Before You Begin

Question #42881

Difficulty: 8/10 | 🟢 Free

The Question

A lossless transmission line of length $d = \lambda/8$ (where $\lambda$ is the operating wavelength) has a real characteristic impedance $Z_0 = \sqrt{L_c/C_c}$ , where $L_c$ and $C_c$ are the inductance and capacitance per unit length. The line is located along the $z$ -axis from $z = -d$ to $z = 0$ . It is terminated at $z = 0$ by a purely inductive load $L_0$ such that its reactance at the operating angular frequency $\omega$ is exactly equal to $Z_0$ (i.e., $\omega L_0 = Z_0$ ). A sinusoidal generator is connected at $z = -d$ , establishing a steady-state standing wave on the line. Let $U_E$ and $U_M$ be the total time-averaged electric and magnetic energies stored in the entire length of the transmission line, respectively. What is the ratio $\frac{U_E}{U_M}$ ?

Options

A.
$\frac{\pi + 2}{\pi - 2}$
B.
1
C.
$\frac{\pi + 1}{\pi - 1}$
D.
$\frac{\pi - 2}{\pi + 2}$

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 Distribution in Standing Waves|Phase Relationship|Frequency Sensitivity of Z

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of energy distribution in standing waves and the effect of reactive boundary conditions.

A common misconception is that the total time-averaged electric and magnetic energies in a lossless transmission line (or any standing EM wave) are always equal ( $U_E = U_M$ ), which is only true for an integer number of quarter-wavelengths or a matched line.

By introducing a purely reactive load ( $Z_L = jZ_0$ ), a specific phase shift ( $\pi/2$ ) is forced upon the reflection coefficient.

This shifts the spatial nodes of the E and B fields.

When integrated over a fractional wavelength ( $d = \lambda/8$ ), the spatial offset between the $\cos^2$ and $\sin^2$ energy density distributions results in a net imbalance between stored electric and magnetic energies.

Students who blindly apply $U_E = U_M$ or ignore the complex phase angle of the reflection coefficient will fail to capture this frequency-dependent energy asymmetry.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Energy Distribution in Standing Waves
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free