Quick questions on AC and DC generators: WACE Year 12 Physics Unit 3

As a coil of $N$ turns and area $A$ rotates at angular speed $\omega$ in a field $B$, the flux is $\Phi=BA\cos(\omega t)$, so by Faraday's law the induced emf is

In an AC generator the coil ends connect to two slip rings, each in continuous contact with a brush. The connection never swaps, so as the coil rotates the output reverses direction every half turn, tracing a full sine wave each revolution. This is how mains electricity is produced in power stations.

A DC generator replaces the slip rings with a single split-ring commutator. Each half turn, as the emf in the coil would reverse, the commutator swaps which half of the ring touches which brush. The external connection therefore always sees the same polarity, so the output is a series of positive humps, never going negative. It is direct current, though it pulses rather than being perfectly steady.

The peak emf $\varepsilon_0=NBA\omega$ shows four ways to increase the output: more turns $N$, a stronger field $B$, a larger coil area $A$, or faster rotation $\omega$. Faster rotation also increases the frequency of the AC, since one cycle is produced per revolution.

Examiners frequently give an emf-against-time axis and ask you to sketch or identify the output. For the AC generator the trace is a smooth sinusoid: it crosses zero when the coil plane is perpendicular to the field (flux at a maximum, rate of change zero) and peaks when the plane is parallel to the field (flux changing fastest). One complete sine cycle corresponds to one full revolution, so if the coil turns at $f$ revolutions per second the AC frequency is also $f$.