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Question #43166

Difficulty: 8/10 | 🟢 Free

The Question

A plane electromagnetic wave propagates in vacuum in the $+x$ direction in the laboratory frame $S$ . Its electric field is given by $\vec{E} = E_0 \cos(kx - \omega t) \hat{y}$ , where $E_0$ is the amplitude, $k$ is the wave number, and $\omega$ is the angular frequency. An observer in frame $S'$ moves with a constant velocity $\vec{v} = \frac{\sqrt{3}}{2} c \hat{y}$ relative to $S$ , where $c$ is the speed of light. Let $\alpha$ be the angle that the electric field vector $\vec{E}'$ makes with the $+x'$ axis in $S'$ , and let $u'$ be the time-averaged energy density of the wave in $S'$ . Which of the following correctly gives $\tan \alpha$ and the ratio $u'/u$ (where $u$ is the time-averaged energy density in $S$ )?

Options

A.
$\tan \alpha = \frac{1}{\sqrt{3}}, \quad \frac{u'}{u} = 4$
B.
$\tan \alpha = \sqrt{3}, \quad \frac{u'}{u} = 2$
C.
$\tan \alpha = -\sqrt{3}, \quad \frac{u'}{u} = 1$
D.
$\tan \alpha = \infty, \quad \frac{u'}{u} = 7 - 4\sqrt{3}$

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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

Transverse Nature of EM Waves|Frame of Reference|Dot Product Geometry

Why This Question Is Tricky

Common Mistakes

This question targets the misconception that the polarization plane of a transverse electromagnetic wave remains invariant when observed from a frame moving perpendicular to the wave's propagation.

Students often erroneously assume $\vec{E}' = E_y \hat{y}$ by neglecting the magnetic field's contribution ( $\vec{v} \times \vec{B}$ ) to the transformed electric field, leading to an incorrect polarization angle ( $\tan \alpha = \infty$ ).

Furthermore, they frequently misapply the energy density transformation, either assuming it is a Lorentz invariant ( $u'/u = 1$ ) or incorrectly applying the longitudinal relativistic Doppler shift formula.

The problem rigorously tests the simultaneous transformation of both $\vec{E}$ and $\vec{B}$ fields to demonstrate the rotation of the polarization plane and the correct scaling of the energy-momentum tensor components.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Waves - Class 12
Concept	Transverse Nature of EM Waves
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free