Electromagnetic induction
The generation of an electromotive force in a circuit whenever the magnetic flux through it changes. Discovered by Faraday and Henry in 1831, it is the foundation of every electric generator, transformer, and inductive sensor ever built.
Definition
Electromagnetic induction is the phenomenon Michael Faraday and Joseph Henry independently discovered in 1831: whenever the magnetic flux threading a closed circuit changes, an electromotive force appears in the circuit that drives a current. The flux can change in three qualitatively different ways — the circuit can move through a non-uniform magnetic field, the magnet can move past a stationary circuit, or the field itself can change in time — and in every case the induced EMF equals the rate of change of flux: EMF = −dΦ/dt. The minus sign (Lenz's law) enforces energy conservation: the induced current always flows in the direction whose own magnetic field opposes the change driving it.
The discovery transformed physics. Before 1831, electricity and magnetism were coupled only in one direction: a current creates a magnetic field (Ørsted, Ampère). After 1831, the coupling was bidirectional: a changing field creates a current, and the interplay between the two became the foundation of Maxwell's dynamical electromagnetism. Induction is what makes possible every technology that converts between mechanical and electrical energy: alternators in power plants (rotating magnets inducing AC voltage in stator coils), transformers (changing primary current inducing secondary voltage via a linked iron core), induction motors (stator fields inducing rotor currents that feel force back on the stator), inductive cooktops (kHz magnetic fields inducing eddy currents in the pan), wireless charging (resonant coupling between phone and pad), and the dynamo in your bicycle light.
The sign convention and the choice of surface integration for Φ require care (right-hand rule, orientation of the circuit loop, which side of the surface is "positive"), but once set up consistently the law is breathtakingly universal. It applies to single-turn loops, multi-turn coils (where Φ is replaced by the flux linkage λ = NΦ), rotating machines, and even free electrons in an applied time-varying vector potential. The integral form ∮E·dℓ = −dΦ_B/dt is one of Maxwell's four equations, and the differential form ∇×E = −∂B/∂t is its pointwise statement: any time-varying magnetic field creates a circulating electric field, anywhere in space, circuit or no circuit.