Simple form: The amount of electricity produced is proportional to the rate of change in magnetism. The amount of magnetism produced is proportional to the rate of change in electricity.

Complete form: Look up Maxwell's equations.

If you have a wire loop (the more wires and the more loops, the better) moving past a magnet, or if you have the magnet move past the coil, you get electricity produced in the wire, because the magnetic field strength changes as things move. The hamster turns the cage, which rotates the generator, which moves something past something else, and the change in magnetic field strength inside the coil makes the electricity.

To get more power:

Bigger magnets
More wire
Faster rotation

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