Question

The region between two concentric spheres of radii ${}^{\prime }{a}^{\prime }$ and ${}^{\prime }{b}^{\prime }$, 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. $\frac{2Q}{\pi a^2}$

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 Ar ^{2}-2 \pi Aa ^{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 $A=\frac{Q}{2\pi a^2}$