Question

A sphere of radius $R$ is having charge $Q$ uniformly distributed over it. The energy density of the electric field in the air, at a distance $r(r>R)$ is given by (in J/m${}^3$) :

• A. $\frac{Q^2}{32{\pi }^2{ϵ}_o{R}^2{r}^2}$
• B. $\frac{Q^2}{32{\pi }^2{ϵ}_o{r}^4}$
• C. $\frac{Q^2}{32{\pi }^2{ϵ}_o{R}^4}$
• D. $\frac{Q^2}{16{\pi }^2{ϵ}_o{r}^4}$

Answer 1

The correct option is B $\frac{Q^2}{32{\pi }^2{ϵ}_o{r}^4}$

The energy density$\left(ED\right)$ is given by,

$ED=1/2\times {ϵ}_0\times E_r^2$

Therefore,

$ED=1/2\times {ϵ}_0\times k^2{Q}^2/r^4={ϵ}_0/2\times Q^2/\left(16{\pi }^2{ϵ}_0^2{r}^4\right)=Q^2/\left(32{\pi }^2{ϵ}_0{r}^4\right)$