Define mutual inductance and give its SI unit. Derive an expression for the mutual inductance of two long coaxial solenoids, and show that M₁₂ = M₂₁.

1. 1 · Define. N₁Φ₁ = M₁₂I₂ and ε₁ = −M₁₂ dI₂/dt: M is the flux linkage with one coil per unit current in the other. Unit henry.
2. 2 · Describe the solenoids. Inner S₁: radius r₁, n₁ turns per metre; outer S₂: n₂ turns per metre; common length l.
3. 3 · Draw them. [Two coaxial solenoids, inner one shaded; r₁ and l marked; current I₂ in the outer]
4. 4 · Find M₁₂. B₂ = μ₀n₂I₂; N₁Φ₁ = n₁l × πr₁² × μ₀n₂I₂; M₁₂ = μ₀n₁n₂πr₁²l.
5. 5 · Find M₂₁. B₁ = μ₀n₁I₁ exists only inside S₁; N₂Φ₂ = n₂l × πr₁² × μ₀n₁I₁; M₂₁ = μ₀n₁n₂πr₁²l.
6. 6 · Conclude. M₁₂ = M₂₁ = M; with a core, multiply by μr.
7. 7 · Real life. Transformers and wireless chargers work by mutual induction.
8. 8 · Close in one line. So, M depends only on the geometry and core of the pair, not on the current.

Scoring tip: Use the inner radius r₁ in both cases and say why: the field of S₁ is zero outside it.