An explicit representation for the axisymmetric solutions of the free Maxwell equations

TL;DR

This paper proves that linear combinations of two classes of axisymmetric solutions fully describe all totally propagating time-harmonic free Maxwell fields, providing an explicit representation useful for modeling astrophysical electromagnetic phenomena.

Contribution

It establishes that these combined solutions form a complete set for axisymmetric Maxwell fields, offering a new explicit representation.

Findings

All totally propagating axisymmetric Maxwell fields can be expressed as linear combinations of the two classes.

The representation applies to time-harmonic fields and extends to all frequencies.

Potential applications include modeling interstellar radiation fields as exact solutions.

Abstract

Garay-Avenda\~no \& Zamboni-Rached (2014) defined two classes of axisymmetric solutions of the free Maxwell equations. We prove that the linear combinations of these two classes of solutions cover all totally propagating time-harmonic axisymmetric free Maxwell fields -- and hence, by summation on frequencies, all totally propagating axisymmetric free Maxwell fields. It provides an explicit representation for these fields. This will be important, e.g., to have the interstellar radiation field in a disc galaxy modelled as an exact solution of the free Maxwell equations. Keywords: Maxwell equations; axial symmetry; exact solutions; electromagnetic duality