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Question

A solid nonconducting sphere of radius R has a uniform charge distribution of volume charge density, ρ=ρ0rR, where ρ0 is a constant and r is the distance from the centre of the sphere. Show that:
(a) the total charge on the sphere is Q=πρ0R3
(b) the electric field inside the sphere has a magnitude given by,
E=KQr2R4

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Solution

a) : Total charge , Q=R0ρdV=R0ρ0rR4πr2dr=4πρ0R[R4/4]=πρ0R3
b) By Gauss's law, the field inside sphere, E.4πr2=Qenϵ0
Here, Qen=r0ρdV=r0ρ0rR4πr2dr=4πρ0R[r4/4]=πρ0r4/R=Qr4/R4
So, E.4πr2=Qr4R4ϵ0
or E=KQr2R4 where K=14πϵ0

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