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 ${\mu }_{0}Na|Ii|[1/s-1/(s+a)]/(2\pi )$. 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 ${\mathcal{M}}^2{\omega }^2{I}_0^2/(2R)$. A fixed coil has no mechanical work despite its time-dependent force.