A coil with N turns and total area A is mounted on a rigid frame so that it can rotate freely about a fixed axis in a uniform magnetic field of magnitude B. The axis of rotation lies in the plane of the coil. The coil is driven by a motor so that it rotates at a constant angular speed ω about that axis.
Ignore any electrical resistance in what follows.
(a) Express the magnetic flux through the coil as a function of time, choosing t=0 when the normal to the plane of the coil is aligned with the magnetic field.
(b) Using your expression from part (a), obtain an analytic expression for the instantaneous induced emf.
(c) Show that the induced emf is sinusoidal in time and identify its peak (maximum) value in terms of N, B, A, and ω. State the phase difference, if any, between the emf and the magnetic flux.
(d) Derive an expression for the root-mean-square (rms) value of the induced emf in terms of the same variables.
(e) For a coil with 300 turns and total area 15 cm² rotating in a 0.10 T magnetic field at 60 Hz, calculate the numerical value of the rms emf.