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Question

A region in space has a total charge Q distributed spherically such that the volume charge density ρ(r) is given by
ρ(r)=3ar(2R) for rR2
ρ(r)=a[1(rR)2] for R2rR
ρ(r)=0 for r>R
Find the electric field at ρ(R2<x<R)
151260.png

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Solution

Charge on hypothetical sphere of radius x is given by

Qenclosed=R/203αr2R(4πr2dr)+xR/2α[1(rR)2]4πr2dr
Qenclosed=3α2R(4π)R/20r3dr+4παR/2x[1r2R2]r2dr
=3α2R(4π)×(R2)444παR33×8+4παx33+4παx55R24παR332×5
=3απR3324παR324+4παx33+4παx55R24παR3160
=απ(3R332R36+4x334x55R2140R3)
E.ds=QenclosedE(4πx2=απ(3R332R36+4x334x55R2R340)
E=[3αR3128x2+αR324x2+αx3+αx35R2R3160πx2]

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