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 capacitor,

$\mathrm{U}=\frac{1}{2} \frac{\mathrm{Q}^{2}}{\mathrm{C}}$

$=\frac{1}{2} \frac{(\sigma \mathrm{A})^{2}}{1} \times \frac{d}{\epsilon_{0} \mathrm{~A}} \text { where } \mathrm{Q}=\sigma \mathrm{A} \text { and } \mathrm{C}=\frac{\epsilon_{0} \mathrm{~A}}{d}$

$=\frac{\sigma^{2} \mathrm{~A} d}{\epsilon_{0}}$

$\text { but } \frac{\sigma}{\epsilon_{0}}=\mathrm{E}$

$\mathrm{U}=\frac{1}{2} \mathrm{E}^{2} \epsilon_{0} \times \mathrm{A} d$

But $\mathrm{A} d$ is the volume of the region between the plates.

$\therefore \frac{\mathrm{U}}{\mathrm{Ad}}=\frac{1}{2} \epsilon_{0} \mathrm{E}^{2}$ is a energy per unit volume.

It is denoted by $\rho_{\mathrm{E}}$ or ' $u$ '. $\therefore$ Energy per unit volume, $\rho_{\mathrm{E}}=\frac{1}{2} \epsilon_{0} \mathrm{E}^{2}$