The Galilean limits of Maxwell's equations

TL;DR

This paper explores three Galilean limits of Maxwell's equations, including electric, magnetic, and an instantaneous limit, highlighting the role of the speed of light in different nonrelativistic regimes.

Contribution

It introduces and analyzes three distinct Galilean limits of Maxwell's equations, clarifying their physical interpretations and the role of the speed of light.

Findings

Electric and magnetic limits are nonrelativistic with $|f{v}| \\ll c$

The instantaneous limit is obtained by letting $c \\to \\infty$

The instantaneous limit differs in interpretation from the other two limits

Abstract

We show that if Maxwell's equations are expressed in a form independent of specific units, at least three Galilean limits can be extracted. The electric and magnetic limits can be regarded as nonrelativistic limits because they are obtained using the condition $|\bf{v}| \ll c$ and restrictions on the magnitudes of the sources and fields. The third limit is called the instantaneous limit and is introduced by letting $c\to \infty$. The electric and instantaneous limits have the same form, but their interpretation is different because the instantaneous limit cannot be considered as a nonrelativistic limit. We emphasize the double role that the speed of light $c$ plays in Maxwell's equations.