Lesson 8 of 10 · 6 min
Self-inductance and magnetic energy
NCERT §6.7.2
Arjun opens the switch on his solenoid, now with a soft-iron core inside, while 2 A flows. A spark jumps across the switch. Where does the energy for that spark come from?
The lesson in notes
In short
A changing current in a coil changes the coil's own flux, inducing an emf in the same coil: self-induction. The flux linkage is NΦB = LI, where L is the self-inductance.
The self-induced emf is ε = −L dI/dt. It opposes any change in the current, rise or fall, and is called the back emf.
For a long solenoid of cross-section A, length l and n turns per metre, L = μ₀n²Al. With a core of relative permeability μr, such as soft iron, L = μrμ₀n²Al.
L behaves like inertia, the electrical analogue of mass: it resists both the growth and the decay of current.
Work done against the back emf while the current builds from 0 to I is stored as magnetic energy: W = ½LI², just as a mass stores ½mv².
With two coils carrying currents, the emf in coil 1 is ε₁ = −L₁ dI₁/dt − M₁₂ dI₂/dt.
Example 6.9: for a solenoid the stored energy is (B²/2μ₀)Al, so the energy per unit volume is B²/2μ₀. This matches the electric energy density ½ε₀E²: both grow as the square of the field, and both hold for any region of space.