Lecture 6

Capacitance and CapacitorsElectrostatic Potential Energy

Prof. Viviana Vladutescu

Capacitance

• A conductor in electrostatic field is equipotential and charges distribute themselves on the surface such way that E=0 inside the conductor Q on the surface is producing V

dsR

Vs

s

04

1

k

QV

C

QV

linear dependence k=C

V

CFC SI

Depends on –the geometry of the conductors -the dielectric constant of the medium between conductors

Capacitance (of the isolated conducting body) - is the electric charge that is added to the body per unit increase in its electric potential (is a constant of proportionality)

Capacitors

Electrolytic capacitors

Determine capacitance

1- assume Vab Q (in terms of Vab)

use boundary conditions

2- assume Q Vab (in terms of Q)

Q Vab

• Step 1Chose coordinate system for given geometry

• Step 2 Assume +Q and –Q on the conductors

• Step 3 Q E from D=εE=ρs or

• Step 4 E

• Step 5 C=Q/Vab

s

QdsE

a

b

ab ldEV

Example

z

z

aS

QE

aD

E

S

QsQ ss

ED

S

QD s

Step 1

Step 2

Step 3

d

z

a

b

zab S

Qddz

S

Qadza

S

QV

0

d

S

d

S

SQdQ

V

QC r

r

ab

0

0

Step 4

Step 5

Vab Q

• Step 1Chose coordinate system for given geometry

• Step 2 Assume Vab between plates

• Step 3 Vab E D (from Laplace’s equation)

• Step 4 Boundary conditions at each plate

conductor –dielectric boundary:

ρs Q

. Step 5 C=Q/Vab

sND

Example

z

Step 1

Step 2 Vab

Step 3 Laplace’s equation to find the potential everywhere in the dielectric

02 V

There is no φ and z variation

BrArVr

A

r

VA

r

Vr

r

Vr

rr

Vr

rr

ln)(

0

001

ba

VA

b

aAVbAaAV ab

abab

lnlnlnln

b

r

ab

V

b

r

ba

VbAr

ba

VrV ababab ln

lnln

lnlnln

ln)(

BbAbV

BaAVVaV abab

ln0)(br

ln)( arfor

Step 4

rabab

r

a

ab

r

VE

br

b

ab

V

arVr

VE

ln1

1

ln

)( that know but we

ab

a

Va

ab

r

VED abr

arsrabr

r

lnln

000

abVL

aL

ab

a

VSds abrabr

ln

2)2(

lnQ 00

s

s

s

Q on the inner conductor

Step 5

ab

L

VabVL

V

QC r

ab

abr

ab ln

2

ln

2 00

Series Connected Capacitors

nsr CCCCC

1..........

1111

321

Parallel Connected Capacitors

nCCCCC ...........321

Electrostatic Potential Energy

Electric potential at a point in an electric field is the work required to bring a unit positive charge from infinity (at reference zero potential) to that point.

)(2

1

4

11222

11120

12222

VQVQW

VQR

QQVQW

Now suppose we want to bring Q3 at R13 from Q1

and R23 from Q2

230

2

130

1333 44 R

Q

R

QQVQW

23

32

13

31

12

21

02 4

1

R

QQ

R

QQ

R

QQWWWtotal

)44

4444(

2

1

230

2

130

13

230

3

120

12

130

3

120

213

R

Q

R

QQ

R

Q

R

QQ

R

Q

R

QQWWtotal

3322112

1VQVQVQWt

-can be negative -represents only interaction energy

(eV)or )( 2

1

1N

kke JVQW

dvVWv

e 2

1

For a group of N discrete charges at rest

For a continuous charge distribution of density ρ

C

QQVCVWe 22

1

2

1 22

Electrostatic energy in terms of field quantities

v

e VdvDW2

1

VDDVDV

VdvDdvDVWv v

e 2

1

2

1

dvEDdsaDVv

n

s

2

1

2

1

Substitute ρ

And by using

Electrostatic Energy Density

v

e

v v

e

dvD

W

dvEdvEDW

(J) 2

1

2

1

2

1

2

2

3

22

mJ

22

1

2

1

DEEDwdvwW

v

eee

Equipotential surfaces are at right angles to the electric field. Otherwise a force would act and work would be done on the path A to B.For a uniform electric field, equipotentials form planes perpendicular to the field.

Along AB, W = -q∆V = zero!

Example