Question

Within a spherical charge distribution of charge density $\rho \left(r\right)$, N equipotential surfaces of potential $V_0,V_0+\Delta V,V_0+2\Delta V,.....,V_0+N\Delta V(\Delta V>0)$, 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. $\rho \left(r\right)=$ constant
• B. $\rho \left(r\right)\propto \frac{1}{r^2}$
• C. $\rho \left(r\right)\propto \frac{1}{r}$
• D. $\rho \left(r\right)\propto 1$

Answer 1

The correct option is C $\rho \left(r\right)\propto \frac{1}{r}$

Considering a spherical gaussian surface and applying Gauss law,

$E\left(4\pi r^2\right)=\frac{{\int }_0^r\rho \left(4\pi r^2\right)dr}{{ϵ}_0}$

$E=\frac{{\int }_0^r\rho r^{2}dr}{{ϵ}_0{r}^2}$

It is given that $\frac{dV}{dr}$ is constant.

But $E=-\frac{dV}{dr}$

$⟹E$ is constant

$⟹\int \rho r^{2}dr\propto r^2$

$⟹\rho r^2\propto r$

$⟹\rho \propto \frac{1}{r}$