CAPACITANCE III

RECAP

The electrostatic energy stored in the capacitor

22

0 2

1

2

1VC

C

Qdq

c

qW

Q

• Capacity increases if the dielectric material is introduced.

• If the battery is connected

• If the capacitor is charged, disconnected from battery, and then dielectric is introduced

qkq e

ek

EE 0

Capacities with the dielectric

• Parallel plate capacitor

• Spherical capacitor

• Cylindrical capacitor

d

AkC e0

ab

abkC e

04

ab

LkC e

ln2 0

Electrostatic pressure

• Force on a plate of capacitor

202

1E

A

FP

Today’s plan

• Energy stored before and after the dielectric is filled.

• Force on a dielectric

Energy stored before and after

• Before

• Since C = kC0

• Why U´< U ?? Where does this energy go?

0

2

2

1

C

QU

ek

UU

Reason

Dielectric, when inserted, gets pulled into the device. External agent do negative work to prevent the dielectric from accelerating.

Work = U-U

The nonuniform electric field (fringing field) near the edges causes a dielectric to be pulled into the capacitor.

Fringing Field

The bound charges tend to accumulate near the free charges of opposite sign.

If no external agent works, slab will be accelerated.

+ + + + + + + + + + + +

- - - - - - - - - - - - - -

+ + + + + + + + + + + ++ + + + + + + + + + + +

- - - - - - - - - - - - - -

+ + + + + + + + + + + +

- - - - - - - - - - - - - -

+ + + + + + + + + + + +

- - - - - - - - - - - - - -

+ + + + + + + + + + + +

- - - - - - - - - - - - - -

+ + + + + + + + + + + +

- - - - - - - - - - - - - -

+ + + + + + + + + + + +

- - - - - - - - - - - - - -

Slab oscillates between the ends

+ + + + + + + + + + + +

- - - - - - - - - - - - - -

Slab oscillates between the ends

+ + + + + + + + + + + +

- - - - - - - - - - - - - -

Slab oscillates between the ends

Slab oscillates between the ends

+ + + + + + + + + + + +

- - - - - - - - - - - - - -

Slab oscillates between the ends

+ + + + + + + + + + + +

- - - - - - - - - - - - - -

To calculate the force due to electric field on the dielectric

material

Let the external agent pulls the dielectric out by a infinitesimal displacement dx

Fext

dW = Fext dx Fext = dW/dx

Electric force on the dielectric = -Fext

x

Plate area is L x L

+ + + + + + + + + + + +

- - - - - - - - - - - - - -

Charge on the plates is constant

C

QW

2

2

1

d

xLkL

d

LxC

)(00

kLkxd

LC )1(0

dx

dC

CQ

dx

dWFext 2

2 1

2

1

exkLd

LC

0

dx

dCVFext

2

2

1

ed

L

dx

dC 0

20

2V

d

LF eext

ed

L

dx

dC 0

20

2V

d

LFF eextE

If the battery maintains a constant potential

VdQdxFdW ext

VdQdxFdW ext

dx

dCVF

dx

dQVF

dx

dCV

dx

dWextext

22

2

1

dx

dCVFext

2

2

1

The force simply depends only upon the fringing field and free and bound charges

Two coaxial metal tubes stand vertically in a tank of dielectric oil (susceptibility e and mass density . Tubes are maintained at a potential difference of V. To what height (h) does the oil rise in the space between the tubes.

Problem

Griffiths Problem 4.28, page 196 vol 3.

abk

h

ab

hlC e

ln

2

ln

2)( 00

hkl

ab

C )1(ln

2 0

mgdh

dCVFE 2

2

1

ghab

ab

V e )(

ln

2202

Calculate the electrostatic energy stored between the plates of the cylindrical capacitor using the relation

dEU 2

2

1

dzrddrrL

qU

2

22

1

C

qU

2

2

1

r: a to b

: 0 to 2

Z: 0 to L

Problem 4.21 Griffiths• A certain coaxial cable consists of a copper

wire , radius a , surrounded by a concentric copper tube of inner radius c. The space between is partially filled (from b out to c) with a material of dielectric constant ke. Find

the capacitance per unit length of this cable.

a

b

c

Find the capacitance per unit length of this cable

a

b

cDielectric Material

a

b

cV

QC

rdEVVVc

a

ca

rdErdEVc

b

b

a

drrk

qdr

Lr

qV

c

b e

b

a

00 22

b

c

Lk

q

a

b

L

qV

e

ln2

ln2 00

bc

kab

LC

e

ln1

ln

2 0

A parallel plate capacitor is filled with a dielectric of dielectric constant ke. The ke

varies parallel to an edge as

• Where x is the distance from the left end. Calculate the capacitance.

xkke 0

d

xaxkC

)( 00

dxd

axkC

a

0

00 )(

Find the Ceq

A

B

Twelve capacitors, each have capacity C are connected to form a cube.

A

B

E

F

BFFEEABA VVVVVVVV

BAeq VV

QC

A

B

E

F

Q/3

Q/6

Q/3

Isolated system

Total charge zero

A

B

E

F

Q/3

Q/6

Q/3

BFFEEABA VVVVVVVV

C

Q

C

Q

C

Q

C

Q

eq

1

3

1

6

1

3

C

Q

C

Q

C

Q

C

Q

eq

1

3

1

6

1

3

CCeq 5

6