Compact complex expressions for the electric field of two-dimensional elliptical charge distributions

Abstract

We present a formula for the analytic calculation of the electric field in complex form for two-dimensional charge distributions with elliptical contours, in the absence of boundary conditions except at infinity. The formula yields compact and practical expressions for a significant class of distributions. The fact that the electric field vanishes inside an elliptical shell follows as a straightforward consequence of Cauchy’s theorem. The known expressions for the field inside and outside a uniformly charged ellipse are recovered in simple, concise form. Similarly, the expression for the field of a Gaussian distribution is found in a straightforward way as a special case of the more general formula. We also present a brief discussion of more complicated distributions.