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Question #43198

Difficulty: 8/10 | 🟢 Free

The Question

A flat boundary separates two extensive regions, Region 1 and Region 2. Region 1 contains a stationary ideal gas where the speed of sound is $c$ . Region 2 contains another stationary ideal gas where the speed of sound is $2c$ . The boundary is moving with a constant velocity $v = c$ perpendicular to itself, advancing into Region 1. A high-frequency sound source is stationary in Region 1 and emits plane sound waves towards the boundary. Let $\theta$ be the angle of incidence, defined as the angle between the incident wave vector and the normal to the boundary pointing into Region 2. Find the critical angle of incidence $\theta_c$ for which total internal reflection occurs at the moving boundary.

Options

A.
$\frac{\pi}{3}$
B.
$\sin^{-1}\left(\frac{4}{5}\right)$
C.
$\frac{\pi}{6}$
D.
$\sin^{-1}\left(\frac{2}{3}\right)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Total Internal Reflection (TIR)|Relative Velocity in 2D|Projectile Path Equation

Why This Question Is Tricky

Common Mistakes

This question tests the understanding of wave propagation and Total Internal Reflection (TIR) in moving media.

A common trap is to assume that the critical condition for TIR occurs when the normal component of the phase velocity (or wave vector $k_{2x}'$ ) is zero, as is the case in stationary media.

However, energy transport is governed by the group velocity.

In a moving medium, the phase and group velocities are not parallel.

Setting $k_{2x}' = 0$ incorrectly ignores the convective drag of the medium, which can still pull energy across the boundary.

The correct physical principle is to apply the velocity addition theorem to the wave-front normal and set the normal component of the group velocity to zero in the boundary frame.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Total Internal Reflection (TIR)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free