question_answer

The region between two concentric spheres of radii 'a' and 'b', respectively (see figure), has volume charge density \[\rho =\frac{A}{r},\]where A is a constant and r is the distance from the centre. At the centre of the spheres is a point charge Q. The value of A such that the electric field in the region between the spheres will be constant, is :-

[JEE MAIN - I 3-4-2016]

A) \[\frac{2Q}{\pi {{a}^{2}}}\]    

B) \[\frac{Q}{2\pi {{a}^{2}}}\]

C) \[\frac{Q}{2\pi ({{b}^{2}}-{{a}^{2}})}\]

D) \[\frac{2Q}{\pi ({{a}^{2}}-{{b}^{2}})}\]

Correct Answer: B

Solution :

[b] Gaussian surface at distance r from center

\[\frac{Q+\int\limits_{a}^{r}{\frac{A}{r}4\pi {{r}^{2}}dr}}{{{\in }_{0}}}=E4\pi {{r}^{2}}\]

\[E=\frac{Q+2\pi A{{r}^{2}}-2\pi A{{a}^{2}}}{4\pi {{r}^{2}}{{\in }_{0}}}\]

make E independent of r then

\[Q-2\pi {{a}^{2}}A=0\Rightarrow A=\frac{Q}{2\pi {{r}^{2}}}\]