Classical Interaction of a Magnet and a Point Charge: The Shockley-James Paradox

TL;DR

This paper critiques a common misunderstanding of the Shockley-James paradox, emphasizing the importance of internal electromagnetic momentum in a classical magnet-charge system and correcting previous misconceptions.

Contribution

It clarifies the role of internal electromagnetic momentum in the classical interaction between a magnet and a point charge, challenging prior reliance on hidden mechanical momentum.

Findings

Internal electromagnetic momentum opposes external electromagnetic momentum.

Detailed calculation for a two-charge magnet model confirms the internal momentum.

Critique of Coleman and Van Vleck's interpretation of the paradox.

Abstract

It is pointed out that Coleman and Van Vleck make a major blunder in their discussion of the Shockly-James paradox by designating relativistic hidden mechanical momentum as the basis for resolution of the paradox. This blunder has had a wide influence in the current physics literature, including erroneous work on the Shockley-James paradox, on Mansuripur's paradox, on the motion of a magnetic moment, on the Aharonov-Bohm phase shift, and on the Aharonov-Casher phase shift. Although hidden mechanical momentum is indeed dominant for non-interacting particles moving in a closed orbit under the influence of an external electric field, the attention directed toward hidden mechanical momentum represents a fundamental misunderstanding of the classical electromagnetic interaction between a multiparticle magnet and an external point charge. In the interacting multiparticle situation, the external charge induces an electrostatic polarization of the magnet which leads to an internal electromagnetic momentum in the magnet where both the electric and magnetic fields for the momentum are contributed by the magnet particles. This internal electromagnetic momentum for the interacting multiparticle situation is equal in magnitude and opposite in direction compared to the familiar external electromagnetic momentum where the electric field is contributed by the external charged particle and the magnetic field is that due to the magnet. In the present article, the momentum balance of the Shockley-James situation for a system of a magnet and a point charge is calculated in detail for a magnet model consisting of two interacting point charges which are constrained to move in a circular orbit on a frictionless ring with a compensating negative charge at the center.