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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 centre of the ball and n is a constant. What should be n such that the electric field inside the ball is directly proportional to the square of distance from the centre?

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Solution


By Gauss's law,

E.dA=qenclosedϵ0=ρdVϵ0 {ρ=chargevolume}

E×4πr2=r0krn×4πr2drϵ0

E×4πr2=4πkϵ0r(n+3)(n+3)

E=kϵ0r(n+1)(n+3)

Given that, Er2

On comparing,

n+1=2

n=1

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