What is the force on a magnetic dipole?

TL;DR

This paper clarifies that the standard formula for the force on a magnetic dipole remains valid even with time-dependent fields, emphasizing the conservation of total momentum without the need for hidden momentum or internal forces.

Contribution

It definitively confirms the correctness of the standard force formula for magnetic dipoles in dynamic fields, correcting misconceptions about hidden momentum.

Findings

The standard force formula ${\bf F=\nabla(\bmu\cdot B)}$ is valid in all cases.

Total momentum, including electromagnetic and mechanical, is conserved.

Attempts to modify the force formula using hidden momentum are incorrect.

Abstract

We show that attempts to modify the force on a magnetic dipole by introducing either hidden momentum or internal forces are not correct. The standard textbook result ${\bf F=\nabla(\bmu\cdot B)}$ is correct even in the presence of time dependent electromagnetic fields. Using this expression for the force, overall momentum (the sum of mechanical and electromagnetic momentum) is conserved in changing electromagnetic fields.