*If the rate of flow of current in the primary coil is unit then the coefficient of mutual induction will be equal to the induced electro-motive force $\left( emf \right)$ in the secondary coil.*

### Mutual Induction Phenomenon and its Coefficient

Mutual Induction:

When two coils are placed near each other then the change in current in one coil ( Primary Coil) produces electro-motive force $\left( emf \right)$ in the adjacent coil ( i.e. secondary coil). This phenomenon is called the principle of Mutual Induction.

The direction of electro-motive force $\left( emf \right)$ depends or can be found by "Lenz's Law"

Mathematical Analysis of Coefficient of Mutual Induction:

Let us consider that two coils having the number of turns are $N_{1}$ and $N_{2}$. If these coils are placed near to each other and the change in current of the primary coil is $i_{1}$, then linkage flux in the secondary coil will be
$N_{2}\phi_{2} \propto i_{1}$

$N_{2}\phi_{2} = M i_{1} \qquad(1)$

Where $M$ $\rightarrow$ Coefficient of Mutual Induction.

According to Faraday's law of electromagnetic induction. The electro-motive force $\left( emf \right)$ in the secondary coil is

$e_{2}=-N_{2}\left( \frac{d \phi_{2}}{dt} \right)$

$e_{2}=-\frac{d \left(N_{2} \phi_{2} \right)}{dt} \qquad(2)$

From equation $(1)$ and equation $(2)$

$e_{2}=-\frac{d \left(M i_{1} \right)}{dt} $

$e_{2}=-M \left(\frac{d i_{1}}{dt} \right) $

$M = \frac{e_{2}}{\left(\frac{d i_{1}}{dt} \right)}$

If $\left(\frac{d i_{1}}{dt} \right) = 1$

Then

$M = e_{2}$

The above equation shows that