Question

Within a spherical charge distribution of charge density $\rho(r),N$ equipotential surfaces of potential $V_0 + \Delta V , V_0 + 2 \Delta V, ......V_0 + N \Delta V (\Delta V >0)$, are drawn and have increasing radii $r_0, r_1, r_2,....r_N$,respectively. If the difference in the radii of the surfaces is constant for all values of $V_0$ and $\Delta V$ then :

• A. $\rho (r) \, \alpha r$
• B. $\rho (r) = $ constant
• C. $\rho (r) \alpha \frac{1}{r}$
• D. $\rho(r) \alpha \frac{1}{r^2}$

Answer 1

Answer: (C)

$\frac{\Delta V }{\Delta r} \rightarrow$ constant
$\Rightarrow $ uniform $E$. field.
(E) $\left(4 \pi r^{2}\right)=\frac{1}{\varepsilon_{0}} \int \rho d V$
(E) $\left(4 \pi r^{2}\right)=\frac{1}{\varepsilon_{0}} \int\limits_{0}^{r} \rho 4 \pi r^{2} d r$
(E) $\left(4 \pi r^{2}\right)=\frac{1}{\varepsilon_{0}} 4 \pi \int\limits_{0}^{r} \rho r^{2} d r$
after integral on $RHS$
We must obtain $r^{2}$
$\Rightarrow \rho \propto \frac{1}{ r }$