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Question #43094

Difficulty: 8/10 | 🟢 Free

The Question

A uniform plane electromagnetic wave of angular frequency $\omega$ and electric field amplitude $E_0$ is traveling in vacuum ( $x < 0$ ) in the $+x$ direction. It is normally incident on a semi-infinite, non-magnetic ( $\mu = \mu_0$ ), lossless dielectric medium ( $x \ge 0$ ) with relative permittivity $\epsilon_r = 4$ . Let $\langle u_e \rangle$ and $\langle u_m \rangle$ denote the time-averaged electric and magnetic energy densities, respectively, in the vacuum region. Calculate the ratio $\frac{\langle u_e \rangle}{\langle u_m \rangle}$ at a distance $d = \frac{\pi c}{2\omega}$ from the boundary (i.e., at $x = -d$ ), where $c$ is the speed of light in vacuum.

Options

A.
4
B.
1
C.
$\frac{1}{4}$
D.
2

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Phase Relationship|Energy Density Partitioning|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the 'Equipartition Trap' in electromagnetic waves.

Students often rote-memorize that the time-averaged electric and magnetic energy densities are always equal ( $\langle u_e \rangle = \langle u_m \rangle$ ), which is only true for a pure traveling wave in a homogeneous medium.

By introducing a dielectric boundary, a reflected wave is generated, creating a partial standing wave in the vacuum region.

The superposition of the incident and reflected waves causes a spatial phase mismatch between the total electric and magnetic fields.

Failing to integrate the localized interference terms (the cross terms in the Poynting vector divergence) leads students to incorrectly assume the ratio is 1, ignoring the localized decoupling of energy densities.

Quick Facts About This Question

Question Information

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