Define mutual inductance and write its S.I. unit.

Question

(a) Define mutual inductance and write its S.I. unit.

(b) Derive an expression for the mutual inductance of two long co-axial solenoids of same length wound one over the other.

Answer 1

(a) Mutual inductance of two coils is equal to the e.m.f induced in one coil when rate of change of current through the other coil is unity.

SI unit of mutual inductance is henry.

(b) Consider two long solenoids S₁ and S₂ of same length ‘l’ such that S₂  surrounds S₁ completely.

Let,

n₁  = Number of turns per unit length of S₁

n₂ = Number of turns per unit length of S₂

I₁ = Current passing through solenoid S₁ 

\(\phi_{21}\) = Flux linked with S₂  due to current flowing in S₁
\(\phi_{21} \propto l_1\)

\(\phi_{21} = M_{21}l_1 ; M_{21}\) is the coefficient of mutual inductance of two solenoids.

When current is passed through S₁, emf is induced in S₂ 

Magnetic field inside solenoid S₁ is given by, \(B_1 = \mu_on_1l_1\)

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

Total magnetic flux liked with S₂ is given by,

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

Hence, coefficient of mutual induction between the two long solenoids is given by, 
M = µ_on₁n₂Al