Chapter 30 Capacitance. Capacitors A device that stores charge (and then energy in electrostatic field) is called a capacitor. A cup can store water charge

Chapter 30 Capacitance

Capacitors

A device that stores charge (and then energy in electrostatic field) is called a capacitor.

A cup can store water

charge

The capacitance of an isolated conductor

04q

C aV

aaq

rr

qE ˆ

4 20

a

qdr

r

qV

a0

20 44

What is the capacitance of the Earth, viewed as an isolated conducting sphere of radius R=6370km?

Example)1085.8(14.34 12

)1037.6( 6

F710F101.7 4 μ

qC

V ( , , )F F pF

A capacitor consists of A capacitor consists of two conductorstwo conductors a and b of a and b of arbitrary shape: arbitrary shape:

These two conductors are called These two conductors are called platesplates (no matter (no matter what their shapes).what their shapes).

Symbolically, a capacitor is represented as:

CC

CCor

C stands for the capacitance of the capacitor.

The charge q appears on the capacitor plates

There is a potential difference V between the plates

The charge q is always directly proportional to the potential difference V between the plates

VVVq

VCq

capacitance

Remarks :A capacitor is said to be charged if its plates carry equal and opposite charges +q and -q. q is not the net charge on the capacitor, which is zero.

V

qC

Capacitors in Series and Parallel

1. Capacitors connected in 1. Capacitors connected in ParallelParallel ::

CC11

aabb

CC22

Question: If we identify the above capacitors connected in parallel as a single capacitor,

aa bb

CCeqeq

what is its capacitance?

VV

VV

VCqVCq 2211 ,

21 qqq VCeq VCC )( 21

21 CCCeq

i ieq CC

2. Capacitors connected in Series :

CC11

aa bb

CC22

VV22VV11

qqqqqq qq

VVV 21

eqCCC

111

21

VV

qqqqCCeqeq

aa bb

21 CqCq eqCq

iieq CC

11

The capacitance is a geometrical factor that depends on the size, shape and separation of the capacitor plates, as well as the material that occupies the space between the plates.

The SI unit of capacitance is farad :

1 farad = 1 F = 1 coulomb/volt

1 1 F = 10F = 10-6-6 F F 1 pF = 10-12 F

Calculating the capacitanceProcedure: 1.Suppose that the capacitor is charged, with ±q on the two plates respectively. 2. Find the electric field E in the region between the plates.3. Evaluate the potential difference between the positive and negative plates, by using the formula:

sEVVV

d

4.The expected capacitance is then: 4.The expected capacitance is then:

VqC

A Parallel-plate Capacitor :

EEE

0

A

qdEdV

0

A

q

0

d

A

V

qC 0

A Cylindrical Capacitor :A Cylindrical Capacitor :

brar

rLqE ,

2

ˆ)(

0

sEV

d

b

arr

rL

rqˆˆ

2

d

0

a

b

L

qln

2 0

ab

LVqC

ln2 0

The capacitor has length The capacitor has length LL, , and and LL>>>>aa, , bb. .

q

q

L

b

a

A Spherical Capacitor :

brar

rqE ,

4

ˆ2

0

sEV

d

b

arr

r

qdrˆˆ

4 20

ba

q 11

4 0 ab

abVqC

04

Capacitor with Dielectric

We now consider the effect of filling the interior of a capacitor with a dielectric material

The effect of the dielectric material is to reduce the strength of the electric field in its interior from the initial E0 in vacuum to E =E0/ke.

q

-q

AqE 00

dAC /0

AqEE ee 00 /

AqdEdV e 0

dAVqC e //' 0 CC e'

⊕⊕⊕⊕ ⊕⊕⊕⊕ ⊕⊕⊕⊕ q

qq0E q A


1e2e

AA

dd

1 1 0 1/ /e eE E q A

2 2 0 2/ /e eE E q A

0 1 0 20 1 2

1 1/ 2 / 2 ( )

2e ee e

qdV qd A qd A

A

0 1 2

1 2

2 e e

e e

AqC

V d

⊕⊕⊕⊕ ⊕⊕⊕⊕ ⊕⊕⊕⊕ q

qq02E q A


1e 2eAA

dd

1 1 0 1/ 2 /e eE E q A

2 2 0 2/ 2 '/e eE E q A

1 0 12 / eV qd A

0 11

1 2e Aq

CV d

2 0 22 ' / eV q d A

0 22

2

'

2e Aq

CV d

1 2

01 2 ( )

2 e e

C C C

A

d

q q’q q’

-q --q -q’q’

CqqVqU ddd

If the process is continued until a total charge q has been transferred, the total potential energy is:

C

qqqUU

qU

2/C'd'd

2

00

VqVC 2

1)(

2

1 2

Energy storage

Suppose that at the instant t, the capacitor has been charged with charge q', the voltage between its plates is V´.

