Orientation torque on a flux-conserving loop ($⟨{\tau }_{\theta }⟩=A^2{B}_0^2\sin \theta \cos \theta /(2L)$)

For a loop of area $A$ and self-inductance $L$, negligible resistance conserves $LI+{\Phi }_{\mathrm{ext}}$. With zero trapped total flux and ${\Phi }_{\mathrm{ext}}=A{B}_0\sin (\omega t)\cos \theta$, the electric current is $I=-A{B}_0\sin (\omega t)\cos \theta /L$. The magnetic torque generalized to the angle between loop normal and field is ${\tau }_{\theta }=-IAB\sin \theta$. For an orientation varying slowly over a field cycle, averaging gives the displayed torque and effective potential $U_{\mathrm{av}}=A^2{B}_0^2{\cos }^2\theta /(4L)$. Its stable orientations have loop normal perpendicular to the applied field. Without damping, a freely released loop librates around such an orientation instead of settling there.