All Problems

Electric Oscillations

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 \(I_{1}=I_{0} \cos \omega t\).

Reveal Answer
⟨Fx⟩=ω2L2L˙12I022(R2+ω2L22)∂L12∂x\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}
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