What is the ratio of the electric force to the gravitational force between a proton and an electron separated by \(5.3 \times 10^{-11} \mathrm{~m}\) (the radius of a hydrogen atom)?

We are given that the distance between the proton and electron (r) is \(5.3 \times 10^{-11} \mathrm{~m}\).

We will use Coulomb's law to calculate the electric force between the proton and electron. Coulomb's law is given by: $$F_e = k_e \frac{q_1q_2}{r^2}$$ Where, \(k_e\) is the electrostatic constant (\(8.99 \times 10^{9} \mathrm{Nm^2C^{-2}}\)), \(q_1\) and \(q_2\) are the charges of the proton and electron, each of which is \(1.602 \times 10^{-19} \mathrm{C}\), and \(-1.602 \times 10^{-19} \mathrm{C}\), respectively, \(r\) is the distance between particles (\(5.3 \times 10^{-11} \mathrm{m}\)). Now, substitute these values in the formula: $$F_e = (8.99 \times 10^9) \frac{(1.602 \times 10^{-19})(-1.602 \times 10^{-19})} {(5.3 \times 10^{-11})^2}$$ The negative sign in the electron charge indicates the opposite direction of the force, but since we are calculating the magnitude, we can ignore the negative sign. After calculating, we get: $$F_e \approx 8.20 \times 10^{-8} \mathrm{N}$$

We will use the universal law of gravitation to calculate the gravitational force between the proton and electron. The formula is: $$F_g = G \frac{m_1m_2}{r^2}$$ Where, \(G\) is the gravitational constant (\(6.674 \times 10^{-11} \mathrm{Nm^2kg^{-2}}\)), \(m_1\) and \(m_2\) are the masses of the proton and electron, which are \(1.673 \times 10^{-27} \mathrm{kg}\) and \(9.110 \times 10^{-31} \mathrm{kg}\), respectively, \(r\) is the distance between particles (\(5.3 \times 10^{-11} \mathrm{m}\)). Now, substitute these values in the formula: $$F_g = (6.674 \times 10^{-11}) \frac{(1.673 \times 10^{-27})(9.110 \times 10^{-31})} {(5.3 \times 10^{-11})^2}$$ After calculating, we get: $$F_g \approx 3.63 \times 10^{-47} \mathrm{N}$$

Now, we can find the required ratio by dividing the electric force (\(F_e\)) by the gravitational force (\(F_g\)): $$\frac{F_e}{F_g} = \frac{8.20 \times 10^{-8}}{3.63 \times 10^{-47}}$$ After calculating, we get: $$\frac{F_e}{F_g} \approx 2.26 \times 10^{39}$$ So, the ratio of the electric force to the gravitational force between a proton and an electron separated by \(5.3 \times 10^{-11} \mathrm{~m}\) (the radius of a hydrogen atom) is approximately \(2.26 \times 10^{39}\).