Interaction of a Magnet and a Point Charge: Unrecognized Internal Electromagnetic Momentum

TL;DR

This paper clarifies the concept of internal electromagnetic momentum in a magnet-charge system, challenging the notion of hidden momentum and providing explicit calculations for various interaction scenarios.

Contribution

It introduces an unambiguous definition of internal momentum using relativistic conservation laws and demonstrates its presence in a magnet with moving charges interacting with a point charge.

Findings

Internal electromagnetic momentum exists in a magnet-charge system.

The magnet's internal momentum depends on charge interactions.

The concept of hidden momentum is challenged and clarified.

Abstract

Whereas nonrelativistic mechanics always connects the total momentum of a system to the motion of the center of mass, relativistic systems, such as interacting electromagnetic charges, can have internal linear momentum in the absence of motion of the center of energy of the system. This internal linear momentum of the system is related to the controversial concept of "hidden momentum." We suggest that the term "hidden momentum" be abandoned. Here we use the relativistic conservation law for the center of energy to give an unambiguous definition of the "internal momentum of a system," and then we exhibit this internal momentum for the system of a magnet (modeled as a circular ring of moving charges) and a distant static point charge. The calculations provide clear illustrations of this system for three cases: a) the moving charges of the magnet are assumed to continue in their unperturbed motion, b) the moving charges of the magnet are free to accelerate but have no mutual interactions, and c) the moving charges of the magnet are free to accelerate and also interact with each other. It is noted that when the current-carrying charges of the magnet are allowed to interact, the magnet itself will contain internal electromagnetic linear momentum, something which has not been presented clearly in the research and teaching literature.