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Question #43567

Difficulty: 8/10 | 🟢 Free

The Question

In a model bridging Huygens' principle and Maxwell's equations, a plane electromagnetic wave propagates in vacuum along the $+z$ -axis. At $z=0$ , the electric field is given by $\vec{E}(t) = E_0 \cos(\omega t) \hat{i}$ . Consider a thin planar slice of the wavefront of thickness $\Delta z$ (where $\Delta z \ll c/\omega$ ) centered at $z=0$ . This slice acts as a localized secondary source generating a magnetic field via its displacement current. What is the magnetic field contribution $\Delta \vec{B}(t)$ generated solely by the displacement current of this specific slice at $z = \Delta z/2$ , and what is the phase difference $\Delta \phi = \phi_{\Delta B} - \phi_E$ between this magnetic field contribution and the local electric field $\vec{E}(t)$ ? (Assume $c$ is the speed of light in vacuum, and all quantities are in standard SI units).

Options

A.
$\Delta \vec{B}(t) = \frac{E_0 \omega \Delta z}{2 c^2} \sin(\omega t) \hat{j}$ , $\Delta \phi = -\frac{\pi}{2}$
B.
$\Delta \vec{B}(t) = \frac{E_0 \omega \Delta z}{2 c^2} \cos(\omega t) \hat{j}$ , $\Delta \phi = 0$
C.
$\Delta \vec{B}(t) = -\frac{E_0 \omega \Delta z}{2 c^2} \sin(\omega t) \hat{j}$ , $\Delta \phi = \frac{\pi}{2}$
D.
$\Delta \vec{B}(t) = \frac{E_0 \omega \Delta z}{c^2} \sin(\omega t) \hat{j}$ , $\Delta \phi = -\frac{\pi}{2}$

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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

Huygens Principle Wavefronts|Ampere’s Law Displacement Current|Displacement Current Source

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental mechanism of electromagnetic wave propagation by forcing students to apply the Maxwell-Ampère law to a localized Huygens secondary source.

The primary trap is the assumption that the magnetic field generated by a single wavefront slice is in-phase with the local electric field ( $\Delta \phi = 0$ ), as is true for the macroscopic plane wave.

However, the local source of the magnetic field is the displacement current ( $\propto \partial \vec{E}/\partial t$ ), which introduces an intrinsic $\pi/2$ phase shift.

Students who rely on memorized plane-wave properties without understanding the underlying differential field-coupling mechanisms will incorrectly choose the in-phase option.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Wave Optics - Class 12
Concept	Huygens Principle Wavefronts
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free