Non-relativistic limits of Maxwell's equations

TL;DR

This paper systematically derives the two non-relativistic Galilei-covariant limits of Maxwell's equations, clarifying their conditions and providing a unified framework that recovers previous results and suggests extensions.

Contribution

It offers a simple, systematic derivation of Maxwell's non-relativistic limits using a dimensionless approach and power series expansion, unifying and extending prior findings.

Findings

Recovered known electric and magnetic limits systematically

Provided a clear derivation framework based on dimensionless equations

Suggested possible extensions to the classical limits

Abstract

In 1973, Le Bellac and Levy-Leblond (Nuovo Cimento B 14, 217-234) discovered that Maxwell's equations possess two non-relativistic Galilei-covariant limits, corresponding to E >> cB (electric limit) or E << cB (magnetic limit). Here, we provide a systematic, yet simple, derivation of these two limits based on a dimensionless form of Maxwell's equations and an expansion of the electric and magnetic fields in a power series of some small parameters. Using this procedure, all previously known results are recovered in a natural and unambiguous way. Some further extensions are also proposed.