Power series expansion solution to a classical problem in electrostatics

Abstract

Using a power series expansion and differential calculus, the normal derivative of the magnitude of the static electric field near the surface of a curved conductor is obtained. This provides an alternative way of solving a problem which is usually treated by an application of Gauss’s theorem.

On the dependence of charge density on surface curvature of an isolated conductor

“…Sir J J Thomson gave recognition to this relation in 1891 [4]. Various methods were used by different authors [5], [6], over the past century, to prove Eq. (1).…”

Unexplored aspects of a variational principle in electrostatics

“…where dn denotes a differential length element in the direction of the outward normal at a point on the equipotential surface. This equation has been proved by many authors [11,12,13], often by lengthy approaches and differential geometric techniques. Here we present a simple proof of the theorem using Eq.…”

The Geometric Solution of Laplace's Equation

“…In an arbitrary coordinate system X Ϫ Y, ͳ x , ͳ y are the An alternative approach to the same problem was also direction cosines of the vector E. presented by Mayo [5] and still requires an O(M и N) operation. Very recently, Greenbaum et al [4] presented an algorithm that is based on the Rokhlin method for solving Thomson's formula was later proved by others [14][15][16]. Laplace's equation in multiply connected domains.…”