On the capacity of the circular disc condenser at small separation

Abstract

A pair of identical circular discs, held at equal and opposite potentials, forms a condenser whose capacity C depends on the ratio ε of separation against diameter. The determination of an asymptotic expansion for C when ε is small poses an axisymmetric boundary-value problem for harmonic functions that has engaged the attention of numerous investigators over a long span of time. It is a simple matter to construct a Fredholm integral equation of the first kind for the charge density ± σ on the discs, in terms of which the potential field and the capacity are implicitly determined, but the equation is unsuitable if ε ≪ 1. Integral equations of the second kind and of the dual variety have also been proposed as a means of securing a more manageable formulation of the boundary-value problem. An elementary approximation follows from the hypothesis that the charge density is almost the same as though the discs were of infinite extent, except for a region close to the edges, and leads to the result C ∼ l/8ε as ε → 0. Kirchhoff considerably improved on this crude estimate by suggesting a plausible edge correction which yields two further terms for C, of orders log ∈ and a constant, respectively, and his results have been rigorously established by the more refined analysis of Hutson. In the present work an integral equation of the first kind for the distribution of potential off the discs is derived and utilized to obtain an approximation for C when ε is small, reproducing the result of Kirchhoff and Hutson. Furthermore, an estimate of the error provides explicit details regarding the next term in the asymptotic expansion of C, which is of the order ε(log ε)².