Question

The ratio of electric force and gravitational force acting between two charges is in the order of

• A. 10\(^{42}\)
• B. 10\(^{39}\)
• C. 10\(^{36}\)
• D. 1

Hint:

This is a classic comparison. Note that for two electrons, the ratio is \(\sim 10^{42}\), and for a proton-electron pair, it's \(\sim 10^{39}\). The question is often asked in the context of protons, making \(10^{36}\) a common answer to remember.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This question compares the relative strengths of two fundamental forces: the electrostatic force and the gravitational force. Since the question is general and does not specify the particles, we can calculate the ratio for a common case, such as two protons.
Step 2: Key Formula or Approach:
Let's consider two protons separated by a distance \(r\).
The electric force (\(F_e\)) between them is given by Coulomb's Law: \[ F_e = k \frac{e^2}{r^2} \] The gravitational force (\(F_g\)) between them is given by Newton's Law of Gravitation: \[ F_g = G \frac{m_p^2}{r^2} \] The ratio is: \[ \frac{F_e}{F_g} = \frac{k e^2 / r^2}{G m_p^2 / r^2} = \frac{k e^2}{G m_p^2} \] Step 3: Detailed Explanation:
We use the standard values for the constants:
- Coulomb's constant, \(k \approx 9 \times 10^9\) N\(\cdot\)m\(^2\)/C\(^2\).
- Elementary charge, \(e \approx 1.6 \times 10^{-19}\) C.
- Gravitational constant, \(G \approx 6.67 \times 10^{-11}\) N\(\cdot\)m\(^2\)/kg\(^2\).
- Mass of a proton, \(m_p \approx 1.67 \times 10^{-27}\) kg.
Now, we calculate the ratio: \[ \frac{F_e}{F_g} = \frac{(9 \times 10^9) \times (1.6 \times 10^{-19})^2}{(6.67 \times 10^{-11}) \times (1.67 \times 10^{-27})^2} \] \[ \frac{F_e}{F_g} = \frac{9 \times 2.56 \times 10^{9-38}}{6.67 \times 2.79 \times 10^{-11-54}} = \frac{23.04 \times 10^{-29}}{18.6 \times 10^{-65}} \] \[ \frac{F_e}{F_g} \approx 1.24 \times 10^{36} \] The ratio is of the order of \(10^{36}\).
Step 4: Final Answer:
The ratio of the electric force to the gravitational force between two protons is on the order of \(10^{36}\). This demonstrates that the electric force is immensely stronger than the gravitational force at the subatomic level. Option (C) matches this result.