What is a capacitor?
Any two pieces of metal, which are separated
by an insulator, have a capacitance. The capacitance
of the two pieces of metal tells us how much
voltage we need to apply in order to place a
charge Q on one of the pieces of metal and a
charge -Q on the other. That is,
Q = CV | (1) |
Calculating capacitance
Parallel plates
Consider two square parallel plates with charge Q (positive
Q on one plate and negative Q on the other). The
plates are separated by distance d and have area A = L2.
Consider L>>d, so we can approximate the magnitude of the electric field
between the plates by
.
The potential difference
between the places is
(2) |
(3) |
Coaxial cable
A co-axial cable consists of an inner conducting rod of
radius a and an outer conducting cylindrical shell
of radious b. Here we assume that between these two
radii there is air, however in reality there is a very
good insulator. We want to find the capacitance of this
system, per unit length. Assuming that the
rod and outer shell have equal and opposite charge
densities ,
the magnitude of the electric field in the
regime a<r<b is given by,
(4) |
(5) |
(6) |
An isolated sphere
Since an isolated sphere, of radius R, has a well defined
potential with respect to infinity, we can also
calculate the capacitance of the isolated sphere
with respect to infinity. We have,
(7) |
(8) |
The energy stored in a capacitor
When we talk of the energy stored in a capacitor,
we are refering to the electrostatic potential
energy. The stored electrostatic potential
energy is equal to the work done in
separating the charges +Q and -Q which
are on the two metal parts of the capacitor.
As in the case of setting up a distribution
of point charges, we start with the capacitor
with no charges on the plates. For each
small element of positive charge, dq, that we
move from the negative plate to the positive
plate, we increase the potential energy by
dU = V(q) dq. Since
V(q) = q/C, the
potential energy change is
dU = qdq/CWe continue this process until the
total charge Q is on the capacitor.
The total potential energy is then,
(9) |
The energy in the electric field
An important way of thinking about the energy stored
in a capacitor is that the energy actually goes into
establishing the electric field. This is evident in the case
of the parallel plate capacitor, where we can use V=Ed to
write,
(10) |
(11) |