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Question #43247

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The Question

In a Young's Double Slit Experiment, the slits $S_1$ and $S_2$ are separated by a distance $d = 1 \text{ mm}$ and are illuminated by monochromatic light of wavelength $\lambda = 600 \text{ nm}$ . A screen is placed at a distance $D = 2 \text{ m}$ from the slits. The entire setup is initially in a vacuum. A thin transparent plate of uniform thickness $t = 2 \text{ mm}$ is placed parallel to the plane of the slits, at an exact distance of $D/2 = 1 \text{ m}$ from the slits. The refractive index of this plate varies along the transverse $y$ -direction (parallel to the slit separation) as $n(y) = 1 + \beta y^2$ , where $\beta = 100 \text{ m}^{-2}$ and $y=0$ corresponds to the central axis passing through the midpoint of the slits. Assuming the deviation of light rays from straight lines is negligible, what is the new fringe width on the screen?

Options

A.
$1.5 \text{ mm}$
B.
$1.2 \text{ mm}$
C.
$1.0 \text{ mm}$
D.
$2.0 \text{ mm}$

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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

Fringe Width Invariance|Dielectric Constant (K) Shift|Binomial Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental understanding of optical path length and phase gradients in wave optics, moving beyond the standard geometric approximation $W = \lambda D/d$ .

Students commonly fall into the trap of treating the fringe width as an invariant property of the setup, assuming a medium only shifts the central maximum.

By introducing a transverse spatial gradient in the refractive index ( $n(y) \propto y^2$ ), the optical path difference acquires an additional linear dependence on the screen coordinate $y$ .

This alters the local phase gradient $\frac{d(\Delta \phi)}{dy}$ , which inversely scales the fringe spacing.

Failing to integrate the local optical path length or misapplying the geometric intersection coordinates at the intermediate plate leads to incorrect fringe widths.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Wave Optics - Class 12
Concept	Fringe Width Invariance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free