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Question #41999

Difficulty: 8/10 | 🟢 Free

The Question

In a non-linear dispersive quantum medium, the energy density $u$ (energy per unit volume) of a transverse wave is given by the relation $u = \eta \hbar^\alpha c^\beta G^\gamma \omega^3 k^3$ , where $\omega$ is the angular frequency, $k$ is the wave number, and $\eta$ is a dimensionless constant. The parameters $\hbar, c,$ and $G$ represent the reduced Planck's constant, speed of light in vacuum, and universal gravitational constant, respectively. The dispersion relation of this medium is $\omega^2 = \mu k^5$ , where $\mu$ is a positive dimensional constant. Which of the following represents the correct expression for the energy flux (intensity) $I$ of the wave?

Options

A.
$\frac{5}{2} \eta \hbar^2 c^{-5} G \omega^4 k^2$
B.
\eta \hbar^2 c^{-5} G \omega^4 k^2
C.
$\frac{5}{2} \eta \hbar c^{-4} G \omega^4 k^2$
D.
$\frac{2}{5} \eta \hbar^2 c^{-5} G \omega^4 k^2$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Derived Units via Fundamental Constants|Energy Density in Transverse Waves|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests two critical concepts in advanced wave mechanics: dimensional analysis with fundamental constants and energy transport in non-linear dispersive media.

A common trap is assuming energy travels at the phase velocity ( $v_p = \omega/k$ ), which leads to the incorrect coefficient (Option B).

Furthermore, students must correctly construct the derived unit for energy density by solving a linear system of dimensional equations.

Option C traps students who guess the formula without verifying dimensional consistency, while Option D catches algebraic errors in calculating the group velocity ( $v_g = d\omega/dk$ ).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Units and Measurements - Class 11 Part 1
Concept	Derived Units via Fundamental Constants
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free