Halliday, Resnick & Walker — Induction and Inductance: Problem 31

31 A rectangular coil of \(N\) turns and of length \(a\) and width \(b\) is rotated at frequency \(f\) in a uniform magnetic field \(\vec{B}\), as indicated in Fig. 30-34. The coil is connected to co-rotating cylinders, against which metal brushes slide to make contact. (a) Show that the emf induced in the coil is given (as a function of time \(t\)) by \(\mathcal{E} = 2\pi f NabB \sin(2\pi f t) = \mathcal{E}_0 \sin(2\pi f t)\). This is the principle of the commercial alternating-current generator. (b) What value of \(Nab\) gives an emf with \(\mathcal{E}_0 = 220\text{ V}\) when the loop is rotated at \(60.0\text{ rev/s}\) in a uniform magnetic field of \(0.400\text{ T}\)?