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The region between two concentric spheres of radii $a$ and $b (> a)$ contains volume charge density $ρ (r) = \frac{C}{r}$ , where $C$ is a constant and $r$ is the radial distance as shown in figure. A point charge $q$ is placed at the origin, $r = 0$ .
Find the value of $C$ for which the electric field in the region between the spheres is constant (i.e. $r$ independent).

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$C = \frac{q}{2 π a^{2}}$

$C = \frac{q}{3 π a^{2}}$

$C = \frac{q}{7 π a^{2}}$

$C = \frac{q}{8 π a^{2}}$

answer is A.

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Detailed Solution

Net charge between $r = a$ to $r = r$ would be

$Q = \int_{a}^{r} ρ (4 π r^{2}) d r = \int_{a}^{r} \frac{C}{r} (4 π r^{2}) d r Q = 2 π C (r^{2} - a^{2}) E_{r} = \frac{k (Q + q)}{r^{2}} = \frac{k [2 π C (r^{2} - a^{2}) + q]}{r^{2}}$

From this expression, we can see that it we put

$C = \frac{q}{2 π a^{2}} \Rightarrow E_{r} = \frac{k q}{a^{2}} = constant$

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