Relational electromagnetism and the electromagnetic force

TL;DR

This paper revisits the electromagnetic force through Ludwig Lorenz's relational electromagnetism, providing a non-analogical derivation consistent with Maxwell's laws and invariant light speed.

Contribution

It introduces a new derivation of the electromagnetic force based on relational electromagnetism, avoiding traditional analogies and aligning with Maxwell's transformation laws.

Findings

Relational electromagnetism is consistent with Maxwell's laws.

The derivation supports invariant light speed in all inertial frames.

The approach challenges traditional analogy-based formulations.

Abstract

The force exerted by an electromagnetic body on another body in relative motion, and its minimal expression, the force on moving charges or \emph{Lorentz' force} constitute the link between electromagnetism and mechanics. Expressions for the force were produced first by Maxwell and later by H. A. Lorentz, but their expressions disagree. The construction process was the result, in both cases, of analogies rooted in the idea of the ether. Yet, the expression of the force has remained despite its production context. We present a path to the electromagnetic force that starts from Ludwig Lorenz' relational electromagnetism. The present mathematical abduction does not rest on analogies. Following this path we show that relational electromagnetism, as pursued by the G\"ottingen school, is consistent with Maxwell's transformation laws and compatible with the idea that the "speed of light" takes the same value in all (inertial) frames of reference, while it cannot be conceived on the basis of analogies with material motion.