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Question #42852

Difficulty: 8/10 | 🟢 Free

The Question

Two coherent plane electromagnetic waves, Wave 1 and Wave 2, of wavelength $\lambda$ propagate in the $+z$ direction. Wave 1 has a uniform intensity $I_1 = 9I_0$ and travels through vacuum. Wave 2 has a uniform intensity $I_2 = 4I_0$ and passes through a transparent slab of uniform thickness $d$ placed just before the $z=0$ plane (occupying $-d \le z \le 0$ ). Due to a spatially varying medium density, the refractive index of the slab varies along the $x$ -axis as $n(x) = 1 + \frac{\lambda}{4ad}x$ , where $a$ is a positive constant. An opaque screen with a rectangular aperture is placed at the $z=0$ plane. The aperture is bounded by $0 \le x \le a$ and $0 \le y \le b$ . Assuming both waves have identical phase at $z = -d$ , what is the net time-averaged power (intensity flux) passing through the aperture?

Options

A.
$abI_0 \left( 13 + \frac{24}{\pi} \right)$
B.
$abI_0 \left( 13 + 6\sqrt{2} \right)$
C.
$25 abI_0$
D.
$13 abI_0$

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

Superposition of Non-identical Waves|Flux and Surface Geometry

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to apply the superposition principle in a medium with a spatially varying refractive index, which acts as a transverse phase gradient.

A common trap for students is to treat the interference of two plane waves as producing a uniform intensity distribution, either by calculating the phase difference at a single representative point (like the center or edge of the aperture) or by completely ignoring the spatial dependence of the phase velocity.

By forcing the integration of the local intensity $I(x) = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos(\Delta \phi(x))$ over the aperture's area, the problem evaluates whether the student recognizes that the non-uniform medium density creates a continuous transverse phase shift, fundamentally altering the net intensity flux compared to standard constant-phase superposition.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Superposition of Non-identical Waves
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free