** Mutual Induction**

When the flux produced by a variable current in one coil (usually called primary) links with a closely situated coil (called secondary), there is said to mutual induction.

If I** _{2}** is the current flowing in the secondary coil, the flux linkages with the primary coil is
proportional to the current in the secondary coil. That is,

where M** _{12 }**is the mutual inductance of
the primary with respect to the secondary. It is also referred to as the coefficient of mutual induction.
Similarly, if I1 is the current flowing in the primary coil, the flux linkages with the secondary coil
is proportional to the current in the primary coil. That is,

where M** _{21}** is the mutual inductance of
the secondary with respect to the primary. It is also referred to as the coefficient of mutual induction.

**Mutual Inductance of two circular co-axial concentric coils :**

Let r_{1} and r_{2} (r_{1} << r_{2}) be radii of two co-axial concentric coils, and N_{1} and N_{2} be number of turns
in two coils. See figure 14. Let the secondary (outer) coil carry current I_{2} . The magnetic field at the centre
due to a current I_{2} is given by

Since the primary (inner) co-axial coil has a very small radius, B_{2} may be considered constant over the cross-sectional area of the primary. Hence, the total flux linkages with the primary (inner) coil are given by

But, N_{1}Φ_{1} = M_{12} I_{2} . Therefore, the mutual-inductance of the of the
primary with respect to the secondary is given by

It is not easy to calculate the flux linkage with the secondary
(outer) coil as the magnetic field due to the primary (inner) coil varies
across the cross section of the secondary coil. Therefore, the calculation of M_{21} will also be extremely
difficult in this case. The equality M_{12} = M_{21} = M (say) given by reciprocity theorem is very useful in such
situations. Therefore, the mutual-inductance of the of the secondary with respect to the primary is given
by

**Mutual Inductance of two co-axial solenoids :**

Let r_{1} and r_{2} be radii of inner (let it be primary P) and
outer (let it be secondary S) co-axial solenoids respectively,
and n_{1} and n_{2} be number of turns per unit length of the two
solenoids. Let N_{1} and N_{2} be total number of turns in two
solenoids and each of length l. See figure 15.

Let the secondary solenoid carry current I_{2} . This
current sets up magnetic flux Φ_{1} through the primary (inner)
solenoid. The total flux linkages with the primary solenoid are
given by
N_{1}Φ_{1} = M_{12} I_{2}

where M_{12} is the mutual
inductance of the primary (inner) solenoid with respect to the secondary (outer) solenoid.

The magnetic field at the centre of the secondary solenoid due to a current I_{2} is given by
B_{2} = μ** _{0}**n

The total flux linkages with the primary solenoid are given by

But, N_{1}Φ_{1} = M_{12} I_{2} . Therefore, the mutual-inductance of the of the primary with respect to the
secondary is given by

Similarly, the total flux linkages with the secondary solenoid due to current in primary are given by
**N _{2}Φ_{2} = (n_{2}l) B_{1}A_{1}**

It should be noted here that we are using A_{1} instead of A_{2}. The flux due to the current I_{1} in the inner
solenoid can be assumed to be confined solely inside this solenoid (i. e. primary) since the solenoids are
very long and hence,

**N _{2}Φ_{2} = (n_{2}l) B_{1}A_{1} (μ_{0}Πn_{1}n_{2} / r_{1}^{2}) / 1**

But, N_{2}Φ_{2}= M_{21} I_{1}. Therefore, the mutual-inductance of the of the secondary with respect to the
primary is given by

It is clear that
M_{12} = M_{21} = M (say)

It should be noted here that this equality holds only for long co-axial solenoids. Therefore, for long co-axial solenoids

**M = μ _{0}Πn_{1}n_{2} / r_{1}^{2}**

It is also important to note that the mutual inductance of a pair of coils, solenoids, etc., depends on their separation as well as their relative orientation.

CBSE Electromagnetic Induction ( With Hint / Solution)

Class XII (By Mr. Ashis Kumar Satapathy)

email - [email protected]

Electromagnetic Induction