Question

The electric field in a region is directed outward and is proportional to the distance r from the origin. Taking the electric potential at the origin to be zero,
(a) it is uniform in the region
(b) it is proportional to r
(c) it is proportional to r²
(d) it increases as one moves away from the origin

Answer 1

(c) it is proportional to r²

Given:
E$\propto$r and V = 0 at r =0
$\Rightarrow$E = kr
Also, E = $\frac{-dV}{dr}$
$$\begin{gathered} \therefore V=-\int Edr \\ \Rightarrow V=-\int krdr \\ \Rightarrow V=-k\frac{r^2}{2}+C \end{gathered}$$
Using the condition, V = 0 at r = 0, we get C =0.
Therefore,
$$\begin{gathered} V=-k\frac{r^2}{2} \\ \Rightarrow V\propto r^2 \end{gathered}$$