= Alternating-current induction levitation
A resistive loop with <self-inductance> $L_r$ coupled by <mutual inductance> $M(z)$ to an imposed <alternating current> obeys $L_r\dot I_r+RI_r=-M\dot I_s$. For $I_s=I_0\cos\omega t$, averaging the magnetic <force> $I_sI_rM'(z)$ gives $-I_0^2\omega^2ML_rM'/[2(R^2+\omega^2L_r^2)]$. It is repulsive where $MM'<0$. The <self-inductance> supplies the current component opposite to the source current; a purely resistive, zero-inductance loop has zero average force in this model.
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