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