(a) Define mutual inductance and write its S.I. units.
(b) Derive an expression for the mutual inductance of two long co-axial solenoids of same length wound one over the other.
(c) In an experiment, two coils C1 and C2 are placed close to each other. Find out the expression for the emf induced in the coil C1 due to a change in the current through the coil C2.
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
= Flux linked with S2 due to current flowing in S1
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, 
Magnetic flux linked with each turn of the solenoid = 
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, 
c) It is found that, 
where, I is the strength of current in coil 2, and
is the total amount of magnetic flux linked with coil 1.
Emf induced in the neighboring coil C1 is,
