Self induction of solenoid and toroid

Self induction

When the current in a coil changes, its own magnetic flux changes, and by Faraday's law an emf is induced in the same coil. This emf opposes the change of current (Lenz's law). The phenomenon is self induction. The flux linkage is NΦ = LI, so the emf is ε = −L dI/dt, where L is the self inductance (unit henry).

(a) Long solenoid

Length ℓ, N turns, cross-section area A, current I.

1. Field inside (Ampere's law): B = μ₀ N I/ℓ.
2. Flux through one turn: Φ = BA = μ₀NIA/ℓ.
3. Flux linkage: NΦ = μ₀N²AI/ℓ.
4. L = NΦ/I:

L = μ₀ N² A / ℓ = μ₀ n² A ℓ (n = N/ℓ)

(b) Toroid

Toroid with N turns, inner radius a, outer radius b, height h.

      ____________
     /   b        \
    |   ( a )      |   B is circular, inside the core
     \____________/

1. Ampere's law on a circle of radius r inside the core: B(2πr) = μ₀NI, so B = μ₀NI/(2πr).
2. Flux through the rectangular section: Φ = ∫ₐᵇ B h dr = (μ₀NIh/2π) ln(b/a).
3. L = NΦ/I:

L = μ₀ N² h ln(b/a) / (2π)

L is a property of the coil

In both results L contains only N, the dimensions (A, ℓ, a, b, h) and μ₀ (or μ for a core). There is no I or dI/dt in the expressions. So L depends only on the geometry of the coil and the medium, and not on the current flowing. Therefore inductance is a property of the coil itself, like resistance for a resistor.