A system consists of a uniformly charged sphere of radius R and a surrounding by a medium with volume charge density `rho =a/r`, where `alpha` is a positive constant and r is the distance from the centre of the sphere. If `NalphaR^(2)` is the charge on the sphere when the electric field intensity outside the sphere is independent of r. What is the value of N ?

Answer

`06.28`

Explanation

Using Gauss. theorem for spherical surface of radius r outside the sphere.
`ointE.ds = Q_("max")/epsilon_(0)`

Here, `Q_("enc") = q + q. = q + int_(R)^(r) rho (4pir^(2))dr`
`therefore E = (4pir^(2)) = 1/epsilon_(0) q + int_(R)^(r) rho (4pir^(2))dr`
`E(4pir^(2)) = 1/epsilon_(0)(q + int_(R)^(r) alpha/r (4pir^(2))dr)`
`=1/epsilon_(0) (q + 2pialphar^(2) - 2pi alphaR^(2))`
`therefore E(4pir^(2)) = (q-2pi alpha R^(2))/epsilon_(0) + (2pialpha r^(2))/epsilon_(0)`
For `q - 2pi alpha R^(2) =0`
`E = (2pi alpha r^(2))/(4pi r^(2) epsilon_(0)) = alpha/(2epsilon_(0))`
i.e., £ becomes independent of r.
`therefore` The intensity E does not depend on r if:
`q - 2pialpha R^(2) = 0`
`therefore q = 2pi alpha R^(2) = 6.28 alpha R^(2)`
`therefore N = 6.28`