XI. The electrification of two parallel circular discs

The only problem relating to two electrified circular discs, placed parallel to each other, for which an exact solution has been obtained hitherto, is the classical one of Nobili’s rings. This was solved by Riemann,* by an application of the Bessel-Fourier integral method. In this problem the discs are circular electrodes fixed to two infinite conducting planes, which are themselves connected together by the earth or by a wire at infinity. If the axis of z is that of the two co-axial discs, and perpendicular to the infinite plane conducting sheets, the electrical potential V satisfies Laplace’s equation at all points between the plates, and the further conditions (1) ∂V/∂ z = 0, z = ± a , p > p 1 (2) ∂v/∂ z = A/√(r 1 2 —r 2 ), z = ± a , p < p 1 where A is a constant, 2 a is the distance between the plates, bisected by the origin, p 1 is the radius of either disc, and p is the distance of any point from the axis of z . In fact ( z , p ) are the two cylindrical polar co-ordinates on which V can alone depend.