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. (a) `(2Q)/(pi(a^2-b^2))`
B. (b) `(2Q)/(pia^2)`
C. (c) `(Q)/(2pia^2)`
D. (d) `(Q)/(2pi(b^2-a^2))`

Answer

C

Explanation

Applying Gauss's law
`ointsvecE.vecds=(Q)/(in_0)`
`:.Exx4pir^2=(Q+4piar^2-4piAa^2)/(in_0)`
`rho=(dr)/(dv)`
`Q=rho4pir^2`
`Q=int_a^*A/r4pir^2dr=4piA[r^2-a^2]`
`E=(1)/(4pi in_0)[(Q-4piAa^2)/(r^2)+4piA]`
For E to be independent of `r`
`Q-2piAa^2=0`
`:.A=(Q)/(2pia^2)`