The c equivalence principle and the correct form of writing Maxwell's equations

TL;DR

This paper clarifies the correct form of Maxwell's equations without assuming the $c$ equivalence principle, highlighting the distinction between the SI-defined speed $c_u$ and the observed electromagnetic wave speed $c$.

Contribution

It presents the correct form of Faraday's law and discusses the covariant form of Maxwell's equations without relying on the $c$ equivalence principle.

Findings

Correct form of Faraday's law without $c$ equivalence assumption

Distinction between $c_u$ and $c$ in Maxwell's equations

Discussion of covariant Maxwell's equations without $c$ equivalence

Abstract

It is well-known that the speed $c_u=1/\sqrt{\epsilon_0\mu_0}$ is obtained in the process of defining SI units via action-at-a-distance forces, like the force between two static charges and the force between two long and parallel currents. The speed $c_u$ is then physically different from the observed speed of propagation $c$ associated with electromagnetic waves in vacuum. However, repeated experiments have led to the numerical equality $c_u=c,$ which we have called the $c$ equivalence principle. In this paper we point out that $\nabla\times{\bf E}=-[1/(\epsilon_0\mu_0 c^2)]\partial{\bf B}/\partial t$ is the correct form of writing Faraday's law when the $c$ equivalence principle is not assumed. We also discuss the covariant form of Maxwell's equations without assuming the $c$ equivalence principle.