Define an inductor. Derive the equation for the energy $U = \frac{1}{2}LI^2$ stored in an inductor.

(A) An inductor is a passive electrical component that stores energy in its magnetic field when electric current flows through it,characterized by its self-inductance $L$.
When the current $I$ in an inductor changes,a back electromotive force (emf) $\varepsilon$ is induced,which opposes the change in current according to Lenz's Law. The magnitude of this back emf is given by $|\varepsilon| = L \frac{dI}{dt}$.
To establish a current $I$ in the inductor,work must be done against this back emf. The rate of doing work is given by:
$\frac{dW}{dt} = |\varepsilon| I = L I \frac{dI}{dt}$
Integrating this expression to find the total work done $W$ to increase the current from $0$ to $I$:
$W = \int dW = \int_{0}^{I} L I' dI' = L \left[ \frac{I'^2}{2} \right]_{0}^{I} = \frac{1}{2} LI^2$
This work done is stored as magnetic potential energy $U$ in the inductor:
$U = \frac{1}{2} LI^2$
This expression is analogous to the kinetic energy of a particle,$K = \frac{1}{2}mv^2$,where $L$ acts as electrical inertia.