Fields Due to Electrons on an Analytic Curve

Abstract

Let D be the interior of a simple closed analytic curve C and let $z_{n1} , \cdots ,z_{nn} $, be points on C. Assuming a logarithmic potential, the electrostatic field due to electrons (charges $ - \varepsilon $) at the points $z_{nk} $ may be represented as (the complex conjugate of) \[ \varepsilon E_n (z) = \varepsilon \sum_{k = 1}^n {\frac{1}{{z_{nk} - z}}} .\] The authors give necessary and sufficient conditions under which $E_n (z) \to 0$ as $n \to \infty $ uniformly on the compact subsets of D. For equilibrium distributions of electrons (thinking of C as a conductor) the fields $\varepsilon E_n (z)$ tend to a limit $\varepsilon E(z)$ holomorphic in the closure of D. The limiting field is identically zero if and only if C is a circle. For general analytic C, the limiting field is of the same order of smallness as the field due to a single electron outside D.