Sketch and interpret graphs of e.m.f. against time for simple a.c. generators and relate the position of the generator coil to the peaks, troughs and zeros of the e.m.f.

Published by Patrick Mutisya · 14 days ago

Cambridge IGCSE Physics 0625 – 4.5.2 The a.c. Generator

4.5.2 The a.c. Generator

Objective

To sketch and interpret graphs of e.m.f. against time for simple a.c. generators and to relate the position of the generator coil to the peaks, troughs and zeros of the e.m.f.

1. Principle of operation

A simple a.c. generator consists of a rectangular coil of area A with N turns, mounted on a shaft that rotates with angular speed ω in a uniform magnetic field B. As the coil rotates, the magnetic flux through the coil changes sinusoidally, inducing an alternating e.m.f. according to Faraday’s law:

\$\mathcal{E}(t)= -\frac{d\Phi}{dt}= N B A \, \omega \sin(\theta)\$

where θ = ωt is the angle between the normal to the coil and the magnetic field direction.

2. Shape of the e.m.f.–time graph

The induced e.m.f. varies sinusoidally with time, producing a smooth wave that repeats every full rotation (period T = 2π/ω).

  • Maximum positive e.m.f. (peak) occurs when the coil is parallel to the magnetic field (normal to the field is zero).

  • Maximum negative e.m.f. (trough) occurs when the coil is parallel but rotating in the opposite sense.

  • Zero e.m.f. occurs when the coil is perpendicular to the magnetic field (normal to the field is maximum).

3. Relating coil position to the graph

Consider the coil at four key positions during one quarter‑turn (90°). The table below summarises the relationship between the coil angle, the magnetic flux, and the e.m.f. value.

Coil position (θ)Orientation of coilMagnetic flux, ΦInduced e.m.f., 𝓔Graph point
0° (or 360°)Plane of coil parallel to B; normal ⟂ BΦ = 0𝓔 = 0 (zero crossing, rising)Origin of the sinusoid
90°Plane of coil perpendicular to B; normal ‖ BΦ = N B A (maximum)𝓔 = +N B A ω (positive peak)Maximum positive point
180°Plane of coil parallel to B again; normal ⟂ B (opposite direction)Φ = 0𝓔 = 0 (zero crossing, falling)Mid‑point crossing
270°Plane of coil perpendicular to B; normal opposite to BΦ = –N B A (minimum)𝓔 = –N B A ω (negative peak)Maximum negative point

4. Sketch of the e.m.f.–time graph

The graph is a sine wave that repeats every period T. The key points are marked as follows:

  • At t = 0 s, the coil is at 0°, e.m.f. = 0 (crosses the time axis upward).

  • At t = T/4, the coil reaches 90°, e.m.f. = +𝓔max (peak).

  • At t = T/2, the coil is at 180°, e.m.f. = 0 (crosses downward).

  • At t = 3T/4, the coil is at 270°, e.m.f. = –𝓔max (trough).

  • At t = T, the coil returns to 0°, completing one full cycle.

Suggested diagram: Sketch of a sinusoidal e.m.f. vs. time graph with labelled points (0, +𝓔max, 0, –𝓔max) and corresponding coil angles (0°, 90°, 180°, 270°).

5. Practical implications for the generator

  1. Increasing the number of turns N, the magnetic field strength B, the coil area A, or the rotation speed ω raises the amplitude of the e.m.f.

  2. The frequency of the alternating current is directly proportional to the rotation speed: f = ω/2π.

  3. Reversing the direction of rotation reverses the order of the peaks and troughs, but the absolute shape of the graph remains unchanged.

6. Summary

A simple a.c. generator produces a sinusoidal e.m.f. because the magnetic flux through its rotating coil varies sinusoidally with the coil’s angle. Peaks occur when the coil’s plane is perpendicular to the magnetic field (normal parallel to the field), troughs when it is perpendicular but with opposite polarity, and zeros when the coil’s plane is parallel to the field (normal perpendicular). Understanding this relationship allows students to predict the shape of the e.m.f.–time graph and to relate it to the physical motion of the generator coil.