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

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$\frac{2 Q}{π (b^{2} - a^{2})}$

$\frac{2 Q}{π a^{2}}$

$\frac{Q}{2 π (b^{2} - a^{2})}$

$\frac{Q}{2 π a^{2}}$

answer is B.

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Detailed Solution

Given data: volume charge density $ρ = \frac{A}{r}$

Concept used: gauss law

Detailed solution:

Total charge enclosed in a sphere of radius r,

$Q_{i n} = Q + \int_{a}^{r} \frac{A}{r} 4 π r^{2} dr = Q + 2 π {Ar}^{2} - 2 π {Aa}^{2}$

by gauss law,

$E \times 4 π r^{2} = \frac{Q_{i n}}{ε_{0}}$

So the charge enclosed must be proportional to r².

$\Rightarrow Q - 2 π {Aa}^{2} = 0$

$A = \frac{Q}{2 π a^{2}}$

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