The Electrostatic Potential Field About a Thin Oblate Body of Revolution

Abstract

The electrostatic potential about a thin, perfectly conducting oblate body of revolution is studied. The first three terms are obtained in the uniform asymptotic expansion of the potential, with respect to the thinness parameter of the body. The body, which may be charged or uncharged, is assumed to possess a plane of symmetry perpendicular to the body axis, but is immersed in an arbitrary extenal field. The part of the potential due to the presence of the body is represented as the superposition of potentials, due to point sources and dipoles, which are distributed over a portion of the plane of symmetry and lie entirely inside the body. The boundary condition on the surface of the body leads to a linear integral equation for the densities of the singularity distributions. This equation is solved asymptotically for the densities by using and extending the method of Handelsman and Kelley [J. Fluid Mech, 28 (1967), pp. 131–142]. The extent of the distribution within the body is uniquely determined by requiring that the solution be uniformly valid, i.e., that it be valid near (or even on) the surface of the body, as well as far from the body. The results are then used to present formulas for the capacitance and polarizability of the body.