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Question #42637

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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 $d$ and the screen is placed at a distance $D$ ( $D \gg d$ ). The space between the slits and the screen is filled with a non-homogeneous transparent medium whose refractive index varies with the transverse coordinate $y$ as $\mu(y) = \mu_0 + \alpha y$ , where $\mu_0$ and $\alpha$ are positive constants. The $y$ -axis is parallel to the plane of the slits, with $y=0$ passing through the midpoint of $S_1$ and $S_2$ . Slit $S_1$ is located at $y = d/2$ and $S_2$ is at $y = -d/2$ . Assuming $\alpha D \ll \mu_0$ such that light rays travel in approximately straight lines, what is the $y$ -coordinate of the central maximum on the screen?

Options

A.
$\frac{\alpha D^2}{2\mu_0}$
B.
$\frac{\alpha D^2}{\mu_0}$
C.
$\frac{\alpha d D}{\mu_0}$
D.
$\frac{\alpha D^3}{\mu_0 d}$

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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

Phase Difference vs. Path Difference|Diffraction vs Interference|Calculus (Integration of Phase)

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from the standard discrete path difference formula ( $\Delta x = d \sin \theta$ ) to the continuous integral form of the optical path length (Fermat's Principle) in a non-homogeneous medium.

A common trap is assuming the rays travel entirely in a medium with the refractive index evaluated at their respective slits ( $\mu_1 = \mu_0 + \alpha d/2$ and $\mu_2 = \mu_0 - \alpha d/2$ ), which incorrectly yields a path difference of $\alpha d D$ and leads to the distractor $Y = \alpha D^2 / \mu_0$ .

Students must recognize that the refractive index varies continuously along the ray's path, requiring integration of $\mu(y)$ over the straight-line trajectory.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Phase Difference vs. Path Difference
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free