How is a sinusoidal alternating voltage generated by a coil rotating in a magnetic field?

In one line

When a coil rotates at constant speed in a uniform magnetic field, the flux linking it changes continuously, so by Faraday's law it induces an EMF that varies as \( e = E_m \sin\omega t \).

Explained simply

Think of a bicycle dynamo: spin the wheel and the lamp glows; spin it faster and it glows brighter. Inside, a magnet and coil move relative to each other. As the coil turns, it cuts magnetic lines fast at some positions and not at all at others, so the voltage rises, falls, reverses, and repeats.

Technically, a rectangular coil of \( N \) turns and area \( A \) rotates with angular velocity \( \omega \) in a uniform field \( B \). The flux linkage is \( N\phi = NBA\cos\omega t \) (taking \( \theta = \omega t \) from the position of maximum flux). By Faraday's law \( e = -N\,d\phi/dt = NBA\omega\sin\omega t \).

How it works

1. At \( \theta = 0^\circ \) the coil plane is perpendicular to the field: flux is maximum but its rate of change is zero, so \( e = 0 \).
2. At \( \theta = 90^\circ \) the coil sides move at right angles to the field: flux is zero but changing fastest, so \( e = E_m \).
3. At \( 180^\circ \) EMF is zero again; from \( 180^\circ \) to \( 360^\circ \) the coil sides move in the opposite direction, so the EMF reverses.
4. One revolution of a two-pole machine gives one complete cycle. Slip rings and brushes take the AC out to the load.

Key formula

\( e = E_m\sin\omega t, \qquad E_m = NBA\omega, \qquad \omega = 2\pi f \)

\( e \) = instantaneous EMF (V), \( E_m \) = maximum EMF (V), \( N \) = number of turns, \( B \) = flux density (T, i.e. Wb/m²), \( A \) = coil area (m²), \( \omega \) = angular velocity (rad/s), \( f \) = frequency (Hz), \( t \) = time (s).

Quick example

A 100-turn coil of area 0.02 m² rotates at \( \omega = 314 \) rad/s (50 Hz) in \( B = 0.5 \) T. \( E_m = 100 \times 0.5 \times 0.02 \times 314 = 314 \) V, so \( e = 314\sin 314t \) V. Power stations use the same principle in alternators, usually with a rotating field and stationary armature.

Common mistakes

• Saying EMF is maximum when flux is maximum; EMF is maximum when the rate of change of flux is maximum.
• Using \( \omega \) in rpm instead of rad/s.
• Confusing slip rings (used for AC output) with a commutator (used in DC machines).

How MU asks it

Asked as "Explain the generation of alternating EMF with a neat diagram" (5 marks). For full marks: draw the coil between N and S poles, show the angle \( \theta \), derive \( e = E_m\sin\omega t \) using Faraday's law, sketch one cycle of the sine wave with 0°, 90°, 180°, 270°, 360° marked, and state that \( E_m = NBA\omega \).