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Question #94038

Difficulty: 5/10 | 🟢 Free

The Question

A thin glass prism of refracting angle $A = 0.05 \text{ rad}$ and base length $H = 0.5 \text{ m}$ is placed with its apex at the origin and its first refracting surface along the positive y-axis. The thickness of the prism in the x-direction is given by $t(y) = yA$ . The refractive index of the prism material is non-uniform and varies linearly with the distance $y$ from the apex as $n(y) = n_0 + ky$ , where $n_0 = 1.6$ and $k = 2.0 \text{ m}^{-1}$ . A narrow monochromatic laser beam of wavelength $\lambda = 600 \text{ nm}$ is incident normally on the first face at a distance $y = 0.1 \text{ m}$ from the apex. Assuming the paraxial approximation holds, what is the total angle of deviation $\delta$ of the emergent ray from its initial direction?

Options

A.
$0.04 \text{ rad}$
B.
$0.05 \text{ rad}$
C.
$0.03 \text{ rad}$
D.
$0.06 \text{ rad}$

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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|Integration

Why This Question Is Tricky

Examiner Levers

Extraneous Data

Provides unnecessary variables (Red Herrings) to test conceptual focus.

Difficulty Boost : +3

Common Mistakes

This question tests the student's fundamental understanding of ray deviation in non-homogeneous media, specifically targeting the rote memorization of the thin prism formula $\delta = (n-1)A$ .

Students commonly fall into the trap of simply substituting the local refractive index $n(y)$ into the standard formula, yielding an incorrect deviation of $(n(y)-1)A = 0.04 \text{ rad}$ .

However, because the refractive index has a transverse gradient, the wavefront itself undergoes differential retardation, or equivalently, the ray continuously bends inside the prism due to the Eikonal equation $\nabla n = \frac{d}{ds}(n \frac{d\vec{r}}{ds})$ .

By forcing the use of the optical path length derivative $\delta = \frac{d}{dy}(OPL)$ , the problem discriminates between superficial formula application and deep conceptual mastery of Fermat's Principle in gradient-index (GRIN) optics.

The wavelength and base length are provided as extraneous red herrings.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Prism Minimum Deviation
Difficulty	5
Cognitive Load	6
Target Solve Time	150 sec
Access	Free