The Ratio of fc/fg in the 1st Bohr Orbit of Hydrogen Atom

Introduction:
The Bohr model of the hydrogen atom is a simplified representation that describes the behavior of an electron in the atom's orbit. According to this model, the electron orbits the nucleus in discrete energy levels. The first Bohr orbit is the lowest energy level, also known as the ground state.

Formula for the Ratio:
The ratio of fc/fg, where fc represents the centripetal force and fg represents the gravitational force, can be calculated using the following formula:

fc/fg = (mv²/r) / (GmM/r²)

where:
m = mass of the electron
v = velocity of the electron
r = radius of the orbit
G = gravitational constant
M = mass of the nucleus (proton)

Derivation:
To find the ratio of fc/fg, we need to determine the expressions for both forces.

Centripetal Force (fc):
The centripetal force is the force required to keep an object moving in a circular path. In the case of the electron in the first Bohr orbit, the centripetal force is provided by the electrostatic force of attraction between the electron and the nucleus.

The electrostatic force between two charged particles can be calculated using Coulomb's law:

Fe = (1/4πε₀)(e²/r²)

where:
Fe = electrostatic force
ε₀ = permittivity of free space
e = charge of the electron
r = radius of the orbit

Since the centripetal force is equal to the electrostatic force in this case, we have:

fc = Fe = (1/4πε₀)(e²/r²)

Gravitational Force (fg):
The gravitational force between the electron and the nucleus can be calculated using Newton's law of gravitation:

Fg = (GmM)/r²

where:
Fg = gravitational force
G = gravitational constant
m = mass of the electron
M = mass of the nucleus (proton)
r = radius of the orbit

Substituting the Values:
To find the ratio fc/fg, we substitute the expressions for fc and fg into the formula:

fc/fg = [(1/4πε₀)(e²/r²)] / [(GmM)/r²]

Simplifying this expression, we can cancel out the common terms of r²:

fc/fg = (1/4πε₀)(e²) / (GmM)

Conclusion:
The ratio of fc/fg in the first Bohr orbit of the hydrogen atom can be calculated using the formula: fc/fg = (1/4πε₀)(e²) / (GmM). This ratio represents the balance between the electrostatic force of attraction (centripetal force) and the gravitational force between the electron and the nucleus. The values of ε₀, e, G, and M are constants, while the radius of the orbit (r) and the mass of the electron (m) are specific to the first Bohr orbit.