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Question #43096

Difficulty: 8/10 | 🟢 Free

The Question

A small circular loop of radius $a$ is placed at the origin in the $xy$ -plane and carries an alternating current $I(t) = I_0 \cos(\omega t)$ . This loop acts as an oscillating magnetic dipole. Consider a point $P$ in the $xy$ -plane at a distance $r$ from the origin ( $r \gg a$ ). The displacement current density $\vec{J}_d$ at point $P$ due to the electromagnetic fields of the dipole can be expressed in the form $\vec{J}_d = J_{\text{max}} \cos(\omega t + \phi) \hat{\phi}$ , where $\hat{\phi}$ is the azimuthal unit vector, $J_{\text{max}} > 0$ is a time-independent scalar amplitude, and $\phi$ is the phase difference relative to the source current $I(t)$ . Given that the speed of light in vacuum is $c$ , which of the following expressions correctly represents the phase difference $\phi$ ?

Options

A.
$\tan^{-1}\left(\frac{\omega r}{c}\right) - \frac{\omega r}{c}$
B.
$-\frac{\omega r}{c}$
C.
$\tan^{-1}\left(\frac{c}{\omega r}\right) - \frac{\omega r}{c}$
D.
$\frac{\omega r}{c} - \tan^{-1}\left(\frac{\omega r}{c}\right)$

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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

Phase Relationship|Displacement Current Source|Magnetic Dipole Moment of Revolving Electron

Why This Question Is Tricky

Common Mistakes

This question targets the 'Near-Field Approximation' fallacy and tests the student's understanding of retarded potentials in electrodynamics.

Students often incorrectly assume that the magnetic field of an oscillating dipole is simply the instantaneous Biot-Savart field, which would lead to a phase difference of zero (ignoring retardation entirely).

Even if they account for the time delay $t - r/c$ , they might erroneously isolate either the radiation ( $1/r$ ) term or the quasi-static ( $1/r^2$ ) term.

By asking for the displacement current phase at an arbitrary distance $r$ , the student is forced to use the complete retarded vector potential.

The phase shift $\tan^{-1}(\omega r/c)$ arises directly from the superposition of these near-field and far-field components, proving that phase synchronization is distance-dependent and that Maxwell's equations do not permit action-at-a-distance.

Quick Facts About This Question

Question Information

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