Force, Torque, Linear Momentum, and Angular Momentum in Classical Electrodynamics

TL;DR

This paper compares the classical Lorentz and Einstein-Laub formulations of electromagnetic force and torque, highlighting differences in force distribution inside matter and implications for conservation laws and relativity.

Contribution

It clarifies the differences between Lorentz and Einstein-Laub force laws, emphasizing the absence of hidden entities in Einstein-Laub and their measurable distribution differences.

Findings

Total force and torque are independent of the formulation used.

Einstein-Laub law does not require hidden energy or momentum.

Differences in force distribution inside matter are potentially measurable.

Abstract

The classical theory of electrodynamics is built upon Maxwell's equations and the concepts of electromagnetic (EM) field, force, energy, and momentum, which are intimately tied together by Poynting's theorem and by the Lorentz force law. Whereas Maxwell's equations relate the fields to their material sources, Poynting's theorem governs the flow of EM energy and its exchange between fields and material media, while the Lorentz law regulates the back-and-forth transfer of momentum between the media and the fields. An alternative force law, first proposed by Einstein and Laub, exists that is consistent with Maxwell's equations and complies with the conservation laws as well as with the requirements of special relativity. While the Lorentz law requires the introduction of hidden energy and hidden momentum in situations where an electric field acts on a magnetized medium, the Einstein-Laub (E-L) formulation of EM force and torque does not invoke hidden entities under such circumstances. Moreover, total force/torque exerted by EM fields on any given object turns out to be independent of whether the density of force/torque is evaluated using the law of Lorentz or that of Einstein and Laub. Hidden entities aside, the two formulations differ only in their predicted force and torque distributions inside matter. Such differences in distribution are occasionally measurable, and could serve as a guide in deciding which formulation, if either, corresponds to physical reality.