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Question #43885

Difficulty: 8/10 | 🟢 Free

The Question

In an inertial reference frame $S$ , an electromagnetic wave propagates in a vacuum. The electric and magnetic fields of the wave are given by $\vec{E} = E_0 \cos(ky - \omega t) \hat{k}$ and $\vec{B} = B_0 \cos(ky - \omega t) \hat{i}$ , where $E_0$ and $B_0$ are positive constants, and $c$ is the speed of light in vacuum. An observer in an inertial frame $S'$ moves with a constant velocity $\vec{v} = \frac{\sqrt{3}}{2} c \hat{i}$ relative to $S$ . The axes of $S$ and $S'$ are parallel. According to the observer in $S'$ , what is the unit vector $\hat{n}'$ along the direction of propagation of the electromagnetic wave?

Options

A.
$-\frac{\sqrt{3}}{2} \hat{i} + \frac{1}{2} \hat{j}$
B.
$-\sqrt{\frac{3}{7}} \hat{i} + \frac{2}{\sqrt{7}} \hat{j}$
C.
\hat{j}
D.
$\frac{1}{2} \hat{i} + \frac{\sqrt{3}}{2} \hat{j}$

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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

Wave Propagation Direction|Frame of Reference Dependency|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the frame-dependency of electromagnetic wave propagation, specifically targeting the misconception that the direction of the Poynting vector is invariant or transforms via Galilean velocity addition.

Students often incorrectly apply $\vec{c}' = \vec{c} - \vec{v}$ , failing to recognize that at relativistic speeds, the electric and magnetic fields undergo field mixing via Lorentz transformations.

By forcing the evaluation of $\vec{E}' \times \vec{B}'$ , the problem elegantly demonstrates relativistic aberration, showing that the transverse and longitudinal components of the fields scale differently, fundamentally rotating the apparent direction of energy flow.

Quick Facts About This Question

Question Information

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