Faraday's Laws of Electromagnetic Induction

Faraday's first law (qualitative). Whenever the magnetic flux linked with a closed circuit changes, an emf is induced in the circuit; this induced emf lasts only as long as the flux keeps changing, and disappears the instant the flux becomes steady again. If the circuit is closed, this induced emf drives an induced CURRENT around it.

Faraday's second law (quantitative). The MAGNITUDE of the induced emf is directly proportional to the rate at which the flux linkage changes with time:

In SI units this proportionality becomes an exact equality with proportionality constant 1, and, folding in Lenz's law (Section 6.5) for the direction of the emf as a minus sign, the complete statement of Faraday's law is

Here $\mathcal{E}$ is the induced emf in volts, $N$ is the number of turns in the coil, and $\Phi_B$ is the flux linked with ONE turn (in weber), so $N\Phi_B$ is the total flux linkage (in weber-turns). The minus sign is not a mathematical afterthought -- it is Lenz's law written into the formula, and says that the induced emf always acts so as to OPPOSE the very change in flux that produced it (Section 6.5 works out exactly what that means in practice).

Ways to make the rate large. Since only the RATE of change of flux matters, not the flux itself, a large emf can be obtained either from a large change in flux happening quickly, or from even a modest change in flux happening extremely fast -- which is exactly why a bar magnet plunged rapidly into a coil gives a sharp galvanometer kick, while the same magnet eased in slowly over several seconds gives only a feeble, barely-visible deflection, even though the total change in flux is identical in both cases. …