Q.An a.c. generator consists of a coil of 10,000 turns and of area 100 cm$^2$. The coil rotates at an angular speed of 140 rpm in a uniform magnetic field of $3.6\times10^{-2}$ T. Find the maximum value of the emf induced.

Principle. An AC generator (alternator) converts mechanical energy, used to rotate a coil or a field magnet, into electrical energy, working on the same electromagnetic-induction principle throughout this unit: relative motion between a conductor and a field changes the linked flux, inducing an emf (Faraday's law) whose direction follows Fleming's right hand rule.

Deriving the sinusoidal emf. For a coil of N turns, area A, rotating with angular velocity $\omega$ in a field B, starting (t = 0) with the coil's plane perpendicular to B (normal along B, maximum flux linkage $\Phi_m=NBA$), at angle $\theta=\omega t$ the flux linkage is $N\Phi_B=NBA\cos\omega t$, and Faraday's law gives the induced emf

-- a purely sinusoidal (alternating) emf, completing one full cycle per one full mechanical rotation of the coil (or field magnet). This is the fundamental principle behind every AC generator. …