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Question #94137

Difficulty: 5/10 | 🟢 Free

The Question

A standing electromagnetic wave in vacuum is described by the electric field $\vec{E}(x,t) = E_0 \cos(kx) \cos(\omega t) \hat{y}$ and magnetic field $\vec{B}(x,t) = -B_0 \sin(kx) \sin(\omega t) \hat{z}$ . A thin transparent dielectric slab of cross-sectional area $A$ and thickness $d$ (where $d \ll \lambda$ ) is placed perpendicular to the x-axis at position $x = \lambda/8$ . The permittivity of the slab is externally modulated such that it varies slowly with time as $\epsilon(t) = \epsilon_0 (1 + \alpha t)$ , where $\alpha$ is a positive constant ( $\alpha \ll \omega$ ). Assuming the fields inside the slab are approximately equal to the external standing wave fields, what is the time-averaged net electromagnetic power entering the slab over one full optical cycle $T = 2\pi/\omega$ ?

Options

A.
$\frac{3}{8} \epsilon_0 \alpha E_0^2 A d$
B.
$\frac{1}{4} \epsilon_0 \alpha E_0^2 A d$
C.
$\frac{3}{4} \epsilon_0 \alpha E_0^2 A d$
D.
$\frac{1}{8} \epsilon_0 \alpha E_0^2 A d$

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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

Power Combination|Poynting Vector and Intensity|Energy Distribution in Standing Waves

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the application of Poynting's theorem in a medium with a time-varying permittivity, a scenario that breaks standard static-medium assumptions.

Students commonly fall into the trap of assuming that the term $\vec{E} \cdot \frac{\partial \vec{D}}{\partial t}$ only contributes through the explicit derivative of permittivity ( $\dot{\epsilon}E^2$ ), ignoring the non-zero time average of $\epsilon \vec{E} \cdot \dot{\vec{E}}$ over a cycle.

Because $\epsilon(t)$ grows linearly, the energy stored in the electric field increases, requiring a net influx of power.

Furthermore, it tests spatial awareness in standing waves; candidates must correctly evaluate the local field amplitude at $x = \lambda/8$ rather than blindly using the peak amplitude $E_0$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Power Combination
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free