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Question

A non-conducting spherical ball of radius R contains a spherically symmetric charge with volume charge density ρ=krn, where r is the distance from the center of the ball and n is a constant. The value of n such that the electric field inside the ball is directly proportional to square of distance from the centre is

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Solution



By Gauss law, we know that
E.dA=qinϵo=ρdVϵo
dV=4πr2dr for the shell element. The element is shown in the figure at a distance r from the centre O
E×4πr2=krn×4πr2drϵo=4πkϵorn+3(n+3)
E=k(n+3)ϵo(r(n+1))
For the electric field to be directly proportional to the square of distance from the centre,
n+1=2n=1

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