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

By the law of gravition ,` (m_e v^2 )/(r ) = (Gm _e m_p ) /(r^2)`
`i.e., m_e v^2 r= gm_e m_p `………(1)
By bohr.s condition `m_e vr =(nh )/( 2 pi)`
`m_(e)^(2) V^2 r^2 = (n^2 h^2)/( 4 pi ^2) `………(2)
for ` 1^(st)` orbit ,`n=1 , h=6.626 xx 10^(-34) Js`
` g= 6.67 xx 10^(-11) Nm ^2 // kg ^2`
` m_e = 9xx 10^(-11) Kg , m_p = 1.67 xx 10^(-27 ) kg`
Eq (2) `div ` (1) ` implies m_e r = (n^2 h^2)/( 4 pi ^2 Gm_e m_p)`
` r=(n^2 h^2)/( 4 pi ^2 Gm_e^(2) m_p)`
Substiting these values we get `r= 1.21 xx 10^(29)` m
this value of .r. is much greater than the size of the universe .