Electromagnetic Induction: Numerical Problems Set (9)

Question

A rectangular coil of $200$ turns and area $0.05\text{ m}^2$ rotates at $50\text{ rev/s}$ in a uniform magnetic field of $0.4\text{ T}$. The coil’s axis of rotation is perpendicular to the field. Find (a) the peak EMF, (b) the RMS EMF, and (c) the EMF at the instant the plane of the coil is parallel to the field.

Solution — Step by Step

Why This Works

Faraday’s law says induced EMF is $-d\Phi/dt$. With $\Phi = NBA\cos(\omega t)$, differentiation gives $\varepsilon = NBA\omega\sin(\omega t)$. Peak occurs when $\sin = 1$, which is exactly when $\cos = 0$ — when flux is zero, its rate of change is largest.

This is the geometric reason students sometimes get backwards: max flux $\ne$ max EMF.

Alternative Method

Energy argument: the kinetic energy of rotation is converted to electrical energy at the rate at which the coil “cuts” field lines. The cutting rate is proportional to the velocity of the wire perpendicular to the field, which peaks when the coil’s plane lies along the field direction.

Common Mistake