Question

The dispersion relation for electromagnetic waves travelling in a plasma is given as \( \omega^2 = c^2k^2 + \omega_p^2 \), where \( c \) and \( \omega_p \) are constants. In this plasma, the group velocity is:

• A. proportional to but not equal to the phase velocity
• B. inversely proportional to the phase velocity.
• C. equal to the phase velocity.
• D. a constant.

Hint:

For a plasma with a given dispersion relation, the group velocity is inversely related to the phase velocity.

Answer 1

The Correct Option is B

Step 1: Understanding the group velocity.
The group velocity \( v_g \) is given by \( v_g = \frac{d\omega}{dk} \). Using the given dispersion relation \( \omega^2 = c^2k^2 + \omega_p^2 \), we find \( \omega = \sqrt{c^2k^2 + \omega_p^2} \). Differentiating \( \omega \) with respect to \( k \), we get \( v_g = \frac{d}{dk} \left( \sqrt{c^2k^2 + \omega_p^2} \right) \). The result shows that the group velocity is inversely proportional to the phase velocity \( v_p = \frac{\omega}{k} \). Hence, the correct answer is option (B).