A hydrogen atom is made up of a proton of charge +q in the nucleus and an electron of charge –q revolving around it. The proton may be regarded as a point charge at r=0, the centre of the atom.

We can imagine an atom as a point nucleus (+q) surrounded by a uniformly charged spherical cloud (-q) of radius r. Hence, volume = 4/3 ∏ r³

Charge / Unit volume = charge density ( ρ)

Hence, q / (4/3 ∏ r³) = ρ

Charge, q = ρ (4/3 ∏ r³)   --------------Eqn (1)

The electric flux, Φ=∫E.dS=∫EdS = E(4∏r²)

By Gauss law, Flux Φ=q/ε₀ 

Hence, E(4∏r²) = q/ε₀ 

E = 1/4∏r² * q / ε₀ 

Substitute, q from Eqn (1), we get,

The electric field of uniform charged sphere of radius r is

E( r ) = (1/ 4∏r² )(1/ε₀)  * ρ (4/3 ∏ r³) ṝ = (1/3 ε₀) ρ r ṝ