The gravitational attraction between electron and proton in a hydrogen atom is weaker than the coulomb attraction by a factor of about `10^(-40)`. An alternative way of looking at this fact is to estimate the radius of the first Bohr orbit of a hydrogen atom if the electron and proton were bound by gravitational attraction. You will find the answer interesting.

Explanation

If electron and proton were bounded by gravitational attraction , then force of attraction between them is given by
`F=G(m_em_p)/r^2`
where `m_e`= mass of electron
`m_p`=mass of proton
`r_0`=radius of first Bohr orbit
This force provides necessary centripetal force for the electrons to revolve around the nucleus
`therefore F=(m_ev^2)/r`
where v is the speed of electron in the orbit
`rArr (m_ev^2)/r =(Gm_em_p)/r^2`
`rArr m_ev^2r=Gm_em_p`....(i)
From Bohr.s condition
`m_evr=(nh)/(2pi)`
Squaring both sides , we get
`m_e^2 v^2 r^2 =(n^2 h^2)/(4pi^2)` ....(ii)
Dividing Eq.(ii) by (i) we get
`m_er =(n^2h^2)/(4pi^2Gm_em_p)`
`r=(n^2h^2)/(4pi^2Gm_e^2m_p)`
For `1^(st)` orbit , n=1
So
`r_0=(1xx(6.62xx10^(-34))^2)/(4xx9.86xx6.67xx10^(-11)xx(9.1xx10^(-31))^2xx1.67xx10^(-27))`
`rArr r_0=1.2xx10^29` m
This is much greater than the estimated size of the whole universe .