Self-inductance of a toroid

Self-induction. When the current in a coil changes, the magnetic flux linked with the coil itself changes, and an emf is induced in the same coil which opposes the change in current (Lenz's law). ε = −L di/dt, where L = NΦ/i is the self-inductance (henry).

Toroid

A toroid is a solenoid bent into a ring. Let it have N turns, mean radius r, core cross-sectional area A, and current i.

        _____________
      /  ___________  \
     |  /   (turns)   \ |   <- mean radius r
     |  \_____________/ |
      \________________/

Derivation

1. Choose a circular Amperian loop of radius r inside the core. By symmetry B is tangential and constant along it.
2. Ampere's law: B (2πr) = μ0 N i (the loop encloses N turns, each carrying i)
3. B = μ0 N i / (2πr)
4. Flux through one turn: Φ = B A = μ0 N i A / (2πr)
5. Flux linkage: NΦ = μ0 N² i A / (2πr)
6. Inductance: L = NΦ/i

L = μ0 N² A / (2πr)

(Using the mean radius r, with A ≪ r².) With a core of relative permeability μr, L = μ0μr N² A/(2πr). Outside the toroid the field is zero, since the loop encloses zero net current.

L depends only on the geometry (N, A, r) and the core material, not on i.