Past exam of the physics B course of the University of Cambridge / 2021 / Paper 2 / Section A / A4

For a fixed magnetic dipole moment in an externally imposed uniform magnetic field, the torque is $\mathbf{\tau }=\mathbf{m}\times \mathbf{B}$. Its component in the direction of increasing $\theta$ is $-mB\sin \theta$. Since ${\tau }_{\theta }=-dU/d\theta$, integration giveswhere the arbitrary additive constant has been chosen so that $U=0$ at $\theta =\pi /2$.

The aligned orientation $\theta =0$ has the minimum $U=-mB$ and is stable against small rotations; the anti-aligned orientation $\theta =\pi$ has the maximum $U=+mB$ and is unstable. The graph repeats every $2\pi$. Rotating quasistatically from alignment to anti-alignment requires external work $\Delta U=2mB$.