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The volume charge density of a sphere ca...

The volume charge density of a sphere carrying charge Q and of radius R is proportional to the square of the distance from the centre. The ratio of the magnitude of the electric field at a distance 2R from the centre to that at a distance of `(R )/(2)` from the centre is

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The correct Answer is:
2

given `rho = Cr^(2)` [where `rho` = volume charge density]
`q(r ) = int_(0)^(r ) (4pi r^(2) dr ) rho`
`= int_(0)^(r ) 4pi r^(2) * Cr^(2) dr =(4)/(3) pi Cr^(5)`
`E"|"_(r=2R) =(kq_((2R)))/((2R)^(2))=(k((4)/(5))piCR^(6))/(4R^(2))`
...[ `:.` sphere has radius R, so r `le` R for enclosed charge]
`=(kpi CR^(3))/(5)`
`E"|"_(r=R//2) =(Kq_((R//2)))/(((R )/(2))^(2)) =(k((4)/(5))piC ((R )/(2))^(5))/(((R )/(2))^(2))=(kpiCR^(3))/(5 xx2)`
`rArr (E_((2R)))/(E_((R//2)))=2`
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