Inductance and the AC generator

Mutual inductance: a changing current in one coil changes the flux through a nearby coil, and induces an emf in it:

ε₂ = –M dI₁/dt, where M is the mutual inductance. Unit: henry (H). It depends on the coils' shapes, their turns, their distance and the core. A transformer works this way.

Self inductance: a changing current in a coil also changes the flux through itself, so it induces an emf in itself:

ε = –L dI/dt, where L is the self inductance.

This is a back emf: by Lenz's law it opposes the change in current. An inductor makes current slow to start and slow to stop, like inertia for current.

Example: L = 0.5 H, and a switch drops the current from 2 A to 0 in 0.01 s. ε = 0.5 × 200 = 100 V. This is why switches on motors can spark.

For a long solenoid: L = µ₀n²Al, so more turns per metre and a bigger volume give more inductance; an iron core multiplies it by µ_r.

Energy stored in an inductor: U = ½LI², held in its magnetic field (B²/2µ₀ per unit volume). With L = 2 H and I = 3 A, U = 9 J.

AC generator: a coil of N turns and area A spins at angular speed ω in a field B. The angle changes steadily, so Φ = BA cos ωt, and:

ε = NBAω sin ωt, an emf that swings between +NBAω and –NBAω.

Example: N = 100, B = 0.1 T, A = 0.02 m², spinning 50 times a second (ω = 100π ≈ 314 rad/s): peak emf = 100 × 0.1 × 0.02 × 314 ≈ 63 V.