Question

The electristatic energy stored in electric field per unit volume is called energy density.Then derive the formula for enegy density.

Answer 1

$$\begin{gathered} \text{Suppose}V\left(\overrightarrow{r}\right)\text{is the potential at a point}\overrightarrow{r}.\text{Suppose}\rho \left(\overrightarrow{r}\right)i\text{s the charge} \\ \text{density of that point. Now let us consider a very small volume}dv \\ \text{around}\overrightarrow{r}.\text{When dv is very small}\left(dv\to 0\right)\text{we can consider that the} \\ \text{electric potential and charge density is constant over that volume.} \\ \text{Now total charge of that volume is}q=\rho \left(\overrightarrow{r}\right)dv.\text{So the total elctrostatic} \\ \text{energy stored in that volume is}dE=Vq=V\left(\overrightarrow{r}\right)\rho \left(\overrightarrow{r}\right)dv,\text{so the energy} \\ \text{density is} \\ \frac{dE}{dv}=\frac{V\left(\overrightarrow{r}\right)\rho \left(\overrightarrow{r}\right)dv}{dv}=V\left(\overrightarrow{r}\right)\rho \left(\overrightarrow{r}\right). \end{gathered}$$