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 \(P=1 / 2 \omega^{2} \Phi_{0}^{2} R /\left(R^{2}+\right.\) \(\left.+\omega^{2} L^{2}\right)\)