Capacitance and Geometry

Great mysteries of the universe elucidated by your good friend Gauss…

Capacitance

• Capacitors are charge-storage devices– Capacitance, C, is a

measure of the ability to store charge

• You have to do work to store charge on a capacitor – Energy is stored in the e-

field between the cap’s electrodes

V

QC

1 C/volt = 1 farad [F]

221 CVUE

Parallel-plate capacitors

Consider two plates of opposite charge separated by a gap d and carrying a charge density

The field between the plates is

and the field beyond them is zero. ooo

EEE

22

Parallel-plate capacitors

Recall that

Since and

the potential difference across this capacitor is

Given the definition of capacitance

the capacitance of this geometry is

q

dqE

q

WV

)(

o

E

A

q

V

QC

d

AC o

oA

qdV

Cylindrical capacitors

Inner cylinder carries a charge density of + while the outer carries -

At any arbitrary location

so a

b

b

a

baab EdrqEdr

qq

WVV

1

a

bV ln

2 0

rE

o

2

therefore

Cylindrical capacitors

Since

for this geometry,

LQ

abL

Cln

2 0

Spherical shellsConsider an inner shell

charged at +Q and an outer shell charged at -Q

At any arbitrary location

and a

b

ab EdrVV

ba

QV

11

4 0

24 r

QE

o

enclosed

and for this geometry

therefore

ab

abC

04

Isolated conductors as capacitors

Think of the 2nd electrode as infinitely far away

• At the surface,

• At the 2nd electrode V=0, so

for this geometryaC o4

a

QV

o4