Energy stored in an inductor

Energy stored in an inductor

When current i flows in an inductor of inductance L, the induced emf opposes the rise: ε = −L di/dt. The source must do work against it.

1. Power supplied against the back emf: P = ε_back · i = L i (di/dt)
2. Work in time dt: dW = L i di
3. Total work to raise current from 0 to I:

U = ∫₀ᴵ L i di = ½ L I²

U = ½ L I², stored in the magnetic field.

Magnetic energy density

Take a long solenoid of length l, area A, n turns per unit length. B = μ0 n I and L = μ0 n² A l.

1. U = ½ (μ0 n² A l) I² = (1/2μ0)(μ0 n I)² A l = B²/(2μ0) · (A l)
2. Volume = A l, so energy per unit volume:

u_B = B²/(2μ0)

Thus u_B ∝ B², the square of the flux density. This holds for any magnetic field in vacuum.

Comparison with electric energy density

Both are proportional to the square of the field and exist throughout space where the field is present.