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)

$\frac{\Delta V}{\Delta r}\to$ constant
$\Rightarrow$ uniform $E$. field.
(E) $\left(4\pi r^2\right)=\frac{1}{{\epsilon }_0}\int \rho dV$
(E) $\left(4\pi r^2\right)=\frac{1}{{\epsilon }_0}\underset{0}{\overset{r}{\int }}\rho 4\pi r^{2}dr$
(E) $\left(4\pi r^2\right)=\frac{1}{{\epsilon }_0}4\pi \underset{0}{\overset{r}{\int }}\rho r^{2}dr$
after integral on $RHS$
We must obtain $r^2$
$\Rightarrow \rho \propto \frac{1}{r}$