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Question #43597

Difficulty: 8/10 | 🟢 Free

The Question

A plane wavefront of unpolarized light is incident from vacuum ( $z < 0$ ) onto a non-magnetic dielectric medium occupying the region $z \ge 0$ . The refractive index of the medium varies with depth $z$ as $n(z) = \sqrt{3 - \alpha z}$ for $0 \le z < 3/\alpha$ , where $\alpha = 1 \text{ m}^{-1}$ . The incident light is directed such that the reflected light at the interface $z=0$ is completely plane-polarized. A ray of this light enters the medium at the origin $(0,0,0)$ . Determine the $x$ -coordinate of the ray at the point where its associated wavefront becomes parallel to the $z$ -axis.

Options

A.
$\frac{3\sqrt{3}}{2} \text{ m}$
B.
$\frac{3\sqrt{3}}{4} \text{ m}$
C.
$\frac{9\sqrt{3}}{8} \text{ m}$
D.
$\frac{9}{4} \text{ m}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Brewster's Law|Huygens Principle Wavefronts|Non-Uniform Dielectric Filling

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from discrete boundary optics to continuous gradient-index (GRIN) optics.

The primary trap occurs when students correctly identify the initial angle of refraction ( $\theta_r = 30^\circ$ ) using Brewster's Law and Snell's Law at $z=0$ , but then erroneously assume the ray travels in a straight line within the medium.

This leads them to calculate $x = z \tan 30^\circ = \frac{3\sqrt{3}}{4} \text{ m}$ (Option B).

To succeed, the student must recognize that a non-uniform dielectric density causes continuous wavefront tilting, requiring the application of the eikonal equation (generalized Snell's Law) to integrate the curved ray trajectory.

Furthermore, it tests the fundamental Huygens' principle that wavefronts are strictly orthogonal to ray vectors.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Wave Optics - Class 12
Concept	Brewster's Law
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free