= Force on a resistive coil beside an alternating wire
In a fixed square coil beside a long wire, neglect <self-inductance> compared with resistance. A changing wire current induces coil current proportional to $dI/dt$. <Forces> on the two perpendicular sides cancel, and the unequal parallel-side fields give magnitude $\mu_0Na|Ii|[1/s-1/(s+a)]/(2\pi)$. When $I=I_0\sin\omega t$, the signed <force> alternates at twice the drive <frequency> and has zero cycle mean, while Joule <dissipation> is positive with mean $\mathcal M^2\omega^2I_0^2/(2R)$. A fixed coil has no mechanical work despite its time-dependent <force>.
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