Question

The electric potential inside a charged sphere varies with the distance 'r' from its centre as V = a - br${}^3$, where 'a' and 'b' are positive constant. Then select the correct statement(s) from the following.

(where E = Electric field inside the sphere at a distance ‘r’ from its centre, $\rho$ = volume charge density inside the sphere at a distance ‘r’ from its centre)

• A. $E\propto r$
• B. $E\propto r^2$
• C. $\rho \propto r$
• D. $\rho \propto r^2$

Answer 1

Answer: (B), (C)

The correct options are
B

$E\propto r^2$

C

$\rho \propto r$

$\text{Given}V=a-b{r}^3$

$\therefore E=\frac{-dV}{dr}=3b{r}^2\therefore E\propto r^2$

Using differential from of Gauss's law $\frac{dE}{dr}+\frac{2E}{4}=\frac{\rho }{{ϵ}_0}$

$6br+6br=\frac{\rho }{{ϵ}_0}\therefore \rho =12b{ϵ}_{0}r$

$\therefore \rho \propto r$