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Question #43888

Difficulty: 8/10 | 🟢 Free

The Question

A plane electromagnetic wave propagates in the $+z$ direction through a linear, non-magnetic, dissipative medium. At the plane $z=0$ , the electric field of the wave is given by $\vec{E}(0,t) = E_0 \cos(\omega t) \hat{i}$ , where $E_0 = 200 \, \text{V/m}$ and $\omega$ is the angular frequency. The intrinsic wave impedance of the medium, defined as the ratio of the complex amplitudes of the electric and magnetic fields ( $Z = \tilde{E}_x / \tilde{H}_y$ ), is given by $Z = (100\sqrt{3} + i100) \, \Omega$ . What is the time-averaged magnitude of the Poynting vector $\langle S \rangle$ entering the medium at $z=0$ ?

Options

A.
$50\sqrt{3} \, \text{W/m}^2$
B.
$100 \, \text{W/m}^2$
C.
$\frac{200}{\sqrt{3}} \, \text{W/m}^2$
D.
$50 \, \text{W/m}^2$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Wave Propagation Direction|Resistor Phase Invariance|Series LCR Impedance Vector

Why This Question Is Tricky

Common Mistakes

Students incorrectly apply the free-space electromagnetic wave propagation logic to a dissipative medium.

In a vacuum, the electric and magnetic fields are perfectly in phase, and the time-averaged Poynting vector is simply $E_0^2 / (2Z_0)$ .

However, in a medium with complex impedance, the resistive and reactive components create a phase lag $\phi$ between $\vec{E}$ and $\vec{H}$ .

By treating the Poynting vector as a simple product of magnitudes and ignoring the $\cos\phi$ power factor (analogous to an LCR circuit's active power), students fall into the trap of calculating the apparent power instead of the true time-averaged energy flow.

Quick Facts About This Question

Question Information

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