Question

Obtain an expression for energy density of a medium.

Answer 1

consider a small element ,ds on the surface of charged conductor as shown in figure. if $\sigma$ is the surface charge density and the charge carried by the element ds be dq.
then, $dq=\sigma.ds$

see figure, net electric field normally outward
so, $E=\frac{\sigma}{2\epsilon_0}$

so, force = E.dq
or , F = $\frac{\sigma}{2\epsilon_0}dq$

$F=\frac{\sigma}{2\epsilon_0}\sigma.dS$

$F=\frac{\sigma^2}{2\epsilon_0}dS$

we know, workdone by force is given by,
$W=F.dl$

$W=\frac{\sigma^2}{2\epsilon_0}dl.dS$

$W=\frac{\sigma^2}{2\epsilon_0}dV$

actually, workdone by external force = potential energy

so, $U=\frac{\sigma^2}{2\epsilon_0}dV$

$\frac{U}{dV}=\frac{1}{2}\frac{\sigma^2}{\epsilon_0}$

we also know, $\sigma=\epsilon_0E$

so, energy density = $\boxed{\boxed{\frac{1}{2}\epsilon_0E^2}}$