On electromagnetic momentum of an electric dipole in a magnetic field

TL;DR

This paper critically examines the electromagnetic momentum of an electric dipole in a magnetic field, clarifying when the commonly assumed formula applies and addressing momentum conservation without hidden momentum.

Contribution

It demonstrates that the standard expression for electromagnetic momentum is not universally valid and clarifies the role of fringing fields and momentum conservation.

Findings

The formula $rac12 extbf{B} imes extbf{p}$ is not generally valid.

Electromagnetic momentum contributions from fringing fields are significant.

Mechanical impulse equals the negative change in electromagnetic momentum, without needing hidden momentum.

Abstract

The total linear electromagnetic field momentum $\mathbf P_{\mathrm{em}}$ of a stationary electric dipole $\mathbf p$ in a static magnetic field $\mathbf B$ is considered. The expression $\mathbf P_{\mathrm{em}} = \frac12\mathbf B \times \mathbf p$, which has previously been implied to hold in all static magnetic field situations, is not valid in general. The contribution of the electromagnetic momentum of the fringing fields of the dipole is discussed. It is shown that when either the static magnetic field or the electric dipole moment is changed, the mechanical impulse on the system equals $-\Delta\mathbf P_{\mathrm{em}}$, and hidden momentum does not need to be invoked in order to conserve total momentum.