AC Generator

The basic idea. An AC generator (or alternator) converts mechanical energy into electrical energy by mechanically forcing a coil to rotate inside a magnetic field, using nothing more than Faraday's law of electromagnetic induction (Section 6.3 of the previous sub-topic): as the coil rotates, the flux linked with it changes continuously, and an emf is induced automatically as a direct, unavoidable consequence.

Construction. A rectangular coil of $N$ turns and area $A$ is mounted so that it can rotate about a fixed axis, placed between the pole faces of a strong permanent magnet (or an electromagnet) that provides a uniform magnetic field $B$ perpendicular to the rotation axis. The two ends of the coil are connected to two separate slip rings mounted on the same rotating shaft, which rotate together with the coil; two stationary carbon brushes press lightly against these slip rings, remaining in continuous sliding electrical contact with them as they turn, and it is through these brushes that the generated current is delivered to the external circuit (the figure for this section shows the complete arrangement).

Deriving the emf. Let the coil be turned at a constant angular speed $\omega$ by some external mechanical agent (a turbine, a hand crank, or similar), so that at time $t$ the normal to the coil makes angle $\theta=\omega t$ with the field $B$ (taking $\theta=0$ at the instant the coil's plane is perpendicular to $B$, i.e. the position of maximum flux). The flux linked with the coil at this instant is

By Faraday's law, the induced emf is $e=-d\Phi_B/dt$:

Writing $e_0=NBA\omega$ for the peak emf, this is

exactly the sinusoidal form assumed for every AC source used throughout this chapter -- confirming that a coil rotated at constant angular speed in a uniform field genuinely produces the sinusoidal alternating emf this whole chapter has been built around. …