AC generators and alternators - TCE Physics (Tasmania)

What this dot point is asking

This dot point applies Faraday's law to a rotating coil, the device that produces almost all the world's electricity.

How the EMF is generated

A generator has a coil that is spun inside a magnetic field. As the coil turns, the angle between the field and the coil face changes, so the flux $\Phi = BA\cos\theta$ through the coil rises and falls. Faraday's law says the induced EMF is proportional to the rate of change of flux, so a steadily rotating coil produces a continuously varying EMF.

The EMF is greatest when the coil is moving parallel to the field lines, that is when the coil plane is parallel to the field, because then the flux is changing fastest. The EMF is zero when the coil is perpendicular to the field, where the flux is at a maximum but momentarily not changing. The result is a smoothly varying, sinusoidal output.

The sinusoidal output

For a coil of $N$ turns and area $A$ rotating at angular frequency $\omega$ in a field $B$, the induced EMF varies as:

The peak EMF is $\varepsilon_0 = NBA\omega$, so a faster rotation, a stronger field, more turns or a larger area all raise the peak voltage. The output reverses direction every half turn, which is what makes it alternating current. The angular frequency relates to the rotation rate by $\omega = 2\pi f$, so a generator turning at $50\ \text{Hz}$ has $\omega \approx 314\ \text{rad s}^{-1}$.

Slip rings and the link to motors

A generator connects the coil to the external circuit through slip rings, two continuous rings that each stay connected to one end of the coil. Because the connection never swaps, the output naturally alternates as the coil rotates, giving AC. This contrasts with a DC motor's split-ring commutator, which deliberately swaps the connection every half turn.

A generator is structurally the same as a motor: a coil, a magnetic field and a means of connection. A motor uses an input current to produce rotation; a generator uses input rotation to produce a current. This is why the same machine can often run either way, and why a spinning motor produces a back EMF that opposes the supply.

Peak and RMS values

Because AC voltage varies, we describe it by two numbers. The peak value $\varepsilon_0$ is the maximum, while the root-mean-square (RMS) value is the steady value that would deliver the same average power. For a sinusoidal output they are related by:

Mains voltage quoted as $230\ \text{V}$ is the RMS value, with a higher peak of about $325\ \text{V}$. The RMS value matters because the average power delivered to a resistor is set by $\varepsilon_\text{rms}$, not the peak.

In the exam, link the size of the EMF to how fast the flux is changing, and identify slip rings as the feature that gives alternating output. If asked, state that the EMF is zero when the coil is perpendicular to the field and maximum when the coil plane is parallel to it, and convert between peak and RMS values with the $\sqrt{2}$ factor.