Capacitance of a Circular Disk Placed Over a Grounded Substrate

Abstract

The electrostatic problem of a conducting circular disk resting over a grounded dielectric slab is formulated in terms of dual integral equations. Following Sneddon [9] the dual integral equations are replaced by a more manageable inhomogeneous Fredholm integral equation of the second kind in which the unknown represents the charge density. The solution of the latter equation is expressed in terms of Neumann series whose integral yields the capacitance. For values of slab thickness small compared to the radius of the disk convergence of the series is very slow and numerical evaluation became so tedious that one could not proceed beyond the sixth term. However, by applying Shank and Aitken acceleration methods to the partial sums very accurate results have been realized. Yet another powerful method of solution is based on calculus of variations. A lower bound for the capacitance in terms of charge density has been obtained. Trial function for the charge density consists of two terms, a constant plus one with an appropriate singularity at the edge. These are weighted in such manner that the constant and singular term dominate, respectively, for small and large slab thicknesses which is in keeping with what rnight one expect in reality. Plots of capacitance versus thickness of the grounded slab with dielectric constant as a parameter are provided.