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Question #43316

Difficulty: 8/10 | 🟢 Free

The Question

In a source-free vacuum region, an induced electromagnetic standing wave is established. The electric field vector is given by $\vec{E}(z, t) = E_0 \sin(kz) \cos(\omega t) \hat{i}$ , where $E_0$ is the electric field amplitude in $\text{V m}^{-1}$ , $k$ is the wave number in $\text{rad m}^{-1}$ , and $\omega$ is the angular frequency in $\text{rad s}^{-1}$ . Let $c$ be the speed of light in vacuum in $\text{m s}^{-1}$ , and $\vec{B}(z, t)$ be the corresponding magnetic field vector in $\text{T}$ . Which of the following equations correctly describes the phase-space trajectory of the field vector components $(E_x, cB_y)$ at the spatial coordinate $z = \frac{\pi}{3k}$ as time $t$ varies?

Options

A.
$\frac{4E_x^2}{3E_0^2} + \frac{4(cB_y)^2}{E_0^2} = 1$
B.
$\frac{4E_x^2}{E_0^2} + \frac{4(cB_y)^2}{3E_0^2} = 1$
C.
$E_x + \sqrt{3}cB_y = 0$
D.
$\frac{4E_x^2}{3E_0^2} + \frac{4(cB_y)^2}{3E_0^2} = 1$

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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

Induced Electric Fields|Transverse Nature of EM Waves|Magnitude Ratio Identity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to derive the magnetic field from an electric field using Maxwell's equations from first principles, rather than relying on the memorized traveling wave relation $E = cB$ .

The specific trap is the 'Phase Space Trajectory' misconception: students often conflate the in-phase behavior of traveling waves (where the trajectory in $E-B$ space is a straight line) with standing waves.

In a standing wave, spatial and temporal derivatives introduce a $\pi/2$ phase shift in both space and time.

By evaluating at a specific spatial coordinate that is not a node or antinode, it forces the realization that the local field vector traces an ellipse.

Students who blindly apply $B_y = E_x/c$ will incorrectly choose the linear trajectory, while those who ignore the spatial derivative's effect on the amplitude will choose the incorrect circular or inverted elliptical trajectories.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Induced Electric Fields
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free