Week 4: Capacitance 133
4.1: Capacitance
In the previous chapter we noted thatconductors in electrostatic equilibrium are equipotential. If
you imagine charging up any given conductor, every new bit of charge we add to it spreads itself
out the same way. One expects the field produced at its surface toscale up or down proportional
to the amount of charge on the conductor but not change its basicshape. As a consequence, one
expects thepotentialproduced by the conductor to be proportional to its total charge at all points
in space, in particular inside the equipotential conductor itself.
This has been apparent in all of our Gauss’s Law examples up to now. For example, a conducting
sphere of radiusR, charged with a total chargeQ, has a field:
Er =keQ
r^2(r > R) (209)
= 0 (r < Rinside the conductor) (210)If we integrate this to find the potential everywhere in space we get:
V = −∫r∞kQ
r^2dr= 0keQ
r(r≥R) (211)The conductor isequipotential, so the potential inside is the same as at its surface:
V=keQ
R(r < R) (212)We have seen how justknowingthis solution for spherical shells, or the equivalent solution for
cylindrical shells, can greatly improve our ability to solve problems quickly and easily by using
superposition of these once-and-for-all solutions instead of trying to explicitly integrate the fields
across all the different forms it might take in a problem with several conducting shells, although of
course one will get the same answer either way.
Our discussion of capacitancebeginswith the observation that in this case (and the others we
can solve, and other ”odd” shaped conductors that we cannot) the potential of the conductor is
directly proportional tothe total charge on the conductor, and that the parameters in the potential
besides the charge arekeand things that describe its geometry, such as its physical dimensions and
shape.
We could thus define a quantity we might call the “volticitance” of theconductorVso that (in
the case of this example):
V=VQ (213)
with
V=ke
R
=^1
4 πǫ 0 R(214)
However, we often use conductors in particular arrangements tostore charge. In general, we
would like to be able to store alotof charge on them with only asmallpotential difference. We thus
seek instead a measure of thecapacityof the conductor to store charge at any given voltage:
Q=CV=(
1
V
)
V= (4πǫ 0 R)V (215)where we have introduced thecapacitance, the constant of proportionality that depends only on the
geometry of the conductor.
To be specific, we define thecapacitanceof an arrangement of conductors used to store charge
to be:
C=