How Does a Rotating Coil in a Magnetic Field Generate a Sinusoidal EMF (How Generators Work)?

A rotating coil generates a sinusoidal emf because its magnetic flux linkage changes as the cosine of time; according to Faraday’s law, the induced emf is the rate of change of this flux and therefore varies as the sine of time. This is an HL-only application in D.4 Induction.

The Mechanism

Consider a coil of area $A$, with $N$ turns, rotating at constant angular velocity $\omega$ in a uniform magnetic field of strength $B$. The angle $\theta$ is measured between the magnetic field and the normal to the coil.

The magnetic flux through one turn is

$$\Phi = BA\cos\theta.$$

For uniform rotation, $\theta=\omega t$, so the flux linkage is

$$N\Phi = NBA\cos(\omega t).$$

Using Faraday’s law of induction,

$$\mathcal{E}=-\frac{d(N\Phi)}{dt},$$

which gives

$$\mathcal{E}=NBA\omega\sin(\omega t).$$

Therefore, the emf is sinusoidal, with maximum value

$$\mathcal{E}_0=NBA\omega.$$

The emf is zero when the flux is maximum because the flux is momentarily not changing. It is maximum when the flux is zero because the coil is cutting magnetic field lines at the greatest rate. The negative sign in Faraday’s law represents Lenz’s law: the induced current opposes the change producing it.

For example, doubling the number of turns doubles the peak emf, while doubling the rotational angular velocity doubles both the peak emf and the output frequency. Thus faster rotation makes the alternating emf larger and completes each cycle in less time.

An external torque keeps the coil rotating, converting mechanical energy into electrical energy. Slip rings connect the rotating coil to the external circuit while preserving the alternating polarity, producing alternating current.

A common misconception is that a large magnetic flux automatically produces a large emf. The emf depends on the rate of change of flux, not the flux itself.

Exam Technique

For an IB Show question, write the flux expression, substitute $\theta=\omega t$, differentiate, and identify $\mathcal{E}_0=NBA\omega$. Always define the angle from the coil’s normal and explain the phase difference between flux and emf.