Question

The ratio of the electric force to the gravitational force between an electron and a proton separated by a distance r is approximately

• A. $10^{39}$
• B. $10^{42}$
• C. $10^{36}$
• D. $10^{31}$
• E. $10^{27}$

Hint:

This ratio highlights how incredibly weak gravity is compared to electromagnetism at the atomic scale. Gravity only becomes dominant for massive, electrically neutral objects like planets and stars.

Answer 1

The Correct Option is A

Concept:
Physics - Comparison of Fundamental Forces.
Step 1: State the formulas for both forces.

• Electric Force ($F_e$): $F_e = \frac{k |q_1 q_2|}{r^2}$
• Gravitational Force ($F_g$): $F_g = \frac{G m_1 m_2}{r^2}$

Step 2: Set up the ratio.
Since both forces follow the inverse square law ($1/r^2$), the distance $r$ cancels out: $$ \frac{F_e}{F_g} = \frac{k |q_e q_p|}{G m_e m_p} $$
Step 3: Substitute standard constants.

• $k \approx 9 \times 10^9$ Nm$^2$C$^{-2}$
• $G \approx 6.67 \times 10^{-11}$ Nm$^2$kg$^{-2}$
• $q_e = q_p \approx 1.6 \times 10^{-19}$ C
• $m_e \approx 9.1 \times 10^{-31}$ kg
• $m_p \approx 1.67 \times 10^{-27}$ kg

Step 4: Estimate the order of magnitude.
$$ \text{Ratio} \approx \frac{10^{10} \times (10^{-19})^2}{10^{-11} \times 10^{-31} \times 10^{-27}} \approx \frac{10^{-28}}{10^{-69}} = 10^{41} $$ While precise calculations for a proton-electron pair yield approximately $2.27 \times 10^{39}$, the closest option representing this massive scale of difference is $10^{39}$.