Question

The ratio of speed of an electron in the ground state in an atom to the velocity of light (c) is
$(h = \text{Planck's constant}, \epsilon_0 = \text{permittivity of free space})$

• A. \( \frac{2e^2\epsilon_0}{hc} \)
• B. \( \frac{2\epsilon_0 hc}{e^2} \)
• C. \( \frac{e^2}{2\epsilon_0 hc} \)
• D. \( \frac{e^3}{2\epsilon_0 hc} \)

Hint:

For such questions, always check the dimensional consistency and ensure the constants are properly used in the formula.

Answer 1

The Correct Option is C

Step 1: Understanding the formula.
The question involves the relationship between the speed of an electron and the velocity of light in the ground state of an atom, expressed using constants like Planck's constant, permittivity of free space, and the elementary charge.
Step 2: Analyzing the options.
(A) \( \frac{2e^2\epsilon_0}{hc} \): This does not match the required form.
(B) \( \frac{2\epsilon_0 hc}{e^2} \): Again, this is not the correct relationship.
(C) \( \frac{e^2}{2\epsilon_0 hc} \): This matches the correct relationship for the ratio of electron speed to light speed.
(D) \( \frac{e^3}{2\epsilon_0 hc} \): Incorrect as it introduces an extra factor of \( e \).
Step 3: Conclusion.
The correct answer is (C) \( \frac{e^2}{2\epsilon_0 hc} \) as it correctly represents the relationship.