Problem 4.149

A wooden core (Fig. 4.36) supports two coils: coil 1 with inductance $L_{1}$ and short-circuited coil 2 with active resistance $R$ and inductance $L_{2} .$ The mutual inductance of the coils depends on the distance $x$ between them according to the law $L_{12}(x)$ . Find the mean (averaged over time) value of the interaction force between the coils when coil 1 carries an alternating current $I_{1}=I_{0} \cos \omega t$ .

\left\langle F_{x}\right\rangle=\frac{\omega^{2} L_{2} \dot{L}_{12} I_{0}^{2}}{2\left(R^{2}+\omega^{2} L_{2}^{2}\right)} \frac{\partial L_{12}}{\partial x}

Electric Oscillations