Electromagnetic Induction

$\text{Magnetic flux}$ $\Phi$ through an area is the flux density times the area perpendicular to the field: $\Phi =BA\cos \theta$, measured in webers, where $\theta$ is the angle between the field and the normal to the area.

For a coil of $N$ turns, each turn links the same flux, so the $\text{flux linkage}$ is $N\Phi$, measured in weber-turns.

Flux linkage changes if the field strength changes, if the area changes, or if the coil rotates relative to the field. Any of these induces an e.m.f.

Faraday’s law states that the magnitude of the induced e.m.f. equals the rate of change of flux linkage: $\epsilon =\Delta (N\Phi )/\Delta t$.

Faster change means a larger e.m.f. — moving a magnet quickly into a coil gives a bigger reading than moving it slowly, even though the total flux change is the same.

On a graph of flux linkage against time, the induced e.m.f. is the gradient. A constant flux linkage, however large, induces nothing.

$\text{Lenz’s law}$: the direction of the induced e.m.f. is such that any current it produces opposes the change in flux that caused it. That is the minus sign in Faraday’s law.

Push a north pole into a coil and the induced current makes the near end of the coil a north pole, repelling the magnet. Pull it out and the near end becomes south, attracting it back.

This is conservation of energy. You must do work against the opposing force to move the magnet, and that work becomes the electrical energy of the induced current. If the induced current attracted the incoming magnet instead, the magnet would accelerate, producing more current, and energy would be created from nothing.

A $\text{generator}$ rotates a coil in a magnetic field. The flux linkage varies sinusoidally, so the induced e.m.f. is also sinusoidal — alternating current. Its peak value is $BAN\omega$, so faster rotation gives a larger and more frequent e.m.f.

A $\text{transformer}$ has two coils on a shared iron core. An alternating current in the primary creates a changing flux in the core, which induces an alternating e.m.f. in the secondary. The ratio of voltages equals the ratio of turns.

Transformers only work with alternating current, because a steady current produces a constant flux and induces nothing. For an ideal transformer, power in equals power out, so stepping voltage up steps current down.