Step 1: Understanding the Concept:
We need to compare the electrostatic force and gravitational force between two protons separated by a distance \( r \). We use Coulomb's Law and Newton's Law of Gravitation.
Step 2: Key Formula or Approach:
1. Electrostatic Force: \( F_e = \frac{k e^2}{r^2} \), where \( k = 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \).
2. Gravitational Force: \( F_g = \frac{G m_p^2}{r^2} \), where \( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \).
3. Ratio: \( \frac{F_e}{F_g} = \frac{k e^2}{G m_p^2} \).
Step 3: Detailed Explanation:
Given values:
- Charge of proton \( e = 1.6 \times 10^{-19} \, \text{C} \)
- Mass of proton \( m_p = 1.67 \times 10^{-27} \, \text{kg} \)
Calculate the ratio:
\[ \frac{F_e}{F_g} = \frac{(9 \times 10^9) (1.6 \times 10^{-19})^2}{(6.67 \times 10^{-11}) (1.67 \times 10^{-27})^2} \]
\[ = \frac{9 \times 2.56 \times 10^9 \times 10^{-38}}{6.67 \times 2.79 \times 10^{-11} \times 10^{-54}} \]
\[ = \frac{23.04 \times 10^{-29}}{18.6 \times 10^{-65}} \]
\[ \approx 1.24 \times 10^{-29 - (-65)} \]
\[ \approx 1.24 \times 10^{36} \]
The order of magnitude is \( 10^{36} \).
So, \( n \approx 36 \).
Step 4: Final Answer:
The value of \( n \) is 36.