Consider two coils 1 and 2 placed near each other as shown below in the figure
Let coil 1 be the primary coil and coil 2 be secondary coil
When current is primary coil changes w.r.t time then the magnetic field produced in the coil also changes with time which causes a change in magnetic flux associated with secondary coil
Due to this change of flux linked with secondary coil an emf is induced in it and this phenomenon is known as mutual induction
Similarly change in current in secondary coil induces an emf in primary coil.This way as a result of mutual inductance emf is induced in both the coils
If I_{1} is the current in primary coil at any instant ,than the emf induced in secondary coil would be proportional to the rate of change of current in primary coil i.e.
Where M is a constant known as coefficient of mutual induction and minus sign indicates that direction of induced emf is such that it opposes the change of current in primary coil
Unit of mutual inductance is Henry
We know that a magnetic flux is produced in primary coil due to the flow of current I_{1}.If this is the magnetic flux associated with secondary coil then from faraday's law of EM induction ,emf induced in secondary coil would be
comparing equation (24) and (25) we get
Thus coefficient of mutual induction of secondary coil w.r.t primary coil is equal to magnetic flux linked with secondary coil when 1 Ampere of current flows in primary coil and vice-versa
Similarly ,if I_{2} is the current in secondary coil at any instant then flux linked with primary coil is
where M_{12} is coefficient of mutual induction of primary coil with respect to secondary coil
EMF induced in primary coil due to change of this flux is
For any two circuits
M_{12}=M_{21}=M
In general mutual inductance of two coil depends on geometry of the coils (shape ,size, number of turns etc),distance between the coils and nature of material on which the coil is wound
Mutual Inductance of two co-axial solenoids
Consider a long solenoid of length l and area of cross-section A containing N_{p} turns in its primary coil
Let a shorter secondary coil having N_{2} number of turns be wounded closely over the central portion of primary coil as shown below in the figure.
If I_{p} is the current in the primary coil then magnetic field due to primary coil would be
So flux through each turn of secondary coil would be
where A is the area of cross-section of primary coil
Total magnetic flux through secondary coil is
Emf induced in secondary coil is
Thus from equation 24
So
Relation between Mutual inductance and self inductance
Consider two coils of same length l and same area of cross-section placed near each other as shown below in the figure
Let there are N_{1} number of turns in primary coil and N_{2} number of turns in secondary coil
A current I_{1} in the primary coil produces a magnetic field
which in turns gives rise to flux?
in primary coil and
in the secondary coil due to current in primary coil.
By the definition of self induction
and by definition of mutual induction
Reversing the procedure if we first introduce the current I_{2} in secondary coil then we get
So L_{1} is the self inductance of primary coil,L_{2} is the self induction of secondary coil and M_{21}=M_{12}=M is the mutual inductance between two coils
Product of L_{1} and L_{2} is
In practice M is always less than eq due to leakage which gives
Where K is called coefficient of coupling and K is always less then 1.