Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 1 / 16C (Electromagnetism)

For a localized steady current, defineAssume the external field varies negligibly across the current distribution. Expanding it to first order, the uniform term gives no net force because $\int \mathbf{J}dV=0$. Using the supplied integral identity and $\nabla \cdot \mathbf{B}=0$, the first-order term in $\int \mathbf{J}\times \mathbf{B}dV$ reduces toSince $\mathbf{F}=-\nabla U$, the magnetic dipole potential energy is

Hereincreases away from the $z$-axis. A dipole parallel to $\mathbf{B}$ has energy $-m|\mathbf{B}|$ and is drawn toward stronger field, hence outward from the axis. An antiparallel dipole has energy $+m|\mathbf{B}|$ and is driven toward the weak-field axis. The field therefore spatially separates the two orientations.