Faraday's Law of Induction

A magnet sliding into a coil makes a bulb light up with no battery because the moving magnet changes the magnetic flux through the coil's loops, and that change is exactly what creates voltage. Flux, Phi, measures how much magnetic field passes through the coil's area. As the magnet approaches, the field through the coil grows; as it leaves, it shrinks. Either way, the flux is changing, and a changing flux is the only thing that generates emf.

The key word is rate. Faraday's Law states emf = N(dPhi/dt), where N is the number of turns in the coil and dPhi/dt is how fast the flux changes over time. Double the magnet's speed and you halve the time it takes to sweep through the same change in flux, which doubles dPhi/dt and doubles the induced emf. A huge flux sitting still contributes nothing: a magnet resting motionless inside a coil holds Phi constant, so dPhi/dt is zero and no current flows, no matter how strong the magnet is.

This law is the reason generators work: spinning a loop in a magnetic field constantly changes the flux through it, producing a continuous alternating emf. It also sets up the next idea, Lenz's Law, which gives the induced current a direction, chosen so the current opposes the very change in flux that created it.