Question

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.

Answer 1

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

Where,

∈₀ = Permittivity of free space

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

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

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

m_p = 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:

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

Where,

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:

∴F_G = F_C

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

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.