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).

The correct option is A
Ze4πϵ₀[1/r² - r/R³]
Charge on nucleus is = +Ze
total negative charge = -Ze (∵ atoms is electrical neutral)
Negative charge density, ρ = charge/volume = -Ze/(4/3πR³)
i.e., ρ = -(3Ze)/(4πR³).. (i)
Consider a Gaussian surface with radius r
By Gauss's theorem
∮E.dS = E(r) × 4πr² = q/ϵ₀ (ii)
Charge enclosed by Gaussian surface
q' = Ze + (4πr³/3)ρ = Ze - Zer³/R³ [Using (i)]
From (ii)
E(r) = q'/4πϵ₀r² = Ze - Zer³/R³ / 4πϵ₀r² = Ze/4πϵ₀(1/r² - r/R³)