Ratio of Electrostatic Force and Gravitational Force

Electrostatic force and gravitational force are the two fundamental forces of nature. The electrostatic force exists between two charged particles whereas the gravitational force exists between any two objects with mass. The ratio of the magnitude of these two forces for an electron and proton can be calculated as follows:

Formula:

Electrostatic force between two charged particles = kq₁q₂/r²

Gravitational force between two objects = Gm₁m₂/r²

where k is Coulomb's constant, G is the gravitational constant, q₁ and q₂ are the charges on the particles, m₁ and m₂ are the masses of the objects, and r is the distance between them.

Calculation:

In the case of an electron and proton, the charges are opposite in sign and equal in magnitude. Therefore, the electrostatic force between them can be calculated as:

F₁ = kq₁q₂/r² = k(-e)(+e)/r²

where e is the charge on an electron and has a value of -1.6 x 10⁻¹⁹ C.

Similarly, the gravitational force between them can be calculated as:

F₂ = Gm₁m₂/r² = G(me)(mp)/r²

where me is the mass of an electron and has a value of 9.1 x 10⁻³¹ kg, and mp is the mass of a proton and has a value of 1.67 x 10⁻²⁷ kg.

Therefore, the ratio of the magnitude of the electrostatic force to the gravitational force can be calculated as:

F₁/F₂ = [k(-e)(+e)]/[G(me)(mp)] = -1/(4πε₀)(e²/me)(mp/G)

where ε₀ is the permittivity of free space and has a value of 8.85 x 10⁻¹² F/m.

Substituting the values, we get:

F₁/F₂ = 2.4 x 10³⁹

Therefore, the correct option is B, i.e., 2.4 x 10³⁹.