Question

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 :

• A. $\frac{Q}{2\pi a^2}$
• B. $\frac{Q}{2\pi (b^2-a^2)}$
• C. $\frac{2Q}{\pi (a^2-b^2)}$
• D. $\frac{2Q}{\pi a^2}$

Answer 1

Answer: (A)

Let us find total charge enclosed in a sphere of radius r,

$Q^{\prime }=Q+{\int }_a^r\frac{A}{r}4\pi r^{2}dr=Q+2\pi A{r}^2-2\pi A{a}^2$

By Gauss law,

$E\times 4\pi r^2=Q-2\pi A{a}^2+2\pi A{r}^2$

Given, $E$ is independent of $r$

Hence, $Q-2\pi A{a}^2=0$

This gives $A=\frac{Q}{2\pi a^2}$