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

Answer

A

Explanation

Given volumetric charge density

`rho(r)=A/r`
To find the charge enclosed in the region of a < x < b
`q_"in"=int_(x=a)^(x=r) rho(x)dV`
`q_"in"=int_a^r A/x 4pix^2dx`
`q_"in"=4piA[(r^2-a^2)/2]`
Now apply Gasses law at point P
`oint vecE. vec(dS)=q_"in"/epsilon_0 =1/epsilon_0` [Charge at centre + Volumetric charge]
`E.4pir^2=1/epsilon_0 [Q+4piA((r^2-a^2)/2)]`
`E=Q/(4piepsilon_0r^2)+A/(2epsilon_0)[1-a^2/r^2]`
for electric field to remain constant, we can write following :
`Q/(4piepsilon_0r^2)=A/(2epsilon_0)(a^2/r^2)`
`A=Q/(2pia^2)`