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An early model for an atom considered it to have a positively charged point nucleus of charge Ze, surrounded by a uniform density of negative charge upto a radius R. The atom as a whole is neutral. The electric field at a distance r from the nucleus is (r<R).
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A
Ze4πϵ0[1r2rR3]
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B
Ze4πϵ0[1r2rR2]
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C
Ze4πϵ0[1R3rr2]
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D
Ze4πϵ0[1R3+rr2]
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Solution

The correct option is A Ze4πϵ0[1r2rR3]
Charge on nucleus is =+Ze
total negative charge =Ze( atoms is electrical neutral)
Negative charge density, ρ=chargevolume=Ze43πR3
i.e., ρ=34ZeπR3...(i)
Consider a Gaussian surface with radius r
By Gauss's theorem
ϕ=E(r)×4πr2=qε0....(ii)
Charge enclosed by Gaussian surface
q=Ze+4πr33ρ=ZeZer3R3 [Using (i)]
From (ii)
E(r)=q4πε0r2=ZeZer3R34πε0r2=Ze4πε0(1r2rR3)

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