= Rotational damping of a resistive loop in an oscillating field
{title2=$\langle\tau_\theta\rangle=-A^2B_0^2\sin^2\theta\,\dot\theta/(2R)$}
For negligible <self-inductance>, a loop with <electrical resistance> $R$ obeys $I=-\dot\Phi_{\rm ext}/R$. With $\Phi_{\rm ext}=AB(t)\cos\theta$, this gives $I=-(A/R)\dot B\cos\theta+(A/R)B\sin\theta\,\dot\theta$. Its <magnetic torque> is
$$
\tau_\theta=\frac{A^2}{R}B\dot B\sin\theta\cos\theta-\frac{A^2}{R}B^2\sin^2\theta\,\dot\theta.
$$
For $B=B_0\sin\omega t$ 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.
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