Chapter 22

Capacitance, Dielectrics,

Electric Energy Storage

Capacitors & capacitance

2

Capacitor / condenser: device that stores charge

For a given capacitor: Q ∝ V

Q

R

Q CV

C: Capacitance of capacitor

or Q

CV

+Q - QUnit: farad (F), μF & pF

Symbol in diagram:

Capacitors

3

Size, shape, relative position, insulating material

Capacitance depends on

Determine capacitance

4

For a parallel-plate capacitor:

+Q - Q

Sd

0 0

QE

S

0 0

d QdV Ed

S

(ignoring edge effect)

0 SQ

CV d

Cylindrical capacitor

5

Example1: A capacitor consists of two coaxial cylindrical shells (R1, R2), both with length

L>>R2. Calculate the capacitance.

L

R1R21 2

0

, 2

E R r Rr

2

1

212

0 1

ln2

R

R

RV Edr

R

0

12 2 1

2

ln( )

LQC

V R R

Spherical capacitor

6

Question: A capacitor consists of a sphere surrounded by a concentric spherical shell (R1, R2).

Calculate the capacitance.

Q2 Q1

R1

R2

- Q1 Q1+Q2

11 22

0

, 4

QE R r R

r

2

1

112

0 1 2

1 1( )

4

R

R

QV Edr

R R

0 1 2

12 2 1

4

R RQC

V R R

Capacitors in series and parallel

7

1) In series

1 2 ,Q Q Q

1C 2C

C1 2V V V

1 2

1 1 1 C C C

2) In parallel

1 2 ,V V V 1 2Q Q Q

1 2 C C C

1C

2C

C

Capacitor combination

8

Example2: Are the capacitors in series or parallel?

1)1C

2C

2)

Example3: How does C, Q, V change if d → 2d ?

1C 2C

V

d

parallel parallel

1C C Q

2V 1V

Electric energy storage

9

Charged capacitor stores electric energy

dW Vdq

+q - q

dq

21

2

q QW dq

C C

qdq

C

221 1 1

2 2 2

QU CV QV

C

Work required to store charges:

Energy stored:

Energy in the field

10

How does the electric energy distribute?

21

2U CV

Energy per unit volume / energy density:

201( )

2

SEd

d

20

1

2E Sd

20

1

2u E

Electric energy is stored in

the electric field!

d

S

Energy in capacitor

11

Example4: A parallel-plate capacitor is charged at 220V. (a) What is the capacitance; (b) how much electric energy can be stored?

d = 1mm

S =10cm2Solution: (a) 0 S

Cd

128.85 10 F

(b) Energy stored:

21

2U CV 72.14 10 J Maximum U ?

Different methods

12

Example5: Determine the electric energy stored in a spherical capacitor.

0 1 2

2 1

4 R RC

R R

Q R1

R2

- Q

Solution: 2 methods to solve it.

a) By using the capacitance:

Energy stored:21

2

QU

C

22 1

0 1 2

( )

8

Q R R

R R

13

20

1

2u E Q R1

R2

- Q

b) By using the energy density:

Total energy:

U udV2

0 1 2

1 1( )

8

Q

R R

20 2

0

1( )

2 4

Q

r

2

1

2 20 2

0

1( ) 4

2 4

R

R

Qr dr

r

22 1

0 1 2

( )

8

Q R R

R R

Discussion: Isolated conductor sphere? Q

Dielectrics

14

Insulating material in capacitors → dielectric

0 ,C KC

①Harder to break down → V ↗

②Distance between plates ↘

③ Increases the capacitance:

where K is the dielectric constant

Also noted as : relative permittivity r

Permittivity of material

15

If a dielectric is completely filled the space:

0C KC 0K S

d

S

d

: the permittivity of material

In a specific region filled with dielectric:

0K 0All in electric expressions →

00 0

QE

S

Q

ES

Electrostatics in dielectric

16

In a specific region filled with dielectric:

0 E

EK

field is reduced but not zero

The energy density in a dielectric: 21

2u E

Example6: How does C, Q, U change?

C Q U

CV battery disconnected?

Half filled dielectric

17

Question: How does the capacitance change if the capacitor is half filled with dielectric? (S, d, K)

1) 2)

Molecular description (1)

18

1) Polar dielectrics, such as H2O and CO

Have permanent electric dipole moments.

- +

p Ql

2) Nonpolar dielectrics, such as O2 and N2

No permanent electric dipole moments.

There are two different types of dielectrics:

Molecular description (2)

19

Polar Nonpolar

E

----

++++

Polarization

Bound chargeInduced field Eind

0 E

EK

Q -Q

indQ 1(1 )Q

K

Thinking & question

20

Thinking: Why charged objects can attract small paper scraps or water flow?

Question: Move a dielectric plate out of a charged capacitor, the electric energy increases, where does the extra energy come from?

CV

Brief review of Chapter 19-22

21

Coulomb’s law: 1 22

0

1

4

Q QF r

r

Gauss’s law: 0

= inE

QE dS

Electric potential: E V

aV E dl

Capacitance; Electric energy:

21,

2U CV 2

0

1

2u E

Dielectrics:

0

Chapter 23 & 24

22

Some useful concepts and equations:

These chapters should be studied by yourself

Ij

S

Ohm’s law: V = I R Current & resistance

Current density: I j dS

Microscopic statement of Ohm’s law:

/j E E

conductivity & resistivity

(ch23, pr.24)