Rotational damping of a resistive loop in an oscillating field ($⟨{\tau }_{\theta }⟩=-A^2{B}_0^2{\sin }^2\theta \dot{\theta }/(2R)$)

For negligible self-inductance, a loop with electrical resistance $R$ obeys $I=-{\dot{\Phi }}_{\mathrm{ext}}/R$. With ${\Phi }_{\mathrm{ext}}=AB(t)\cos \theta$, this gives $I=-(A/R)\dot{B}\cos \theta +(A/R)B\sin \theta \dot{\theta }$. Its magnetic torque isFor $B=B_0\sin \omega t$ and slow mechanical motion, the first term averages to zero while the second gives the displayed opposing torque. A stationary loop nevertheless experiences an alternating torque at $2\omega$ and generally executes small forced motion. The zero leading mean aligning torque is not a claim of exactly zero instantaneous motion or of no higher-order effect.