Mutual Induction
Electromagnetic Induction of Class 12
MUTUAL INDUCTION
The appearance of an induced emf in one circuit due to changes in the magnetic field produced by a nearby circuit is called mutual induction.
The flux linkage through coil 1 due to current I2 in coil 2 may be written as
Φ12 = M12I2
where the constant of proportionality M12 is called the mutual inductance of the coil 1 with respect to coil 2. Similarly the flux linkage through coil 2 due to current I1 in coil 1 may be written as
Φ21 = M21 I1
The mutual inductance of two circuits depends on their sizes, their shapes, and their relative positions. Intuitively one would expect the mutual inductance to be greater when the coils are near to each other and oriented so that the maximum amount of flux from the coil intercepts the other. The SI unit of mutual inductance is henry (H).
There is one remarkable property of mutual inductance, i.e.
M_{12} = M_{21} = M(4.13)
This is also called the reciprocity theorem
The emf induced in coil 1 due to changes in I2 takes the form
ε12 = M dI_{2}/dt
Example: 4.8 Two concentric coplanar circular loops 1 and 2 are shown in the Fig 4.19. The radii of the loops are r and R. Current I flows in the loop 1. Find the magnetic flux Φ2 through the loop 2 if r << R 

Solution
The direct calculation of the flux Φ2 is very complicated because the magnetic field is not uniform over the big loop 2. However, the application of the reciprocity theorem greatly simplifies the solution of the problem
Let us pass the same current I through the loop 2. Then the magnetic flux Φ1 created by this current through loop 1 can be easily found because magnetic field is more or less uniform in the small loop 1 (because r << R) Thus, magnetic field at the centre of the loops is
B =
and the magnetic flux through the loop 1 is
Φ1 = (πr2) B =
Using reciprocity theorem
Φ2 = Φ1 =
And the coefficient of mutual inductance is
M =