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

Radius of first orbit hydrogen atom in Bohr model.
`r=(n^(2)h^(2) epsi_(0))/(pi me_(e)e^(2))`
Taking n=1 for first orbit,
`r=(h^(2)epsi_(0))/(pi m_(e)e^(2))xx(4pi)/(4pi)`
`=(h^(2)xx4pi epsi_(0))/(4pi^(2)m_(e)e^(2))`
`:. r=(h^(2))/(4pi m_(e)ke^(2)) ....(1) [ :. 4pi epsi_(0)=(1)/(k)]`
Now coulomb force between proton and electron
`F_(e)=(ke^(2))/(r^(2)) " "....(2)`
Now coulomb force between proton and electron
`F_(G)=(Gm_(p)m_(e))/(r^(2)).....(3)`
From equation (2) and (3),
`ke^(2)=Gm_(p)m_(e)`
`r_(G)=(h^(2))/(4pi^(2)m_(e)xxGm_(p)m_(e))`
`:.r_(G)=((6.625xx10^(-34))^(2))/(4xx(3.14)^(2)xx6.67xx10^(-11)xx(1.67xx10^(-27)xx(9.1xx10^(-31))^(2))`
`:.r_(G)=(43.890625)/(36378.46)xx10^(32)`
`=0.01206xx10^(32)`
`=1.21xx10^(29)m`