Torque on slowly moving electric or magnetic dipoles in vacuo

TL;DR

This paper derives the torque on slowly moving electric or magnetic dipoles in vacuum using Lorentz transformations, confirming the Ampère equivalence principle's validity in dynamic conditions and identifying three contributing torque terms.

Contribution

It provides a model-independent derivation of the torque on moving dipoles, demonstrating the validity of the Ampère equivalence principle in dynamical scenarios.

Findings

Torque includes three terms: direct, dual-dipole induced, and inertial torque.

Equations are independent of dipole model (Amperian or Gilbertian).

Confirms the Ampère equivalence principle in slow-motion conditions.

Abstract

The torque on a moving electric or magnetic dipole in slow motion is deduced using the Lorentz transformation of the fields to first order in v/c. It is shown that the obtained equations are independent of the model adopted for the dipole, whether it is of Amperian or Gilbertian type, thus showing the complete validity of the Amp\`ere equivalence principle even in dynamical conditions. The torque is made of three terms: beside that due to the direct torque on the dipole there are two more terms: one due to the torque on the associated perpendicular dual-dipole caused by motion, while the other is the inertial torque due to the displacement of the dipole which carries with it the field linear momentum, or the hidden momentum.