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. 

Radius of the first Bohr orbit is given by the relation,

$r_{1}=\frac{4 \pi \epsilon_{0}\left(\frac{h}{2 \pi}\right)^{2}}{m_{e} e^{2}}\dots(i)$

Where,

$\epsilon_{0}=$ Permittivity of free space

$h=$ Planck's constant $=6.63 \times 10^{-34}\, Js$

$m_{e}=$ Mass of an electron $=9.1 \times 10^{-31} \,kg$

$e=$ Charge of an electron $=1.9 \times 10^{-19}\, C$

$m_{p}=$ Mass of a proton $=1.67 \times 10^{-27} \,kg$

$r=$ Distance between the electron and the proton

Coulomb attraction between an electron and a proton is given as

$F_{c}=\frac{e^{2}}{4 \pi \epsilon_{0} r^{2}}\dots(ii)$

Gravitational force of attraction between an electron and a proton is given as

$F_{G}=\frac{G m_{p} m_{e}}{r^{2}}\dots(iii)$

Where,

$G=$ Gravitational constant $=6.67 \times 10^{-11} \,Nm ^{2} / kg ^{2}$

The electrostatic (Coulomb) force and the gravitational force between an electron ahd a proton are equal, then we can write

$\therefore F_{G}=F_{C}$

$\frac{G m_{p} m_{e}}{r^{2}}=\frac{e^{2}}{4 \pi \epsilon_{0} r^{2}}$

$\therefore \frac{e^{2}}{4 \pi \epsilon_{0}}=G m_{p} m_{e}\dots(iv)$

Putting the value of equation $(iv)$ in equation $(i),$ we get

$r_{1}=\frac{\left(\frac{h}{2 \pi}\right)^{2}}{G m_{p} m_{e}^{2}}$

$=\frac{\left(\frac{6.63 \times 10^{-34}}{2 \times 3.14}\right)^{2}}{6.67 \times 10^{-11} \times 1.67 \times 10^{-27} \times\left(9.1 \times 10^{-31}\right)^{2}}$$\approx 1.21 \times 10^{29} \,m$

It is known that the universe is $156$ billion light years wide or $1.5 \times 10^{27} \,m$ wide. Hence, we can conclude that the radius of the first Bohr orbit is much greater than the estimated size of the whole universe.