Figure 1: Setup to understand the concept of Mutual Inductance.
Definition of Mutual Inductance
Mutual Inductance is defined as the property due to which the e in current through one coil produces an emf in the other coil placed nearby, by induction. The two magnetically coupled coils C1 and C2 in Fig. 1, are said to have mutual inductance. It is denoted by M and measured in Henry. The expression for mutual inductance is,
\[M=\frac{{{N}_{2}}{{\phi }_{2}}}{{{I}_{1}}}\]
Or,
\[M=\frac{{{N}_{1}}\times {{\phi }_{1}}}{{{I}_{2}}}\]
Where N1 and N2 are the number of turns of coils C1 and C2. I1 and I2 are the currents flowing through them. ϕ1 is the flux produced by I1 in C1 and is the flux ϕ2 produced by I2 in C2.
Setup for Mutual Inductance
The setup to understand the concept of mutually induced e.m.f is as shown in Fig. 1. Two coils C1 and C2 are placed near each other. Coil C1 consists of N1 turns and coil C2 consists of N2 number of turns. A switch, battery and a variable resistance R are connected in the circuit consisting of coil C1. Whereas, a galvanometer is connected to coil C2. This galvanometer is connected to sense the current induced in coil C2.
Description :
Due to the battery a current I1 starts flowing through coil C1. This current can be changed by changing the value of resistance R. Due to current I1 flowing through coil C1, flux is produced, which is denoted by ϕ1. A part of this flux completes its path through coil C2 as shown in Fig. 1. This flux is called as the mutual flux. If we change current I1 through coil C1 by changing R, then flux ϕ1 will change. This will change the mutual flux ϕ2. According to the Faraday’s laws there will be induced e.m.f. in coil C2. Due to this emf, current I2 will start flowing through coil C2 which is indicated by the galvanometer “G”.
Thus if the flux produced by one coil gets linked with another coil, and due to the change in this flux produced by the coil C1 if there is induced emf in the second coil then such an emf is known as mutually induced emf. The mutually induced emf in coil C2 will exist as long as the value of current through coil C1 i.e. I1 is changing with respect to time.
Formula for Mutual Inductance :
The expression for mutual inductance is as follows:
\[M=\frac{{{N}_{1}}{{N}_{2}}{{\mu }_{0}}{{\mu }_{r1}}{{a}_{1}}}{{{l}_{1}}}\]
\[M=\frac{{{N}_{1}}{{N}_{2}}{{\mu }_{0}}{{\mu }_{r2}}{{a}_{2}}}{{{l}_{2}}}\]
Where,
N1, N2 = Number of turns of coils C1 and C2
a1, a2 = Cross sectional areas.
l1, l2 = Lengths of the coils.
Coefficient of Coupling (K) :
The two expressions for the mutual inductance are,
\[M=\frac{{{N}_{2}}{{K}_{1}}{{\phi }_{1}}}{{{I}_{1}}}…(1)\]
And,
\[M=\frac{{{N}_{1}}{{K}_{2}}{{\phi }_{2}}}{{{I}_{2}}}…(2)\]
Multiply Equations (1) and (2) to get,
\[{{M}^{2}}=\frac{{{N}_{1}}{{N}_{2}}{{K}_{1}}{{K}_{2}}{{\phi }_{1}}{{\phi }_{2}}}{{{I}_{1}}{{I}_{2}}}\]
Rearrange this expression as follows :
\[{{M}^{2}}={{K}_{1}}.{{K}_{2}}.\left[ \frac{{{N}_{1}}{{\phi }_{1}}}{{{I}_{1}}} \right]\left[ \frac{{{N}_{2}}{{\phi }_{2}}}{{{I}_{2}}} \right]…(3)\]
But,
\[\left[ \frac{{{N}_{1}}{{\phi }_{1}}}{{{I}_{1}}} \right]={{L}_{1}}\] i.e. self inductance of coil C1.
And,
\[\left[ \frac{{{N}_{2}}{{\phi }_{2}}}{{{I}_{2}}} \right]={{L}_{2}}\] i.e. the self inductance of coil C2.
Substituting these values into Equation (3) we get,
\[{{M}^{2}}={{K}_{1}}.{{K}_{2}}.{{L}_{1}}.{{L}_{2}}\]
\[M=\sqrt{{{K}_{1}}.{{K}_{2}}}.\sqrt{{{L}_{1}}.{{L}_{2}}}\]
\[M=K\sqrt{{{L}_{1}}.{{L}_{2}}}\]
Where,
\[K=\sqrt{{{L}_{1}}.{{L}_{2}}}\]
And K is called as the coefficient of coupling.
Mathematical expression for coefficient of coupling is :
\[K=\frac{M}{\sqrt{{{L}_{1}}.{{L}_{2}}}}\]
Definition :
From this expression, we can define the coefficient of coupling K as the ratio of actual mutual inductance (M) present between the coils C1 and C2 to the maximum value of M.
However,
\[{{M}_{\max }}=\sqrt{{{L}_{1}}.{{L}_{2}}}\]
\[K=\frac{M}{{{M}_{\max }}}\]
The maximum value of K is 1 which represents the coupling of all the flux produced by one with the other one. Corresponding to K = 1 the value of the mutual inductance will be maximum and it is given by,
\[{{M}_{\max }}=\sqrt{{{L}_{1}}.{{L}_{2}}}\]
Definition :
Thus, the coefficient of mutual inductance is defined as the property which is responsible for the induced emf in one coil due to change in current flowing through some other coil placed nearby.
Tight coupling and loose coupling :
The coupling between the two coils is said to be a tight coupling if K = 1 and the coupling is called as the loose coupling if K is less than one. The coefficient of coupling is also called as Magnetic Coupling Coefficient.