Before You Begin

Question #43240

Difficulty: 8/10 | 🟢 Free

The Question

In a modified Young's Double Slit Experiment, the space between the slit plane (at $x = 0$ ) and the screen (at $x = D$ ) is completely filled with a non-uniform transparent medium. The refractive index of this medium varies spatially as $n(y) = n_0 + k y$ , where $y$ is the transverse coordinate parallel to the slit plane, $n_0$ is a dimensionless constant, and $k$ is a positive constant in $\text{m}^{-1}$ . The two slits $S_1$ and $S_2$ are located at $y = d/2$ and $y = -d/2$ respectively. Assuming $d \ll D$ and considering only paraxial rays ( $y \ll D$ ) such that the geometric paths can be approximated as straight lines, find the $y$ -coordinate of the central bright fringe on the screen.

Options

A.
$\frac{k D^2}{2 n_0}$
B.
$\frac{k D^2}{n_0}$
C.
0
D.
$\frac{k D^2}{4 n_0}$

Check Your Answer

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

Phase and Path Difference|Optical Path Difference|Calculus/Integration

Why This Question Is Tricky

Common Mistakes

This question tests the transition from algebraic optics to wave optics in a non-homogeneous medium.

Students often fall into the trap of assuming the path difference is generated solely by the local refractive index at the slits, effectively treating the medium as two separate uniform blocks of $n_1 = n_0 + kd/2$ and $n_2 = n_0 - kd/2$ .

This incorrect algebraic approach yields $\Delta x = -kdD + n_0 y d / D = 0$ , leading to an erroneous shift of $k D^2 / n_0$ (Option B).

The correct physical execution requires integrating the spatially varying phase accumulation $\int n(r) dl$ along the wavefront's path, revealing that the average refractive index along each specific ray dictates the true optical path length.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Wave Optics - Class 12
Concept	Phase and Path Difference
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free