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Question #43729

Difficulty: 8/10 | 🟢 Free

The Question

A modified Young's Double Slit Experiment setup consists of two narrow slits $S_1$ and $S_2$ located at coordinates $(0, d/2)$ and $(0, -d/2)$ respectively. A screen is placed at $x = D$ . The space between the slits and the screen ( $0 \le x \le D$ ) is filled with a non-uniform transparent dielectric medium whose refractive index varies with the transverse coordinate $y$ as $\mu(y) = \mu_0 + k y$ , where $\mu_0$ and $k$ are positive constants. Light of vacuum wavelength $\lambda$ is incident on the slits. Assuming $d \ll D$ , $y \ll D$ , $k y \ll \mu_0$ , and that the rays travel in approximately straight lines due to the weak gradient, what is the $y$ -coordinate of the central maximum (where the optical path difference is zero) on the screen?

Options

A.
$\frac{k D^2}{2\mu_0}$
B.
$\frac{k D^2}{\mu_0}$
C.
$\frac{k D^2}{4\mu_0}$
D.
$\frac{2k D^2}{\mu_0}$

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

Intensity Distribution in YDSE|Non-Uniform Dielectric Filling|Optical Path Difference

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to apply the Superposition Principle and calculate phase differences in a continuously varying dielectric medium, which dictates the spatial redistribution of energy (interference pattern).

A common lethal mistake is assuming the optical path difference introduced by the medium is simply the difference in refractive index at the slits multiplied by the distance D (i.e., $\Delta x_{med} = \Delta \mu \cdot D = k d D$ ).

This ignores the fact that the rays converge to a single point on the screen, meaning they traverse a varying refractive index profile.

By failing to integrate $\mu(y)$ along the actual ray paths, students miss the averaging effect that introduces a critical factor of $1/2$ , leading them directly to the trap answer $\frac{k D^2}{\mu_0}$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Wave Optics - Class 12
Concept	Intensity Distribution in YDSE
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free