Equilibrium Charge Density on a Thin Curved Wire

Abstract

This work addresses the electrostatic problem of a thin, curved, cylindrical conductor, or a conducting filament, and shows that the corresponding linear charge density slowly tends to uniformity as the inverse of the logarithm of a characteristic parameter. An alternative derivation of this result directly based on energy minimization is developed. These results are based on a general asymptotic analysis of the electric field components and potential near a charge filament in the limit of vanishing diameter. It is found that the divergent parts of the radial and azimuthal electric field components, as well as the electric potential, are determined by the local charge density while the axial component is determined by the local dipole density. For a straight filament, these results reduce to those for conducting needles discussed in the literature. For curved filaments, there is an additional length scale in the problem arising from the finite radius of curvature of the filament. Remarkably, the basic uniformity result survives the added complications, which include an azimuthal variation in the surface charge density of the filament. These uniformity results yield an asymptotic formula for the capacitance of a curved filament that generalizes Maxwell's original result. The examples of a straight filament with uniform and linearly varying charge densities, as well as a circular filament with a uniform charge distribution, are treated analytically and found to be in agreement with the results of the general analysis. Numerical calculations illustrating the slow convergence of linear charge distribution to uniformity for an elliptical filament are presented, and an interactive computer program implementing and animating the numerical calculations is provided.