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Question #43099

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The Question

A plane electromagnetic wave of angular frequency $\omega$ propagates in the $+x$ direction through a non-magnetic ( $\mu = \mu_0$ ) inhomogeneous lossless dielectric medium. The permittivity of the medium varies spatially as $\epsilon(x) = \epsilon_0 e^{\alpha x}$ , where $\alpha > 0$ is a small constant such that the WKB approximation is valid ( $\alpha \ll \omega \sqrt{\mu_0 \epsilon_0}$ ). Let $\langle u_E \rangle$ and $\langle u_B \rangle$ denote the time-averaged electric and magnetic energy densities, respectively, at a position $x$ . Which of the following expressions correctly represents the fractional difference $\frac{\langle u_B \rangle - \langle u_E \rangle}{\langle u_E \rangle}$ at position $x$ ?

Options

A.
$\frac{\alpha^2}{16 \mu_0 \omega^2 \epsilon(x)}$
B.
$\frac{\alpha^2}{4 \mu_0 \omega^2 \epsilon(x)}$
C.
0
D.
$\frac{\alpha^2}{8 \mu_0 \omega^2 \epsilon(x)}$

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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

Energy Density Partitioning|Poynting Vector and Intensity|Potential Gradient Direction

Why This Question Is Tricky

Common Mistakes

This question tests a deep conceptual understanding of electromagnetic wave propagation in inhomogeneous media, specifically targeting the assumption of constant wave impedance.

Students typically memorize that the time-averaged electric and magnetic energy densities are strictly equal ( $\langle u_E \rangle = \langle u_B \rangle$ ), which is only true for homogeneous media.

In a graded-index medium, the spatial gradient of the permittivity introduces an imaginary component to the local impedance relationship between $\vec{E}$ and $\vec{B}$ .

This phase shift means that while the time-averaged Poynting vector remains constant to conserve energy flux, the localized energy density partitions unequally, favoring the magnetic field.

Choosing option C (0) is the classic trap of blindly applying homogeneous plane wave properties to a continuously varying medium.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Waves - Class 12
Concept	Energy Density Partitioning
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free