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Question #43187

Difficulty: 8/10 | 🟢 Free

The Question

A light ray enters a transparent medium at the origin $(0,0)$ of an $xy$ -coordinate system. The medium occupies the region $y \ge 0$ and has a spatially varying refractive index given by $n(y) = n_0 \sqrt{1 + \alpha y}$ , where $n_0 = 1.5$ and $\alpha = 2 \text{ m}^{-1}$ . The ray enters the medium grazing the $x$ -axis (i.e., its initial velocity vector is in the $+x$ direction). Calculate the optical path length of the ray from the origin to the point where it reaches a vertical height $H = 2 \text{ m}$ .

Options

A.
7 \text{ m}
B.
9 \text{ m}
C.
$\frac{14}{3} \text{ m}$
D.
$\frac{16}{3} \text{ m}$

Check Your Answer

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

Refractive Index Relativity|Snell’s Law Vector Form|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the application of Snell's Law in a gradient-index (GRIN) medium and the critical distinction between geometric and optical path lengths.

A common trap is misidentifying the angle in Snell's Law; students often use the angle with the x-axis instead of the normal (y-axis), leading to the incorrect invariant $n(y)\cos\theta = \text{constant}$ or $n(y)\sin\theta_{x} = \text{constant}$ .

This yields an entirely different trajectory and an incorrect optical path length of 9 m (Option B).

Furthermore, students frequently fail to correctly parameterize the differential arc length $ds$ in terms of $dy$ , mistakenly integrating $n(y)dy$ directly instead of $n(y)\sqrt{1+(dx/dy)^2}dy$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Refractive Index Relativity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free