Unphysical Moment-Method Solutions of an Approximate Integral Equation of Electrostatics [Open Problems in Computational EM]

The electrostatic charge distribution on a conducting cylindrical wire exactly satisfies an integral equation. Many textbooks discuss an approximate form of this equation and solve it using the method of moments (MoM). This approximate electrostatics equation admits MoM solutions that are unphysical, and we will explain why such solutions occur and propose several remedies. We include a rigorous theorem pertinent to the solvability of the equation. Our approach has parallels in recent works on Hallén's and Pocklington's equations (HE/PE), in particular, because we use the wire of infinite length as a means of deducing information about the actual (finite-length) wire. Our analysis of the infinite wire has particular difficulties that are not present in the works on HE/PE.