Question

Define mutual inductance between two long coaxial solenoids. Find out the expression for the mutual inductance of inner solenoid of length l having the radius r₁ and the number of turns n₁ per unit length due to the second outer solenoid of same length and r₂ number of turns per unit length.

Answer 1

Mutual Inductance

The ability of production of induced emf in one coil, due to varying current in the neighbouring coil is called mutual inductance.

Magnetic flux, Φ = MI Where, M is called coefficient of mutual induction

Mutual Inductance of Two Long Solenoids

Consider two long solenoids S₁ and S₂ of same length l, such that solenoid S₂ surrounds solenoid S₁ completely.

Φ₂₁ = M₂₁I₁

Where, M₂₁ is the coefficient of mutual induction of the two solenoids

Magnetic field produced inside solenoid S₁ on passing current through it,

B₁ = μ₀n₁I₁

Magnetic flux linked with each turn of solenoid S₂ will be equal to B₁ times the area of cross-section of solenoid S₁.

Magnetic flux linked with each turn of the solenoid S₂ = B₁A

Therefore, total magnetic flux linked with the solenoid S₂,

Φ₂₁ = B₁A × n₂l = μ₀n₁I₁× A× n₂l

Φ₂₁ = μ₀n₁n₂lAI₁

∴ M₂₁ = μ₀n₁n₂Al

Similarly, the mutual inductance between the two solenoids, when current is passed through solenoid S₂ and induced emf is produced in solenoid S₁, is given by

M₁₂ = μ₀n₁n₂Al

∴M₁₂ = M₂₁ = M (say)

Hence, coefficient of mutual induction between the two long solenoids

`M = mu_0n_1n_2Al`