Question

The energy density in the electric field created by a point charge falls off with the distance from the point charge as
(a) $\frac{1}{r}$

(b) $\frac{1}{r^2}$

(c) $\frac{1}{r^3}$

(d) $\frac{1}{r^4}$.

Answer 1

(d) $\frac{1}{r^4}$

Energy density U is given by

U = $\frac{1}{2}{\in }_0{E}^2$ ...(1)

The electric field created by a point charge at a distance r is given by
E = $\frac{q}{4\mathrm{\pi }{\in }_0{r}^{\mathit{2}}}$

On putting the above form of E in eq. 1, we get

U = $\frac{1}{2}{\in }_0{\left(\frac{q}{4\mathrm{\pi }{\in }_0{r}^{\mathit{2}}}\right)}^2$

Thus, U is directly proportional to $\frac{1}{r^4}$.