Question

Derive an expression for the energy stored per unit volume (energy density) in an electric field. OR Obtain an expression for energy density of a medium.

Answer 1

When a conductor is a charged, during the process of charging work has to be done to bring the charge on the surface of conductor. This work done is stored in the form of surrounding the conductor in the form of electrostatic energy per unit volume is called energy density.
Consider a charged conductor of surface charge density $\sigma$ in a median of dielectric constant K. Consider small area ds of the conductor.
We know that the mechanical force per unit area of a charged conductor =$\frac{1}{2}ϵE^2$
$\therefore$ Mechanical force act ingon area ds,
$F=\frac{1}{2}ϵE^{2}ds$
Assume that the element ds is displaced through a distance dl under the action of mechanical force acting on area ds. During the displacement the work done:
$dw=\frac{1}{2}ϵE^{2}ds.dl$ $\therefore dw=\frac{1}{2}ϵE^{2}dv$
$dv=ds.dl$= Volume swept by ds during displacement dl.
This work done is stored in the electric field in form of electrostatic energy and it is given by:
Electrostatic energy $du=\frac{1}{2}ϵE^{2}dv$
$\therefore$ Energy per unit volume = Electrostatic energy per unit volume
$\frac{du}{dv}=\frac{1}{2}ϵE^2\therefore \frac{du}{dv}=\frac{1}{2}{ϵ}_{0}k{E}^2$
This is the expression for energy density of the medium.
If the charged conductor is placed in air, k= 1.
$\therefore$ Energy density $\frac{du}{dv}=\frac{1}{2}{ϵ}_0{E}^2$