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Question #43196

Difficulty: 8/10 | 🟢 Free

The Question

A light ray enters a non-homogeneous transparent medium at the origin $(0,0)$ of an $xz$ -coordinate system. The $z$ -axis points vertically downwards into the medium, and the $x$ -axis lies along the surface boundary. The refractive index of the medium varies with depth $z$ as $\mu(z) = \mu_0 \sqrt{1 - \alpha z}$ for $z < 1/\alpha$ , where $\mu_0 = 1.5$ and $\alpha = 0.1 \text{ m}^{-1}$ . The ray enters the medium at an angle $\theta_0 = 30^\circ$ with respect to the positive $z$ -axis. The ray curves continuously, reaches a maximum depth, and eventually undergoes total internal reflection to return to the surface $z=0$ . What is the total horizontal distance $x$ traveled by the ray when it emerges back at the surface?

Options

A.
$10\sqrt{3} \text{ m}$
B.
$5\sqrt{3} \text{ m}$
C.
$20\sqrt{3} \text{ m}$
D.
$15 \text{ m}$

Check Your Answer

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

Total Internal Reflection (TIR)|Gravitational Field Variation|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the application of Fermat's Principle and generalized Snell's Law in a medium with a continuous refractive index gradient.

A common trap is treating the gradient as discrete slabs and failing to construct the correct differential equation for the ray path ( $dx/dz = \tan\theta$ ).

Students often incorrectly assume $dx/dz = \sin\theta$ or fail to account for the symmetry of the trajectory, missing the factor of 2 for the return journey (leading to option B).

Advanced students can bypass the integration by recognizing the optomechanical analogy: the ray path is mathematically identical to the parabolic trajectory of a projectile in a uniform gravitational field, where the effective acceleration is derived from the gradient of $\mu^2$ .

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