Obtain the expression for the energy stored per unit volume in a charged capacitor.

Energy stored per unit volume is known as energy density. The energy stored in a capacitor is given by $U = \frac{1}{2} \frac{Q^2}{C}$.
Substituting $Q = \sigma A$ and $C = \frac{\epsilon_0 A}{d}$,we get:
$U = \frac{1}{2} \frac{(\sigma A)^2}{\epsilon_0 A / d} = \frac{1}{2} \frac{\sigma^2 A^2 d}{\epsilon_0 A} = \frac{1}{2} \frac{\sigma^2 A d}{\epsilon_0}$.
Since the electric field between the plates is $E = \frac{\sigma}{\epsilon_0}$,we have $\sigma = \epsilon_0 E$.
Substituting this into the expression for $U$:
$U = \frac{1}{2} \frac{(\epsilon_0 E)^2 A d}{\epsilon_0} = \frac{1}{2} \epsilon_0 E^2 (A d)$.
Here,$A d$ represents the volume $V$ of the region between the plates.
Therefore,the energy stored per unit volume $(u)$ is:
$u = \frac{U}{V} = \frac{U}{A d} = \frac{1}{2} \epsilon_0 E^2$.