Summary

Magnetic flux through a flat surface of area $A$ in a uniform field $\vec{B}$ is $\Phi_B=BA\cos\theta$ (SI unit weber, Wb), maximum when the surface is perpendicular to $\vec{B}$ and zero when it is parallel to $\vec{B}$; for an $N$-turn coil the total flux linkage is $N\Phi_B$.

Faraday's laws state that a changing flux linkage induces an emf, of magnitude $\mathcal{E}=-N\,d\Phi_B/dt$; the flux can change because the field $B$ changes, the area $A$ changes, or the angle $\theta$ changes -- the last case, for a conductor of length $l$ moving with velocity $v$ perpendicular to $B$, gives the motional emf $\mathcal{E}=Blv$.

Lenz's law ($\mathcal{E}=-N\,d\Phi_B/dt$'s minus sign) fixes the DIRECTION of the induced current: it always opposes the change in flux that produced it, and is a direct consequence of the conservation of energy -- any other direction would let a system gain energy from nothing.

Eddy currents are induction currents circulating within the bulk of a solid conductor rather than a thin wire; useful in the induction furnace, induction cooktop, electromagnetic braking and dead-beat galvanometers, but wasteful in transformer/motor cores, where they are minimised by laminating the core.

Self-inductance $L$, defined by $N\Phi_B=LI$, gives the back emf $\mathcal{E}=-L\,dI/dt$ opposing any change in a coil's own current; for a long solenoid, $L=\mu_0 n^2Al$. …