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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. - Physics

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.

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

`r_1 = 4pi in_0 (h/(2pi))^2/(m_e e^2)`   ....(1)


0 = Permittivity of free space

h = Planck’s constant = 6.63 × 10−34 Js

me = Mass of an electron = 9.1 × 10−31 kg

e = Charge of an electron = 1.9 × 10−19 C

mp = Mass of a proton = 1.67 × 10−27 kg

r = Distance between the electron and the proton

Coulomb attraction between an electron and a proton is given as:

`F_C = e^2/(4piin_0 r^2)`   .....(2)

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

`F_G = (Gm_p m_e)/r^2`  ....(3)


G = Gravitational constant = 6.67 × 10−11 N m2/kg2

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


`(Gm_p m_e)/r^2 = e^2/(4piin_0 r^2)`

`:. e^2/(4piin_0) = Gm_p m_e`   ...(4)

Putting the value of equation (4) in equation (1), we get:

`r_1 = (h/(2pi))^2/(Gm_p m_e^2)`

`= (((6.63 xx 10^(-34))/(2xx3.14))^2)/(6.67 xx 10^(-11) xx 1.67 xx 10^(-27) xx (9.1 xx 10^(-31))^2) ~~ 1.21 xx 10^(29) m`

It is known that the universe is 156 billion light years wide or 1.5 × 1027 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.

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NCERT Class 12 Physics Textbook
Chapter 12 Atoms
Q 12 | Page 436
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