The region between two concentric spheres of radii 'a' and 'b', respectively (see figure), have volume charge density `rho=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:

Options

A. `(Q)/(2 pi a^(2))`
B. `(Q)/(2pi (b^(2) - a^(2)))`
C. `(2Q)/(pi (a^(2) - b^(2)))`
D. `(2Q)/(pi a^(2))`

Answer

A

Explanation

Let us consider a sphere of radius `r` and thickness `dr` lying in the region between two concentric spheres.
According to Gauss's theroem, charge inside alone contributes to electric `field//flux`
`:. (kQ)/(a^(2)) = (k [ Q + int_(a)^(b) 4pi r^(2) dr (A)/(r )])/(b^(2))`
or `Q (b^(2))/(a^(2)) = Q +4pi A [(r^(2))/(2)]_(-a)^(b)`
`= Q + 4pi A ((b^(2) - a^(2)))/(2)`
`Q((b^(2))/(a^(2)) - 1) = 2pi A (b^(2) - a^(2))`
`(Q(b^(2) - a^(2)))/(a^(2)) = 2pi A (b^(2) - a^(2))`
`:. A = (Q)/(2pi a^(2))`