Chapter 27 Capacitance and Dielectrics

第二十七章 電容及介電質

Capacitors

The uses of capacitorsCapacitor or condenser is a charge storage device. It is also an energy storage device.

Capacitors have many uses beyond the energy or charge storage, such as in tuning circuits, filter circuits, memories, etc.

What else can you think of?

CapacitanceTwo isolated conductors of any shape comprise a capacitor. We call these two conductors plates. When the capacitor is charged, there are equal amount but opposite charges on the two plates. The amount of charge is proportional to the potential difference between them.

Q CVThe proportionality constant C is called capacitance.

Parallel-plate capacitor

Neglecting the edges, we have

0

E

0

dV Ed

0( )A

Q A Vd

0( )A

Cd

The capacitance of a parallel plate capacitor:

The unit of capacitanceThe SI unit of capacitance is faraday (F). 1 1 /F C V

How large is one faraday?

120 8.85 10 /F m

0( )A

Cd

If d = 10 m, then11

5

101

10

AF

6 210 1 1A m km km Not easy to achieve!

Charging the capacitor

A cylindrical capacitor

02ln( / )

Cb a

A spherical capacitor

04ab

Cb a

An isolated sphere

R

04C R

Sample problem 1A storage capacitor on a random access memory (RAM) chip has a capacitance of 55 fF. If the capacitor is charged to 5.3 V, how many excess electrons are on its negative plate?

1.8 106 electrons

Some circuit symbols

Capacitors in parallel

Capacitors in series

Sample problem 2

Sample problem 3

Energy stored in a capacitorq

dW Vdq dqC

2

0 2

Q q QW dq

C C

The work is stored as the potential energy U in the capacitor.

221

2 2

QU CV

C

The medical defibrillator70C F 5000V V About ¼ of the stored energy is sent to

the patient during a period of 2.0 ms.

Energy stored in an electric field

2 2 200

1 1 1( )( ) ( )

2 2 2

AU CV Ed E Ad

d

Neglecting the edges, Ad is the volume in which the electric field is present.

Thus the energy density of the electric field is:

20

1

2u E

Sample problem 4An 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?

(a) 103 nJ. (b) 25.4 J/m3.

Capacitor with a dielectric

V+ + + +

+ + +

0

AC

d

0

AC

d

airC C

Schematics of real capacitors

Capacitor with a dielectric

Space filled with a dielectric material

For a point charge:2

0

1

4

qE

r

Just outside a conductor:0

E

Sample problem 5A parallel-plate capacitor whose capacitance C is 13.5 pF is charged by a battery to a potential difference V = 12.5 V between its plates. The charging battery is now disconnected and a porcelain slab ( = 6.50) is slipped between the plates. What is the potential energy of the capacitor-porcelain device, both before and after the slab is put into place?

Ui = 1055 pJ and Uf = Ui/ = 160 pJ

Dielectrics: an atomic view

Induced dipole moment

Capacitance: an atomic view

00

E

0

indE

0

0

EE

1ind

Sample problem 6What is the electric field in (a) the gap between the plates and the dielectric slab? (b) the dielectric slab? What is the capacitance?

Metal slab in a capacitor

Capacitor partially filled with a dielectric

Sample problem 7What is the ratio of the capacitance of the capacitor with dielectrics shown below to that of the same capacitor without any dielectric?

Force exerted on a dielectric in a capacitor

Force exerted on a dielectric in a capacitor

Home work

Question ( 問題 ): 1, 6, 19

Exercise ( 練習題 ): 11, 16, 19

Problem ( 習題 ): 10, 20, 24, 38