A system consists of a uniformly charged sphere of radius R and a surrounding medium filled by a charge with the volume density `rho=alpha/r` , where `alpha` is a positive constant and r is the distance from the centre of the sphere. Find the charge of the sphere for which the electric field intensity E outside the sphere is independent of R.

Options

A. `alpha/(2epsilon_0)`
B. `2/(alphaepsilon_0)`
C. `2pialphaR^2`
D. None of these

Answer

C

Explanation

Using Gauss theorem for spherical surface of radius r outside the sphere with the charge q
`int_0^r Eds=1/epsilon_0 int_0^r rho(r)dV`
`rArr E4pir^2 =1/epsilon_0 (q+int_R^r alpha/r (4pir^2)dr)`
`rArr E4pir^2 = ((q-2pialphaR^2))/epsilon_0+(4pialphar^2)/(2epsilon_0)`
The intensity E does not depend on R if
`(q-2pialphaR^2)/epsilon_0=0` or `q=2pialphaR^2`