Explain mutual induction and derive the equation for mutual $emf$.

(N/A) Mutual induction is the phenomenon where a change in current in one coil induces an $emf$ in a neighboring coil due to the change in magnetic flux linked with it.
As shown in the figure,when the current $I_{2}$ in coil $C_{2}$ changes,the magnetic flux $\Phi_{1}$ linked with coil $C_{1}$ also changes,inducing an $emf$ in coil $C_{1}$.
If the number of turns in coil $C_{1}$ is $N_{1}$,then the total flux linkage is proportional to the current in $C_{2}$:
$N_{1} \Phi_{1} \propto I_{2}$
$N_{1} \Phi_{1} = M_{12} I_{2}$
where $M_{12}$ is the coefficient of mutual induction.
According to Faraday's law of electromagnetic induction,the induced $emf$ $\varepsilon_{1}$ in coil $C_{1}$ is:
$\varepsilon_{1} = -\frac{d(N_{1} \Phi_{1})}{dt}$
Substituting the expression for flux linkage:
$\varepsilon_{1} = -\frac{d}{dt}(M_{12} I_{2})$
Assuming $M_{12}$ is constant:
$\varepsilon_{1} = -M_{12} \frac{dI_{2}}{dt}$
Similarly,for coil $C_{2}$,the induced $emf$ is:
$\varepsilon_{2} = -M_{21} \frac{dI_{1}}{dt}$