Problem 4.148

A ring of thin wire with active resistance $R$ and inductance $L$ rotates with constant angular velocity $\omega$ in the external uniform magnetic field perpendicular to the rotation axis. In the process, the flux of magnetic induction of external field across the ring varies with time as $\Phi=\Phi_{0} \cos \omega t .$ Demonstrate that (a) the inductive current in the ring varies with time as $I=$ $=I_{m} \sin (\omega t-\varphi),$ where $I_{m}=\omega \Phi_{0} / \sqrt{R^{2}+\omega^{2} L^{2}}$ with tan $\varphi=$ $=\omega L / R$ (b) the mean mechanical power developed by external forces to maintain rotation is defined by the formula $P=1 / 2 \omega^{2} \Phi_{0}^{2} R /\left(R^{2}+\right.$ $\left.+\omega^{2} L^{2}\right)$