question_answer

A spherically symmetric charge distribution is characterized by a charge density having the following variations:

\[\rho (r)={{\rho }_{o}}\left( 1-\frac{r}{R} \right)\]for r < R\[\rho (r)=0\]for \[r\ge R\]

Where r is the distance from the centre of the charge distribution \[{{\rho }_{o}}\]is a constant. The electric field at an internal point (r < R) is:[JEE ONLINE 12-04-2014]

A) \[\frac{{{\rho }_{o}}}{4{{\varepsilon }_{0}}}\left( \frac{r}{3}-\frac{{{r}^{2}}}{4R} \right)\]

B) \[\frac{{{\rho }_{o}}}{{{\varepsilon }_{o}}}\left( \frac{r}{3}-\frac{{{r}^{2}}}{4R} \right)\]

C) \[\frac{{{\rho }_{o}}}{3{{\varepsilon }_{o}}}\left( \frac{r}{3}-\frac{{{r}^{2}}}{4R} \right)\]

D) \[\frac{{{\rho }_{o}}}{12{{\varepsilon }_{o}}}\left( \frac{r}{3}-\frac{{{r}^{2}}}{4R} \right)\]

Correct Answer: B

Solution :

[b] Let us consider a spherical shell of radius x and thickness dx.

Charge on this shell

\[dq=\rho .4\pi {{x}^{2}}dx={{\rho }_{0}}\left( 1-\frac{x}{R} \right).4\pi {{x}^{2}}dx\]

\[\therefore \]Total charge in the spherical region from centre to r (r < R) is

\[q=\int_{{}}^{{}}{dq=4\pi {{\rho }_{0}}}\int\limits_{0}^{r}{\left( 1-\frac{x}{R} \right){{x}^{2}}dx}\]

\[=4\pi {{\rho }_{0}}\left[ \frac{{{x}^{3}}}{3}-\frac{{{x}^{4}}}{4R} \right]_{0}^{r}\]\[=4\pi {{\rho }_{0}}\left[ \frac{{{r}^{3}}}{3}-\frac{{{r}^{4}}}{4R} \right]\]

\[=4\pi {{\rho }_{0}}{{r}^{3}}\left[ \frac{1}{3}-\frac{r}{4R} \right]\]

\[\therefore \]Electric field at \[E=\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{q}{{{r}^{2}}}\]

\[=\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{4\pi {{\rho }_{0}}{{r}^{3}}}{{{r}^{2}}}\left[ \frac{1}{3}-\frac{r}{4R} \right]\]

\[=\frac{{{\rho }_{0}}}{{{\varepsilon }_{0}}}\left[ \frac{r}{3}-\frac{{{r}^{2}}}{4R} \right]\]