Surface charge density of current-carrying conductors: An exact analytical solution for infinitely thin wires of arbitrary shape

Abstract

An exact asymptotic expression is derived for the surface charge density associated with a steady current in closed loops or wire segments of infinitesimal cross section connected to a battery. Except for vanishingly thin boundary layers at the ends of the wire, and irrespective of the conductor shape, the charge density varies linearly with arc length, as does the electrostatic potential along the curve. This proportionality generalizes earlier results and provides a clear physical picture of charge accumulation in electrical circuits.