Computer Solution for Time-Invariant Electric Fields

Abstract

A general method suitable for computerized solution is described for finding the electric and/or current density fields due to time-invariant sources. A charge distribution satisfying the boundary conditions is specified as the solution of an integral equation. The latter is approximated as a system of linear algebraic equations and solved by the computer. The field at any point is then obtained by Coulomb's law and superposition from the boundary charge distribution and any impressed fields that exist. The method will, within the practical restrictions imposed by the computer's capacity to solve simultaneous equations, solve any inhomogeneous, linear, isotropic, steady current flow and/or static field problem with any of the various boundary conditions which uniquely define a field. While the derivation is given here in terms of electric field quantities, it applies as well to the several analogous fields of engineering and physics. In addition to the theoretical formulation, data are given to guide in the selection of the most efficient and accurate formulas with which to approximate the integral equation. Several examples and results are presented together with data bearing on questions of accuracy and convergence. Two example results are combined to derive a quasi-empirical formula for the capacitance of widely spaced square parallel plates.