Explain self-induction and obtain the equation for the self-induced $emf$ in a coil.

(N/A) Self-induction is the phenomenon where an $emf$ is induced in a single isolated coil due to a change in magnetic flux through it,caused by varying the current flowing through the same coil.
In this case,the magnetic flux linkage through a coil of $N$ turns is proportional to the current $I$ flowing through it:
$N \phi_{B} \propto I$
$N \phi_{B} = LI \quad \dots (1)$
Here,the constant of proportionality $L$ is called the self-inductance of the coil,also known as the coefficient of self-induction.
According to Faraday's law of induction,the induced $emf$ $(\varepsilon)$ is given by the negative rate of change of magnetic flux linkage:
$\varepsilon = -\frac{d(N \phi_{B})}{dt}$
Substituting equation $(1)$ into this expression:
$\varepsilon = -\frac{d(LI)}{dt}$
Since $L$ is constant for a given coil:
$\varepsilon = -L \frac{dI}{dt} \quad \dots (2)$
Thus,the self-induced $emf$ always opposes any change (increase or decrease) in the current flowing through the coil.