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Question #43213

Difficulty: 8/10 | 🟢 Free

The Question

A transparent solid slab of length $L$ is placed in air (refractive index $= 1$ ). The refractive index of the slab varies with the vertical distance $y$ from its bottom edge as $n(y) = n_0 \sqrt{1 + k y}$ , where $n_0$ and $k$ are positive constants. A light ray enters the slab at the origin $(x=0, y=0)$ grazing the x-axis (i.e., the initial velocity vector of the light ray is along the positive x-axis). The ray travels through the slab and exits from the opposite vertical face at $x=L$ . Assuming the slab is sufficiently high so that the ray does not exit from the top face, what is the sine of the angle of deviation $\delta$ of the emergent ray with respect to its initial direction?

Options

A.
$\frac{1}{2} n_0 k L$
B.
$n_0 k L$
C.
$\frac{k L}{2}$
D.
$\frac{n_0 k L}{\sqrt{1 + k L}}$

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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

Prism Minimum Deviation|Snell’s Law Vector Form|Calculus Integration

Why This Question Is Tricky

Common Mistakes

Students often memorize the paraxial GRIN formula $\theta \approx \int \frac{1}{n} \frac{dn}{dy} dx$ or assume the deviation is simply the angle $\theta$ the ray makes inside the medium at the exit face.

This ignores the final refraction at the medium-air boundary.

Furthermore, applying the standard thin prism formula $\delta = (\mu-1)A$ is impossible here.

The problem tests the fundamental understanding of generalized Snell's law ( $n(y) \cos \theta = \text{const}$ ) and forces students to correctly apply boundary conditions at the exit face, distinguishing between the internal trajectory angle and the true external deviation angle.

Quick Facts About This Question

Question Information

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