Alternating-current induction levitation

A resistive loop with self-inductance $L_r$ coupled by mutual inductance $M(z)$ to an imposed alternating current obeys $L_r{\dot{I}}_r+R{I}_r=-M{\dot{I}}_s$. For $I_s=I_0\cos \omega t$, averaging the magnetic force $I_s{I}_r{M}^{\prime }(z)$ gives $-I_0^2{\omega }^{2}M{L}_r{M}^{\prime }/[2(R^2+{\omega }^2{L}_r^2)]$. It is repulsive where $M{M}^{\prime }<0$. 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.