Perturbation Approach to the Electrostatic Problem for Irregularly Shaped Conductors

Abstract

A simple and straightforward perturbation method for treating the electrostatic problem of a charged, irregularly shaped conductor is presented. The perturbation solution is generated starting from the zero‐order solution for a charged sphere. The method consists of expanding the boundary condition in a Taylor series, which in effect transforms the boundary condition at the irregular boundary into a succession of boundary conditions to be satisfied at the surface of a sphere. The simplicity of the formalism consists further in applying, in a consistent manner, sufficient rather than necessary conditions on the successive correction potentials. First‐ and second‐order expressions for the potential, surface charge density, and capacitance of irregularly shaped conductors, are derived explicitly, and an elementary theorem for the first‐order capacitance is obtained. A perturbation expansion for the capacitance valid to all orders is presented. The application of the method is illustrated by calculating the capacitance of several irregularly shaped conductors. Possible generalizations to more complicated boundary‐value problems are indicated.