Such a pair of concentric spheres constitute a condenser (see LEYDEN
JAR), and it is obvious that by making R2 nearly equal to R1, we may
enormously increase the capacity of the inner sphere. Hence the name
_condenser_.
Capacity of two coaxial cylinders.
The other case of importance is that of two coaxial cylinders. Let a
solid circular sectioned cylinder of radius R1 be enclosed in a
coaxial tube of inner radius R2. Then when the inner cylinder is at
potential V1 and the outer one kept at potential V2 the lines of
electric force between the cylinders are radial. Hence the electric
force E in the interspace varies inversely as the distance from the
axis. Accordingly the potential V at any point in the interspace is
given by
_
/
E = -dV/dR = A/R or V = -A | R^-1 dR, (12),
_/
where R is the distance of the point in the interspace from the axis,
and A is a constant. Hence V2 - V1 = -A log R2/R1. If we consider a
length l of the cylinder, the charge Q on the inner cylinder is Q =
2[pi]R1l[sigma], where [sigma] is the surface density, and by
Coulomb's law [sigma] = E1/4 [pi], where E1 = A/R1 is the force at the
surface of the inner cylinder.
Accordingly Q = 2[pi]R1lA/4[pi]R1 = Al/2. If then the outer cylinder
be at zero potential the potential V of the inner one is