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

`v=(me^(4))/((4pi)^(3)epsilon_(0)^(2)(h//2pi)^(3))[(1)/((n-1)^(2))-(1)/(n^(2))]=(me^(4)(2n-1))/((4pi)^(3)epsilon_(0)^(2)(h//2pi)^(3)n^(3))`
Orbital frequency `v_(c)=(v//2pir)." In Bohr model v"=(n(h//2pi))/(mr), and `r=(4pi epsilon_(0)(h//2pi)^(2))/(me^(2))n^(2)`. This gives `v_(c)=(n(h//2pi))/(2pinv^(2))=(me^(4))/(32pi^(3)epsilon_(0)^(2)(h//2
pi)^(3)n^(3))`
which is same as v for large n.