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

Answer 1

Answer: (A)

Let us find total charge enclosed in a sphere of radius $r$,
$$
Q ^{\prime}= Q +\int_{ a }^{ r } \frac{ A }{ r } 4 \pi r ^{2} dr = Q +2 \pi Ar ^{2}-2 \pi Aa ^{2}
$$
By Gauss law,
$E \times 4 \pi r ^{2}= Q -2 \pi Aa ^{2}+2 \pi Ar ^{2}$
Given, $E$ is independent of $r$
Hence, $Q -2 \pi Aa ^{2}=0$
This gives $A=\frac{Q}{2 \pi a^{2}}$