An infinite, thin, plane sheet of conducting material has a circular hole of radius a cut in it. A thin, flat disc of the same material and slightly smaller radius lies in the plane, filling the hole, but separated from the sheet by a very narrow insulating ring. The disc is maintained at a fixed potential V, while the infinite sheet is kept at zero potential.

(a) Using appropriate cylindrical coordinates, find an integral expression involving Bessel functions for the potential at any point above the plane.

(b) Show that the potential a perpendicular distance z above the center of the disc is

Ф₀(z) = V(1 – z /√a² + z²)

(c) Show that the potential a perpendicular distance z above the edge of the disc is

Фa(z) = V/2p[1 – kz/πa k(k)]

Where k = 2a/(z² + 4a²)^1/2, and K(k) is the complete elliptic integral of the first kind.