In hydrogen like system the ratio of coulombian force and gravitational force between an electron and a proton is in the order of:

The correct answer is 10³⁹.

In a hydrogen-like system, an electron and a proton interact via two fundamental forces: the Coulombian (electrostatic) force and the Gravitational force. We need to find the ratio of these two forces.

Step 1: Write the expressions for both forces

Let r be the separation between the electron and proton. Since both forces follow an inverse-square law with the same r, it cancels out in the ratio.

Coulombian Force:

F_C=\frac{k{e}^2}{r^2}

Gravitational Force:

F_G=\frac{G{m}_e{m}_p}{r^2}

Step 2: Write the ratio

\frac{F_C}{F_G}=\frac{k{e}^2}{G{m}_e{m}_p}

Notice that r² cancels out, so the ratio is independent of distance.

Step 3: Substitute known values

The constants needed are:

- Coulomb's constant: k=9\times {10}^9 N·m²/C²
- Charge of electron: e=1.6\times {10}^{-19} C
- Gravitational constant: G=6.67\times {10}^{-11} N·m²/kg²
- Mass of electron: m_e=9.11\times {10}^{-31} kg
- Mass of proton: m_p=1.67\times {10}^{-27} kg

Step 4: Calculate the numerator (Coulombian)

k{e}^2=9\times {10}^9\times {(1.6\times {10}^{-19})}^2

=9\times {10}^9\times 2.56\times {10}^{-38}

=23.04\times {10}^{-29}=2.304\times {10}^{-28}\text{N·m²}

Step 5: Calculate the denominator (Gravitational)

G{m}_e{m}_p=6.67\times {10}^{-11}\times 9.11\times {10}^{-31}\times 1.67\times {10}^{-27}

=6.67\times 9.11\times 1.67\times {10}^{-11-31-27}

=101.5\times {10}^{-69}\approx 1.015\times {10}^{-67}\text{N·m²}

Step 6: Compute the final ratio

\frac{F_C}{F_G}=\frac{2.304\times {10}^{-28}}{1.015\times {10}^{-67}}

=2.27\times {10}^{-28+67}=2.27\times {10}^{39}

Conclusion: The ratio of Coulombian force to gravitational force between an electron and a proton is approximately 2.27 × 10³⁹, which is of the order of 10³⁹.

This enormous ratio shows that the electrostatic (Coulombian) force is approximately 10³⁹ times stronger than the gravitational force between subatomic particles. This is why gravity is completely negligible in atomic and subatomic physics — electrostatic interactions completely dominate at these scales.