On the problem of the electrified disc

Abstract

When a perfectly conducting uniform thin circular disc is kept at a potential V0 in an external electrostatic field of potential Φ, electric charge is induced on the surface of the disc; the problem is to find the surface-density σ of this induced charge and its potential V so that the total potential V + Φ has the constant value V0 on the surface of the disc. This problem was first discussed by Green in 1832, and the solution in the case when there is no external field was deduced by Lord Kelvin from the known formula for the gravitational potential of an elliptic homoeoid. The problem is still of interest since similar ideas occur in the theory of diffraction by a circular disc and in the theory of the generation of sound waves by a vibrating disc when the wave-length is large compared with the radius of the disc.