On the Generalized Potential Problem for a Surface of Revolution

Abstract

The following generalized potential problem is considered: given an arbitrary potential distribution on a surface of revolution, the charge density is to be determined. The problem is called generalized due to the assumption that the point charge potential diminishes with the distance $R$ as ${R^{ - 1 - v}}$, $\left | v \right | < 1$. The particular case of $v = 0$ corresponds to the electrostatic potential problem. A closed form solution to the problem is obtained for a certain class of surfaces of revolution due to a special integral representation of the kernel of the governing integral equation. Three examples are considered: the uniform potential distribution over a spherical cap, the case of an earthed conducting spherical cap in a uniform external field acting in two different directions, namely along the $0z$ axis and the $0x$ axis, respectively. The expressions for the charge density distributions are determined for each of the examples. The general results presented in this paper may also be applied to the solution of the mathematically identitical problems in hydrodynamics, thermal conductivity, etc.