2 Capacitor and Capacitance• A capacitor consists of two isolated conductors (the plates) with charges +

q and - q. Its capacitance C is defined fromq=CV

Where V is the potential difference between the conductors.

• To find the capacitance, we have to find the electric field E, then the potential difference V. Gauss Law is used

25-3 Calculating the Capacitance

f

i

dsEV

• Parallel-Plate CapacitorThe plate area is A and the distance between the plates is d.

dEV AEq oV

qC

d

AC o

• Cylindrical Capacitor

br

aro

f

i

drrL

qdsEV

1

2

)a

bln(

L2

qV

o

)ln(

L2

V

qC

ab

o

• Spherical Capacitor

br

ar2

o

f

i

drr

1

2

qdsEV

ab

ab

4

qV

o

ab

ab4

V

qC o

• The Capacity of an Isolated ConductorThe isolated conductor is a sphere of conducting material which is isolated such that it keeps its charge. It is considered to be a spherical capacitor of an outer sphere with infinitely large radius b=∞

ab

ab4C o

bao 1

a4C

and substituting about b=infinity, and a=R, the radius of the inner sphere,we get

R4C o

Sample Problem 25-1In Fig. 25-7 a, switch S is closed to connect the uncharged capacitor of capacitance C =0.25 mF to the battery of potential difference V= 12V. The lower capacitor plate has thickness L=0.50 cm and face area A=2.0 x 10-4 m2, and it consists of copper, in which the density of conduction electrons is n= 8.49 x 1028 electrons/m3. From what depth d within the plate (Fig. 25-7b) must electrons move to the plate face as the capacitor becomes charged?

25-4 Capacitances in Parallel and SeriesParallel Connection

Series Connection

Sample Problem 25-2a. Find the equivalent capacitance for the combination of

capacitances shown in Fig a, across which potential difference V is applied. Assume C1 = 12.0 mF, C2 = 5.30 mF, and C3 = 4.50 mF.

b. The potential difference applied to the input terminals in Fig. a is V=12.5 V. What is the charge on C1?

Sample Problem 25-3Capacitor 1, with C1 =3.55 mF, is charged to a potential difference Vo= 6.30 V using a 6.30 V battery. The battery is then removed, and the capacitor is connected as in the figure to an uncharged capacitor 2, with C2=8.95 mF When switch S is closed, charge flows between the capacitors. Find the charge on each capacitor when equilibrium is reached.

25-5 The Energy Stored in a Capacitor

The potential energy U stored in a capacitor due to holding the charge q, is given by

qV2

1U 2CV

2

1U

C2

qU

2

The energy density uIn case of the parallel plate capacitor, the potential energy per unit volume between the plates is given by

Ad2

CV

Ad

Uu

2

d

AC o

2

2

o d

V

2

1u

2oE

2

1u

The capacity of this spherical conductor is given by

Sample Problem 25-5An isolated conducting sphere whose radius R is 6.85 cm has a charge q = 1.25 nC.a. How much potential energy is stored in the electric field of

this charged conductor?b. What is the energy density at the surface of the sphere?

R4C o

and the potential energy is given byC2

qU

2

J10X110X9)0685.0(2

)10X25.1(U 79

29

The energy density u is given by2

oE2

1u

2mJ52

2o

o 10X5.2)r

q

4

1(

2

1u