Exact solution to Maxwell's equations for the infinite ideal solenoid with a time-dependent surface current

Abstract

Very little previous literature has considered the *exact* solution to Maxwell's equations for an infinite ideal cylindrical solenoid with an arbitrary time-dependent azimuthal surface current $K(t) \hat{\bf \phi}$. Most of the previous literature has focused on special cases and has approached the problem by calculating the magnetic vector potential ${\bf A}$, which requires performing some very complicated surface integrals over the cylinder. In this article, we take a simpler approach and directly tackle Maxwell's equations without ever invoking a vector potential. The high symmetry of the geometry allows us to reduce Maxwell's equations to just two coupled partial differential equations for two functions of two real variables, which can be readily solved numerically. We find the general analytic solution to these PDEs and derive the Green's functions for the electromagnetic fields, which allow us to calculate the fields directly from the surface current $K(t)$. We also briefly discuss a family of exact formal solutions that (the author believes) has not appeared in the previous literature because it corresponds to a current $K(t)$ that does not have a Fourier transform.