Question

Which one of the following is correct for the phase velocity \( v_p \) and group velocity \( v_g \)? (c is the speed of light in vacuum)

• A. For matter waves in the relativistic case, \( v_p v_g>\frac{c^2}{2} \)
• B. For electromagnetic waves in a medium, \( v_p \) represents the speed with which energy propagates
• C. For electromagnetic waves in a medium, both \( v_p \) and \( v_g \) can be more than \( c \)
• D. For matter waves in free space, \( v_p \neq v_g \)

Hint:

For matter waves in free space, remember that the phase velocity \( v_p \) and group velocity \( v_g \) typically differ, especially in relativistic cases.

Answer 1

The Correct Option is D

Let’s analyze the question step-by-step to arrive at the correct answer.

Step 1: Definitions of phase velocity and group velocity.
The phase velocity (\( v_p \)) is the velocity at which the phase of the wave, such as the wave crest, propagates.
The group velocity (\( v_g \)) is the velocity at which the energy or information carried by the wave propagates.

Step 2: Analyzing the correct option.
Option (D) is correct: For matter waves in free space, the phase velocity \( v_p \) is not equal to the group velocity \( v_g \) in most cases. This is because the two velocities describe different properties of the wave. In free space, the phase velocity and group velocity can differ, especially for matter waves, where relativistic effects become significant.

Step 3: Analyzing other options.
Option (A): For matter waves in the relativistic case, \( v_p v_g > \frac{c^2}{2} \): This is incorrect because the product of the phase and group velocities does not necessarily exceed \( \frac{c^2}{2} \) in the relativistic case.
Option (B): For electromagnetic waves in a medium, \( v_p \) represents the speed with which energy propagates: This is incorrect because it is the group velocity \( v_g \) that represents the speed at which energy propagates, not the phase velocity \( v_p \).
Option (C): For electromagnetic waves in a medium, both \( v_p \) and \( v_g \) can be more than \( c \): This is incorrect because neither the phase velocity nor the group velocity can exceed the speed of light \( c \) in a vacuum. In a medium, both are typically less than or equal to \( c \).

Step 4: Conclusion.
The correct answer is option (D), because for matter waves in free space, the phase velocity and group velocity are typically not equal.