The volume charge density in a spherical ball of radius R varies with distence r from the centre as `p(r)=p_(0)[1-((r)/(R))^(3)]`, where,`p_(0)` is a constant. The radius at which the field would be maximum is

Options

A. `(R)/(2^(1//3))`
B. R
C. `(R)/(2)`
D. `(R^(1//3))/(2)`

Answer

A

Explanation

Given, `rho(r)=rho_(0)[1-((r)/(R))^(3)]`
So, effective charge at r is q ( r ) `=intrho` ( r ) .V
`= rho_(0)[1-((r)/(R))^(3)](4)/(3)pir^(3)`
For maximum value of field,
`(dq(r))/(dr)=0`
`rArr (4)/(3)pirho_(0)[3r^(2)(1-(r^(3))/(R^(3)))+r^(3)(-(3r^(2))/(R^(3)))]=0`
`rArr 3r^(2)-(3r^(5))/(R^(3))-(3r^(5))/(R^(3)) = 0`
`rArr r^(3)=(R^(3))/(2) rArr r = (R)/(2^(1/3))`