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Question #43061

Difficulty: 8/10 | 🟢 Free

The Question

A uniform plane electromagnetic wave of angular frequency $\omega$ propagates along the $+z$ -direction in a non-magnetic ( $\mu = \mu_0$ ) lossy dielectric medium. The medium is characterized by a complex permittivity $\epsilon^* = \epsilon_0 (4 - 3j)$ , where $j = \sqrt{-1}$ . In electromagnetic theory, the complex Poynting vector is defined as $\vec{S} = \frac{1}{2} \vec{E} \times \vec{H}^*$ , where $\vec{E}$ and $\vec{H}$ are the complex phasor representations of the electric and magnetic fields. The real part of $\vec{S}$ represents the time-averaged active (real) power density, while its imaginary part represents the reactive (wattless) power density. What is the ratio of the magnitude of the reactive power density to the time-averaged active power density at any point in the medium?

Options

A.
0
B.
$\frac{1}{3}$
C.
$\frac{3}{4}$
D.
$\frac{4}{3}$

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

Wattless Current|Energy Density Partitioning|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of energy transport in lossy media, specifically the partitioning of electromagnetic power into active (real) and reactive (wattless) components.

A common misconception is that the Poynting vector only represents real power flow, leading students to assume the reactive component is zero (Option A).

Another major trap is incorrectly assuming the power ratio directly equals the loss tangent of the medium ( $\tan\delta = \epsilon''/\epsilon' = 3/4$ , Option C).

By introducing complex permittivity, students must recognize that the magnetic field lags the electric field by an angle $\theta_\eta = \delta/2$ .

The complex Poynting vector $\vec{S} = \frac{1}{2}\vec{E} \times \vec{H}^*$ captures this phase difference, yielding a reactive-to-active power ratio of $\tan(\delta/2)$ , not $\tan\delta$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Wattless Current
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free