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Self induction and inductance of a coil

Short answer

Self-induction is the emf induced by a coil’s own changing current; inductance L=NΦ/I. For a long solenoid, L=μN²A/ℓ=μ₀μ_r N²A/ℓ, proving L depends on the permeability μ of the core and the coil geometry (N, A, ℓ).

What is self induction? Define inductance of a coil. Show by calculation that the inductance of a coil depends on the permeability of the medium and the geometry of the coil.

Self induction

When the current through a coil changes, the magnetic flux linked with the coil changes. An emf is induced in the same coil which opposes the change (Lenz's law). This is self induction: ε = −N dΦ/dt = −L dI/dt.

Inductance

The inductance of a coil is the flux linkage per unit current: L = NΦ/I. One henry is the inductance for which a current changing at 1 A/s induces 1 V. L is also the emf per unit rate of change of current: L = −ε/(dI/dt).

L depends on permeability and geometry

Take a long solenoid with N turns, length ℓ, cross-section A, filled with a core of permeability μ = μ₀μ_r.

  1. Field inside (Ampere's law): B = μ N I/ℓ.
  2. Flux through each turn: Φ = BA = μ N I A/ℓ.
  3. Flux linkage: NΦ = μ N² A I/ℓ.
  4. Inductance: L = NΦ/I

L = μ N² A / ℓ = μ₀ μ_r N² A / ℓ

Reading the result:

  • L ∝ μ: an iron core (μ_r ~ hundreds or thousands) raises L by that factor compared with air.
  • L ∝ N²/ℓ and ∝ A: it depends on the number of turns, length and area, that is, the geometry.
  • I does not appear. L is fixed by the medium and the shape of the coil alone.

✓ Answer checked by Claude Opus 5.5 (high effort)

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