Consider the case of two concentric spheres, a solid one enclosed in a hollow one. Let R1 be the radius of the inner sphere, R2 the inside radius of the outer sphere, and R2 the outside radius of the outer spherical shell. Let a charge +Q be Capacity of two concentric spheres. given to the inner sphere. Then this produces a charge −Q on the inside of the enclosing spherical shell, and a charge +Q on the outside of the shell. Hence the potential V at the centre of the inner sphere is given by V = Q/R1 − Q/R2 + Q/R3. If the outer shell is connected to the earth, the charge +Q on it disappears, and we have the capacity C of the inner sphere given by
C = 1/R1 − 1/R2 = (R2 − R1) / R1R2
(11).
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.
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 Capacity of two coaxial cylinders. 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 = 2πR1lσ, where σ is the surface density, and by Coulomb’s law σ = E1/4π, where E1 = A/R1 is the force at the surface of the inner cylinder.
Accordingly Q = 2πR1lA / 4πR1 = Al/2. If then the outer cylinder be at zero potential the potential V of the inner one is
V = A log (R2/R1), and its capacity C = l/2 log R2/R1.