During the next time interval [t, t+dt], if an additional charge dq' is added on the plates, then the increase of the electrostatic energy is,

CqV dq′

q′ΔV´

Why do we say that the energy is stored in the electric field between the capacitor plates?

Take the parallel-plate capacitor as an example.

dAC 0

charged with q, then: AqE 0

q

q

C

qU

2

2

A

dq

0

2

2

AdA

q2

002

1

ΩE 2

02

1

q

q

2'

CC

UU 2double the volume

double the energy



dAC 0


q

q

C

qU

2

2

A

dq

0

2

2

AdA

q2

002

1

ΩE 2

02

1

2'

CC

UU 2energy density

202

1E

Ω

Uu

AA 2'



dAC 0


q

q

C

qU

2

2

A

dq

0

2

2

AdA

q2

002

1

ΩE 2

02

1

2'

CC

UU 2energy density

202

1E

Ω

Uu

AA 2' ?2' UU ?2/' UU

q

q

q2

q2

?2' UU

Energy storage

Charge storage

Electric Potential

Energy storage

Charge storage

Electric field

Energy storage

VqVCqC

U 2

1)(

2

1

2

1 22 ΩEU 202

1

An isolated conducting sphere of radius R carries a charge q.

R

qRqCqU

0

2

022

8)4(22

Example

How much energy is stored?

V

N

iii qVVqU d

2

1

2

1

1

0

1'

2 4

qU dq

R

0

1

2 4

qds

R

2

0

442

1R

R

q

R

q

0

2

8

00

''

4

q qU dq

R

2

0 0

'

8

qq

R

R

q

0

2

8

An isolated conducting sphere of radius R carries a charge q.

Example

How much energy is stored?

202

1E

Ω

Uu udU dE 2

02

1

rRrqrE ,4)( 20

22

0 2 2 40

14 d

2 16R

qU r r

r

R

q

0

2

8

What is the radius b of an imaginary spherical surface such as one thirds of the stored energy lie within it?

bRqU 0

2 243

rrrqb

Rd432 24

022

b

Rrrru d4)( 2

rrqb

Rd8 2

02

bRq 118 02

bRR

11

3

1

23Rb

R

qU

0

2

8

VVqq

qq

e

U

CC e

'2'

2

C

qU ?

If the potential difference between the capacitor plates are the same, the electric fields inside the capacitor are the same also.

AqE 0 0( )eq A qq e

eq

eq

22 ( )''

2 ' 2e

e

qqU

C C

eU

qq

qq

e

U

CC e

'2'

2

C

qU

If the potential difference between the capacitor plates are the same, the electric fields inside the capacitor are the same also.

AqE 0 0( )eq A qq e22 ( )'

'2 ' 2

e

e

qqU

C C

eU

' / eE E

' / eV V 2'( ')

'2

C VU

e

U

Dielectrics and Gauss’ Law

)(

d

00

000

AqE

qAEAE

EbbdEsEV

)(d 0

V

qC

ebEbdE

q

/)( 00

)1/1(0

ebd

A

qsdE

00

A

qE

00

qsdEe

0

A

qE

e 0

Example

e

? C


)(

d

00

000

AqE

qAEAE

A dielectric slab is inserted,

)()( d 000 AqAqEqqAE

q ’ is induced surface charge.

A

q

A

q

A

q

A

qEE

eee 0000

0 ,

)11

( e

qq

/d0 eqAE

d0 qAEe

Gauss’ law should be amended as:

kkee instead of instead of

. . The charge The charge qq contained within the Gauss surface contained within the Gauss surface is taken to be the is taken to be the free chargefree charge only. only.


Electric polarization vectorElectric polarization vector

0 ep E

0 + D E p Electric displacement vectorElectric displacement vector

0 0+ eE E

0(1+ ) e E

0 E

'0 qqAdE

')'( 00 qqAdEE

qAdE

00

''0 qAdE

Adp

Adp

qAdpE

)( 00 qAdD

0D

E

0E

e

E

0

ExercisesP695 P695 27 27 ProblemsP 696~699 P 696~699 6, 9, 20, 24 6, 9, 20, 24

Documents

Chapter 30 Capacitance. Capacitors A device that stores charge (and then energy in electrostatic field) is called a capacitor. A cup can store water charge