Before You Begin

Question #93864

Difficulty: 5/10 | 🟢 Free

The Question

A plane electromagnetic wave of angular frequency $\omega$ is normally incident on a semi-infinite good conductor occupying the region $z > 0$ . The conductor is characterized by a dynamic surface resistance $R_s$ (in $\Omega$ ) and a real permittivity $\epsilon$ . The condition for a good conductor ( $\sigma \gg \omega\epsilon$ ) holds, where $\sigma$ is the conductivity. Let $\langle u_E \rangle$ denote the time-averaged electric energy density of the wave evaluated just inside the surface of the conductor ( $z=0^+$ ). Which of the following expressions correctly represents the total time-averaged electromagnetic power dissipated per unit area inside the conductor?

Options

A.
$\frac{\langle u_E \rangle}{\epsilon R_s}$
B.
$\frac{2\langle u_E \rangle}{\epsilon R_s}$
C.
$\frac{\sqrt{2}\langle u_E \rangle}{\epsilon R_s}$
D.
$\frac{4\langle u_E \rangle}{\epsilon R_s}$

Check Your Answer

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

Energy Density Partitioning|Dynamic Resistance (AC Resistance)|Frequency Sensitivity of Z

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

The core trap is the misapplication of the free-space Poynting vector formula $\langle S \rangle = E_0^2 / (2Z)$ to a lossy medium.

In free space, the electric and magnetic fields are in phase, and the impedance is purely real.

In a good conductor, the complex impedance $Z_c = R_s(1+i)$ introduces a $\pi/4$ phase shift between the fields.

Students who ignore this phase shift and incorrectly assume a purely real impedance will use $\langle S \rangle = E_0^2 / (2R_s)$ , overestimating the power dissipation by a factor of 2 (a 100% error).

Furthermore, using the magnitude of the impedance $|Z_c| = \sqrt{2}R_s$ without extracting the real part of the power flow leads to another common incorrect prefactor.

This question rigorously tests the understanding of energy partitioning and phase relations in dynamic, non-linear resistive media.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Waves - Class 12
Concept	Energy Density Partitioning
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free