Mutual inductance

The flux a current sets up through a coil grows in step with that current. For N closely wound turns, the flux linkage NΦB divided by I is a fixed number for the coil, its inductance.

Inductance depends only on geometry and on the material inside, much as capacitance depends on plate area, gap and dielectric. It is a scalar with dimensions [M L² T⁻² A⁻²]; its SI unit is the henry (H).

Two long coaxial solenoids of length l: inner S₁ with radius r₁ and n₁ turns per metre, outer S₂ with n₂. A current I₂ in S₂ gives field μ₀n₂I₂, and the flux linkage with S₁ is N₁Φ₁ = M₁₂I₂ with M₁₂ = μ₀n₁n₂πr₁²l.

Working the other way, the flux of S₁ lies only inside S₁, and M₂₁ comes out the same. In general M₁₂ = M₂₁ = M, which helps when one direction is hard to calculate.

With a core of relative permeability μr, M = μrμ₀n₁n₂πr₁²l. M also depends on how far apart the coils are and how they are oriented.

Example 6.8: a small loop of radius r₁ at the centre of a large coaxial loop of radius r₂ (r₁ ≪ r₂) sits in a nearly uniform field μ₀I₂/2r₂, so M = μ₀πr₁²/2r₂.

A changing current in one coil induces an emf in the other: ε₁ = −M dI₂/dt. This explains the kicks in Experiment 6.3.