The experiments of Faraday and Henry
Around 1830, Michael Faraday and Joseph Henry found that a changing magnetic field can produce an electric current.
Moving a magnet towards or away from a coil produces a current, detected by a galvanometer. A stationary magnet produces none.
The faster the magnet moves, the larger the current.
Switching the current on or off in one coil induces a current in a nearby coil.
The common factor in all these cases is a change in magnetic flux through the circuit.
Magnetic flux
The magnetic flux through a plane area A in a uniform field B is
Φ_B = B · A = B A cos θ
where θ is the angle between B and the normal to the area. The SI unit is the weber (Wb): 1 Wb = 1 T m².
Faraday's law
The magnitude of the induced emf in a circuit equals the rate of change of magnetic flux through it.
For a coil of N turns:
ε = −N dΦ_B / dt
Flux can change in three ways: by changing B, by changing the area A, or by changing the angle θ (rotating the coil).
Example. A 100-turn coil has its flux reduced uniformly from 0.04 Wb to 0 in 0.2 s. The induced emf is 100 × 0.04 / 0.2 = 20 V.
Lenz's law
The induced current flows in a direction that opposes the change in flux that produced it.
This is the reason for the minus sign in Faraday's law. If the north pole of a magnet is pushed towards a coil, the induced current makes the near face of the coil a north pole, which repels the approaching magnet.
Lenz's law is a consequence of conservation of energy. If the induced current helped the change, energy would be created from nothing. Instead, you must do work against the opposition, and that work appears as electrical energy.
Motional emf
A straight conductor of length l moving with velocity v perpendicular to a uniform field B has an emf induced across its ends:
ε = B l v
The Lorentz force on the free charges in the moving rod pushes them to one end.
Example. A 0.5 m rod moves at 4 m/s perpendicular to a 0.2 T field. ε = 0.2 × 0.5 × 4 = 0.4 V.
Eddy currents
When a bulk piece of metal experiences a changing flux, circulating currents called eddy currents are induced in it. They oppose the change (by Lenz's law) and heat the metal.
Unwanted: in transformer cores they waste energy. Cores are therefore made of thin laminated sheets, insulated from each other, to reduce eddy currents.
Useful: magnetic braking in trains, induction furnaces, induction cooktops, and the damping of galvanometer needles.
Inductance
Mutual inductance. A changing current in one coil induces an emf in a nearby coil: ε₂ = −M dI₁/dt. M depends on the shapes, sizes, separation and orientation of the coils and on the core material.
Self-inductance. A changing current in a coil induces an emf in the same coil, opposing the change: ε = −L dI/dt. The unit of inductance is the henry (H).
For a long solenoid with n turns per unit length, cross-sectional area A and length l:
L = μ₀ n² A l
An inductor resists changes in current, much as mass resists changes in velocity. The energy stored in an inductor carrying current I is
U = ½ L I²
The AC generator
A coil of N turns and area A rotates with angular speed ω in a uniform field B. The flux through it is Φ = NBA cos ωt, so the induced emf is
ε = N B A ω sin ωt = ε₀ sin ωt, with peak emf ε₀ = NBAω
This is a sinusoidal alternating emf. Mechanical energy (from steam, water or wind turbines) is converted into electrical energy.