A geometric decomposition of the Coulomb integral for uniform planar objects

Abstract

We present a vector-calculus derivation of the electrostatic potential of uniformly charged planar objects: the circular disk, the arbitrary triangular plate, the rectangular plate and the regular polygon. By geometrically decomposing the Coulomb integral, the potential is expressed as a combination of a solid-angle term and a boundary line integral. For the disk, this gives closed-form off-axis expressions in terms of complete elliptic integrals and the Heuman Lambda function; for the finite plates only elementary functions are needed. Continuity of the potential and discontinuity of the axial field inside and at boundary of the disk are analyzed, including edge and corner behavior for the plates. Exact self-energy expressions are derived for the disk, the equilateral triangle, the rectangle and numerically for regular polygon under this decomposition. For regular n -gon, compact expressions are obtained that approaches the disk results analytically in the large n limit.