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Question

A solid sphere of radius R has a charge Q distributed in its volume with a charge density ρ=Kra, where K and a are constants and r is the radial distance from its centre. If the electric field at r=R2 is 18 times that at r=R, the value of a is (answer upto two decimal places)


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Solution

Given ρ=Kra

E(r=r2)=18E(r=R)
qenclosedR/24πϵ0(R/2)2=18Q4πϵ0(R)2
32qenclosedR/2=Q ...(1)

https://s3-us-west-2.amazonaws.com/infinitestudent-images/ckeditor_assets/pictures/454902/original_Capture.PNGqenclosed=ρdv=Kra4πr2dr
qenclosedR/2=R/20Kra4πr2dr=4Kπ(a+3)(R2)(a+3)
Q=qenclosedR=R0Kra4πr2dr=4Kπ(a+3)(R)(a+3)
QqenclosedR/2=2a+3
putting in equation (1),
2(a+3)=32
a=2


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