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Question #43086

Difficulty: 8/10 | 🟢 Free

The Question

In a source-free region of vacuum, an electromagnetic standing wave has an electric field given by $\vec{E}(x,t) = E_0 \cos(kx) \sin(\omega t) \hat{j}$ , where $E_0$ , $k$ , and $\omega$ are positive constants. Consider a rectangular loop $C$ in the $xz$ -plane with vertices traversed in the order $(0, 0, 0) \rightarrow (0, 0, L) \rightarrow (x_0, 0, L) \rightarrow (x_0, 0, 0) \rightarrow (0, 0, 0)$ . Which of the following expressions correctly represents the circulation of the magnetic field, $\oint_C \vec{B} \cdot d\vec{l}$ , around this loop at time $t$ ?

Options

A.
$\mu_0 \varepsilon_0 \frac{\omega}{k} E_0 L \sin(kx_0) \cos(\omega t)$
B.
$\mu_0 \varepsilon_0 \omega k E_0 L \sin(kx_0) \cos(\omega t)$
C.
$\mu_0 \varepsilon_0 \frac{\omega}{k} E_0 L \cos(kx_0) \sin(\omega t)$
D.
$\mu_0 \varepsilon_0 \frac{k}{\omega} E_0 L \sin(kx_0) \cos(\omega t)$

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

Displacement Current Source|Dimensional Consistency and Transcendental Functions|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of the Ampere-Maxwell Law and dimensional consistency when dealing with transcendental spatiotemporal fields.

A common trap (Option B) occurs when students erroneously differentiate the spatial component $\cos(kx)$ instead of integrating it over the area, yielding a factor of $k$ rather than $1/k$ .

This not only produces the wrong magnitude but violates dimensional homogeneity, as the resulting units fail to match those of magnetic circulation ( $T \cdot m$ ).

Another trap (Option C) tests the phase relationship; students may forget to differentiate the time component $\sin(\omega t)$ , missing the $\pi/2$ phase shift inherent in the displacement current mechanism.

Option D represents an error in time integration versus differentiation.

Recognizing that $\mu_0 \varepsilon_0 \omega / k = 1/c$ ensures the final expression correctly carries the dimensions of magnetic circulation.

Quick Facts About This Question

Question Information

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