A system consits of a ball of radus `R` carrying spherically symmetric charge and the surrounding space filled with a charge of volume density `rho = alpha//r` , wehre `alpha` is a constant, `r` is the distance from the centre of the ball. Find the ball's the charge at whcih the magnitude of the electric field strength vector is independent of `r` outside the ball. How high is this strength ? The permittives of the ball and the surrounding space are assumed to be equal to unity.

Explanation

Let the charge carried by the sphere be `q`, then using Gauss's therorem for a spherical surface having radius `r gt gt R`, we can write.
`E 4pi r^(2) = (q_("inclosed"))/(epsilon_(0)) = (q)/(epsilon_(0)) + (1)/(epsilon_(0)) int_(R )^(r ) (alpha)/(r ) 4pi r^(2) dr`.
On integrating we get,
`E 4pi r^(2) = ((q-2pi alpha R^(2)))/(epsilon_(0))+(4pi alpha r^(2))/(2epsilon_(0))`
This intensity E does not depend on r when the expression in the parentheses is equal to zero. Hence
`q = 2pi alpha R^(2)` and `E = (alpha)/(2epsilon_(0))`