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Question #43121

Difficulty: 8/10 | 🟢 Free

The Question

A monochromatic light ray is incident from vacuum onto a transparent parallel-sided slab of thickness $d$ (in meters) at a constant angle of incidence $\theta_0$ . The refractive index of the slab is spatially uniform but varies slowly with time as $n(t) = n_0 + \beta t$ , where $\beta$ is a positive constant (in $\text{s}^{-1}$ ). Assume the variation is slow enough that the transit time of light through the slab is negligible (adiabatic approximation). Which of the following expressions correctly represents the instantaneous optical path length $L_{opt}(t)$ of the ray strictly inside the slab?

Options

A.
$\frac{n^2(t) d}{\sqrt{n^2(t) - \sin^2\theta_0}}$
B.
$\frac{n(t) n_0 d}{\sqrt{n_0^2 - \sin^2\theta_0}}$
C.
$\frac{n(t) d}{\sqrt{n^2(t) - \sin^2\theta_0}}$
D.
$\frac{n(t) d}{\cos\theta_0}$

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

Snell’s Law Vector Form|Optical Path Difference|Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple temporal modulation with spatial geometry in optics.

The primary trap (Option B) assumes that the physical trajectory of the light ray is static and fixed by the initial refractive index $n_0$ , leading students to simply multiply the static physical path length by the time-varying $n(t)$ .

However, because $n(t)$ changes, the angle of refraction $\theta(t)$ must also change dynamically according to Snell's Law.

Consequently, the physical path length $s(t)$ is itself a function of time.

Option C is another trap where students correctly calculate the dynamic physical path length but forget to multiply by $n(t)$ to find the optical path length, confusing the two concepts.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Snell’s Law Vector Form
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free