Question

Derive an expression for the mutual inductance of a system of two long coaxial solenoids of same length l, having turns N₁ and N₂ and of radii r₁ and r₂ (> r₁).

Answer 1

Consider two co-axial long Solenoids S₁ and S₂

Let I = length of each solenoid.

r₁, r₂ = radii of two solenoids

`A = pir_1^2`

= area of cross section of S₁

N₁, N₂ = no. of turns in S₁ and S₂ respectively

Let I₂, a time-varying current, be passed through S₂.

So magnetic field `B_2 = mu_0eta_2I_2; eta_2 = N_2/l`

Total magnetic flux linked with S₁ is,

Φ₁ = B₂AN₁ = μ₀η₂I₂.AN₁

∴ Mutual inductance of coil −1 w.r.t. coil −2 is

`M_12 = phi_1/I_2 = mu_0eta_2AN_1`

= `(mu_2N_1N_2A)/l`

Similarly, flux linked with the outer solenoid S₂ due to current I₁ in S₁,

Φ₂ = B₁AN₂ = μ₀η₁I₁.AN₂

⇒ `phi_2 = (mu_0N_1N_2AI_1)/l`

Now mutual inductance of coil −2 w.r.t coil −1 is

`M_21 = phi_2/I_1 = (mu_0N_1N_2A)/l`

So, M₁₂ = M₂₁ = M(Let)

∴ `M = (mu_0N_1N_2A)/l`

= = `mu_0eta_1eta_2Al`

= `mu_0eta_1eta_2pir_1^2l`