On momentum and energy of a non-radiating electromagnetic field

TL;DR

This paper investigates the momentum and energy of non-radiating electromagnetic fields, examining the validity of different momentum expressions, the concept of hidden momentum, and proposing a unified energy-momentum tensor.

Contribution

It introduces a new form of the energy-momentum tensor that unites mechanical and electromagnetic masses and analyzes the discrepancies in momentum expressions for dynamic processes.

Findings

Electromagnetic momentum expressions coincide in quasi-static cases.

Discrepancies arise in dynamic processes between different momentum expressions.

A new energy-momentum tensor uniting mechanical and EM masses is proposed.

Abstract

This paper inspects more closely the problem of the momentum and energy of a bound (non-radiating) electromagnetic (EM) field. It has been shown that for an isolated system of non-relativistic mechanically free charged particles a transformation of mechanical to EM momentum and vice versa occurs in accordance with the requirement PG=const, where PG is the canonical momentum. If such a system contains bound charges, fixed on insulators then, according to the assumption of a number of authors, a so-called "hidden" momentum can contribute into the total momentum of the system. The problem of "hidden momentum" (pro and contra) is also examined in the paper, as well as the law of conservation of total energy for different static configurations of the system "magnetic dipole plus charged particle". Analyzing two expressions for electromagnetic momentum of a bound EM field, qA and the Poynting expression, we emphasize that they coincide with each other for quasi-static configurations, but give a discrepancy for rapid dynamical processes. We conclude that neither the first, nor the second expressions provide a continuous implementation of the momentum conservation law. Finally, we consider the energy flux in a bound EM field, using the Umov vector. It has been shown that Umov vector can be directly derived from Maxwell equations. A new form of the momentum-energy tensor, which explicitly unites the mechanical and EM masses, has been proposed.