question_answer

Within a spherical charge distribution of charge density\[\rho (r)\], 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 surface is constant for all values of \[{{V}_{0}}\] and \[\Delta V\] then:                                      [JEE ONLINE 10-04-2016]

A) \[\rho (r)\alpha r\]

B) \[\rho (r)\alpha \frac{1}{{{r}^{2}}}\]

C) \[\rho (r)\alpha \frac{1}{r}\]

D) \[\rho (r)\]= constant 

Correct Answer: C

Solution :

[c]

\[\frac{\Delta V}{\Delta r}\to \] constant

\[\Rightarrow \] uniform E. field.

\[(E)\,(4\pi {{r}^{2}})=\frac{1}{{{\varepsilon }_{0}}}\int{\rho dV}\]

\[(E)\,(4\pi {{r}^{2}})=\frac{1}{{{\varepsilon }_{0}}}\int\limits_{0}^{r}{\rho 4\pi {{r}^{2}}dr}\]

\[(E)\,(4\pi {{r}^{2}})=\frac{1}{{{\varepsilon }_{0}}}4\pi \int\limits_{0}^{r}{\rho {{r}^{2}}dr}\]

after integral on RHS

We must obtain \[{{r}^{2}}\]

\[\Rightarrow \,\rho \,\propto \frac{1}{r}\]