Question 212 of 311

Within a spherical charge distribution of charge density $\rho \left(r\right),$ N equipotential surfaces of potential ${V}_{0},{V}_{0}+∆V,{V}_{0}+2∆V,\ldots {V}_{0}+N∆V\left(∆V\gt 0\right),$ are drawn and have increasing radii ${r}_{0},{r}_{1},{r}_{2},\ldots {r}_{N},$ respectively. If the difference in the radii of the surfaces is constant for all values of ${V}_{0}and∆V$ then :

A$\rho \left(r\right)=$ constantB$\rho (r)\propto \dfrac{1}{{r}^{2}}$C$\rho (r)\propto \dfrac{1}{r}$D$\rho (r)\propto r$

Show answer and solution▾

Answer: (C) $\rho (r)\propto \dfrac{1}{r}$

$E=-\dfrac{\Delta V}{\Delta r}$

$∴$ △v and △r are constant

E = constant

$E=\dfrac{Kq}{{r}^{2}}=c$

$\Rightarrow q\propto {r}^{2}$

$∴q={\int }_{0}^{r}\rho 4\pi {r}^{2}dr$

$\Rightarrow \rho \propto \dfrac{1}{r}$