Capacity of the circular plate condenser: analytical solutions for large gaps between the plates

Abstract

A solution of Love's integral equation (Love E R 1949 Q. J. Mech. Appl. Math. 2 428), which forms the basis for the analysis of the electrostatic field due to two equal circular co-axial parallel conducting plates, is considered for the case when the ratio, ?, of distance of separation to radius of the plates is greater than 2. The kernel of the integral equation is expanded into an infinite series in odd powers of 1/? and an approximate kernel accurate to is deduced therefrom by terminating the series after an arbitrary but finite number of terms, N. The approximate kernel is rearranged into a degenerate form and the integral equation with this kernel is reduced to a system of N linear equations. An explicit analytical solution is obtained for N = 4 and the resulting analytical expression for the capacity of the circular plate condenser is shown to be accurate to . Analytical expressions of lower orders of accuracy with respect to 1/? are deduced from the four-term (i.e., N = 4) solution and predictions (of capacity) from the expressions of different orders of accuracy (with respect to 1/?) are compared with very accurate numerical solutions obtained by solving the linear system for large enough N. It is shown that the approximation predicts the capacity extremely well for any ? ? 2 and an approximation gives, for all practical purposes, results of adequate accuracy for ? ? 4. It is further shown that an approximate solution, applicable for the case of large distances of separation between the plates, due to Sneddon (Sneddon I N 1966 Mixed Boundary Value Problems in Potential Theory (Amsterdam: North-Holland) pp 230?46) is accurate to for ? ? 2.