question_answer

Let \[P(r)=\frac{Q}{\pi {{R}^{4}}}r\] be the charge density distribution for a solid sphere of radius \[R\] and total charge Q. For a point 'p' inside the sphere at distance r, from the centre of the sphere, the magnitude of electric field is:

A) \[\frac{Q}{4\pi {{\in }_{0}}r_{1}^{2}  }\]

B) \[\frac{Qr_{1}^{2}}{4\pi {{\in }_{0}}{{R}^{4}}  }\]

C) \[\frac{Qr_{1}^{2}}{3\pi {{\in }_{0}}{{R}^{4}}  }\]

D) 0

Correct Answer: B

Solution :

Let us consider a spherical shell of thickness \[dx\]and radius\[x\]. The volume of this spherical shell\[4\pi {{x}^{2}}dx\]. The charge enclosed within shell

\[=\left[ \frac{Q.x}{\pi {{R}^{4}}} \right]\left[ 4\pi {{x}^{2}}dx \right]=\frac{4Q}{{{R}^{4}}}{{x}^{3}}dx\]

The charge enclosed in a sphere of radius \[{{r}_{1}}\] is

\[=\frac{4Q}{{{R}^{4}}}\int\limits_{0}^{{{r}_{1}}}{{{x}^{3}}dx=\frac{4Q}{{{R}^{4}}}\left[ \frac{{{x}^{4}}}{4} \right]_{0}^{{{r}_{1}}}=\frac{Q}{^{{{R}^{4}}}}}{{r}_{1}}^{4}\]

\[\therefore \]The electric field at point p inside the sphere at a distance\[{{r}_{1}}\], from the centre of the sphere is

\[E=\frac{1}{4\pi {{\in }_{0}}}\frac{\left[ \frac{Q}{{{R}^{4}}}{{r}_{1}}^{4} \right]}{{{r}_{1}}^{2}}\]

\[=\frac{1}{4\pi {{\in }_{0}}}\frac{Q}{{{R}^{4}}}{{r}_{1}}^{2}\]