What is mutual induction? Derive an expression for the mutual inductance of two long coaxial solenoids of the same length.

1. 1 · Define it. Mutual induction is the induction of an emf in a coil when the current in a nearby coil changes; φ₂ = MI₁ and ε₂ = −M dI₁/dt.
2. 2 · Draw the solenoids. [A long outer solenoid with n₁ turns per metre; a narrower inner solenoid of radius r₂ and N₂ turns on the same axis; current I₁ marked in the outer one]
3. 3 · Field of the outer solenoid. B₁ = μ₀n₁I₁, uniform inside it.
4. 4 · Flux through the inner solenoid. φ₂ = N₂ × B₁ × πr₂² = μ₀n₁N₂πr₂²I₁.
5. 5 · Find M. M = φ₂/I₁ = μ₀n₁N₂πr₂², or μ₀n₁n₂πr₂²l with N₂ = n₂l.
6. 6 · Close in one line. So, M depends only on the turns, the inner area and the core, not on the current.

Scoring tip: Use the area of the inner solenoid, πr₂², because the field links only that area of the inner turns.