Electromagnetic Induction

Faraday's law — the magnitude of the induced e.m.f. is equal to the rate of change of flux linkage:

ε = − Δ(NΦ) / Δt

Lenz's law — the minus sign. The induced e.m.f. acts in the direction that opposes the change producing it.

Lenz's law is conservation of energy — if the induced effect helped the change along, the system would accelerate itself and generate energy from nothing. Opposition means you must do work against the induced effect, and that work is what becomes electrical energy. The minus sign is not a detail; it is the reason a generator needs turning.

Example: drop a magnet down a copper tube and it falls absurdly slowly. The changing flux induces circulating currents in the tube, and by Lenz's law those currents oppose the magnet's motion. The magnet's gravitational potential energy is converted to heat in the copper instead of kinetic energy. Nothing is touching it; the resistance is entirely electromagnetic.

Three ways to induce an e.m.f., all of which change the flux linkage:

Move a conductor through a field, cutting field lines.

Change the field strength through a stationary coil, for instance by switching a current on or off in a nearby coil.

Change the area or orientation of the coil, as in a rotating generator.

What matters is the rate of change — a large flux that is steady induces nothing at all. A tiny flux changing very fast induces a large e.m.f. This is why induction demonstrations involve sudden movements and why a stationary magnet near a coil does nothing.

Fleming's right-hand rule gives the direction of the induced current for a moving conductor: first finger Field, thuMb Motion, seCond finger Current. Right hand for generating, left hand for motors.

A straight conductor moving through a field — sweeping out area A = Lv in time t gives

ε = B L v

when the motion, field and conductor are mutually perpendicular.

Graphs — the induced e.m.f. is the gradient of a flux linkage against time graph. A linear change in linkage gives a constant e.m.f.; a sinusoidal linkage gives a sinusoidal e.m.f. a quarter cycle ahead.