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Question #43103

Difficulty: 8/10 | 🟢 Free

The Question

An electromagnetic wave propagates in the $+x$ direction through a non-magnetic inhomogeneous dielectric medium ( $\mu = \mu_0$ ). The medium has a spatially varying permittivity $\epsilon(x)$ . At a specific instant $t=0$ , the electric and magnetic fields of the wave are given by: $\mathbf{E}(x,0) = E_0 e^{-\alpha x} \cos(kx) \hat{j}$ $\mathbf{B}(x,0) = B_0 e^{-\alpha x} \cos(kx) \hat{k}$ where $E_0, B_0, \alpha,$ and $k$ are positive constants with appropriate SI units. Calculate the instantaneous rate of change of the total electromagnetic energy density, $\frac{\partial u}{\partial t}$ , at the position $x = \frac{\pi}{4k}$ at $t=0$ .

Options

A.
$\frac{E_0 B_0}{\mu_0} e^{-\frac{\pi \alpha}{2k}} (\alpha + k)$
B.
$\frac{2 E_0 B_0}{\mu_0} e^{-\frac{\pi \alpha}{2k}} (\alpha + k)$
C.
$\frac{E_0 B_0}{\mu_0} e^{-\frac{\pi \alpha}{2k}} \alpha$
D.
$\frac{E_0 B_0}{\mu_0} e^{-\frac{\pi \alpha}{2k}} k$

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

Poynting Vector and Intensity|Energy Density Locality|Dielectric Polarization Logic|Vector Cross-Product

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to recognize and apply Poynting's Theorem as a universal conservation law, bypassing missing constitutive data.

The primary trap is the illusion of missing information: students often attempt to calculate energy density using $u = \frac{1}{2}\epsilon E^2 + \frac{1}{2\mu_0}B^2$ , but get stuck because the spatially varying permittivity $\epsilon(x)$ is deliberately omitted.

A secondary trap lies in the differentiation of the Poynting vector; students accustomed to uniform plane waves may ignore the spatial attenuation factor $\alpha$ or fail to apply the product rule correctly, leading to incorrect options C or D.

The problem forces the abandonment of steady-state uniform intensity assumptions and demands rigorous application of local differential energy conservation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Waves - Class 12
Concept	Poynting Vector and Intensity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free