Question

A long solenoid of length L and radius r₁ having N₁ turns is surrounded symmetrically by a coil of radius r₂ (> r₁) having N₂ turns (N₂ ≪ N₁) around its mid-point. Derive an expression for the mutual inductance of solenoid and coil. Is M₁₂ = M₂₁ valid in this case?

Answer 1

Mutual inductance (M) = `"Flux linked with secondary"/"Current in primary"`

Magnetic field inside a long solenoid (B) = µ₀nI

= `mu_0 N_1/L I_1`

The field is uniform inside the solenoid and negligible outside.

The magnetic field exists only inside the solenoid of radius r₁. Area contributing to flux:

A = `pi r_1^2`

Flux through one turn of the outer coil (Φ) = BA

= `mu_0 N_1/L I_1 * pi r_1^2`

Outer coil has N₂ turns:

Φ = N₂ϕ

= `N_2 mu_0 N_1/L I_1 pi r_1^2`

Mutual inductance (M₁₂) = `phi/I_1`

= `mu_0 (N_1 N_2)/L pi r_1^2`

In general M₁₂ = M₂₁

This is a fundamental property of mutual inductance, independent of geometry (as long as the medium is linear and isotropic).

Thus, M₁₂ = M₂₁ is valid.