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>>Electric Charges and Fields
>>Applications of Gauss Law
>> $The region between two concentric sphere$

Question

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 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 :

472845

A

$\frac{Q}{2 π a ^{2}}$

B

$\frac{Q}{2 π ( b ^{2} - a ^{2} )}$

C

$\frac{2 Q}{π ( a ^{2} - b ^{2} )}$

D

$\frac{2 Q}{π a ^{2}}$

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JEE Mains

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Updated on : 2022-09-05

Solution

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Correct option is A)

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

$Q^{'} = Q + \int_{a}^{r} \frac{A}{r} 4 π r^{2} d r = Q + 2 π A r^{2} - 2 π A a^{2}$

By Gauss law,
$E \times 4 π r^{2} = Q - 2 π A a^{2} + 2 π A r^{2}$

Given, $E$ is independent of $r$
Hence, $Q - 2 π A a^{2} = 0$

This gives $A = \frac{Q}{2 π a ^{2}}$

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