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Question #43202

Difficulty: 8/10 | 🟢 Free

The Question

A light ray enters a gradient-index (GRIN) dielectric slab of thickness $d$ (in $\text{m}$ ) at the origin $(0,0)$ . The slab occupies the region $0 \le y \le d$ . The refractive index of the slab varies with the $y$ -coordinate as $n(y) = n_0 \sqrt{1 + \alpha y}$ , where $n_0$ is a dimensionless constant and $\alpha$ is a positive constant with units of $\text{m}^{-1}$ . The ray is incident from a uniform medium of refractive index $n_0$ (occupying $y < 0$ ) at an angle $\theta_0$ with the $y$ -axis. Assuming the optical path length is minimized (Fermat's Principle), find the lateral shift $x$ (in $\text{m}$ ) of the ray at the point it exits the slab at $y = d$ .

Options

A.
$\frac{2 \sin\theta_0}{\alpha} \left( \sqrt{\cos^2\theta_0 + \alpha d} - \cos\theta_0 \right)$
B.
$\frac{\sin\theta_0}{\alpha} \left( \sqrt{\cos^2\theta_0 + \alpha d} - \cos\theta_0 \right)$
C.
$\frac{2 \cos\theta_0}{\alpha} \left( \sqrt{\sin^2\theta_0 + \alpha d} - \sin\theta_0 \right)$
D.
$\frac{2 \sin\theta_0}{\alpha} \left( \sqrt{1 + \alpha d} - 1 \right)$

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

Lateral Shift in Slab|Conservative Field Path Independence|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the application of Fermat's Principle of Least Time in a gradient-index (GRIN) medium, framing it as an analogue to a conservative potential field.

Students often attempt to apply Snell's Law incrementally without recognizing the underlying variational principle (Euler-Lagrange equation), where the translational symmetry along the x-axis dictates that the optical Hamiltonian (the invariant $n(y)\sin\theta$ ) is conserved.

A common trap is failing to correctly relate the local angle $\theta(y)$ to the spatial derivative $dx/dy$ , or incorrectly assuming the path is a straight line characterized by an 'average' refractive index, completely missing the continuous path deviation caused by the gradient field.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Lateral Shift in Slab
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free