/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 The semi-infinite regions \(z&lt... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 7

The semi-infinite regions \(z<0\) and \(z>1 \mathrm{~m}\) are free space. For \(0

Short Answer

Expert verified
Question: Calculate the standing wave ratios (SWR) for regions 1, 2, and 3, and the location of the maximum electric field magnitude nearest to the \(z=0\) interface for the region \(z<0\), given a uniform plane wave with a frequency of \(\omega=4 \times 10^{8} \mathrm{rad} / \mathrm{s}\) and the following values for permittivity and permeability: Region 1 (\(z<0\)): \(\epsilon_{1}=\epsilon_{0}\), \(\mu_{1}=\mu_{0}\) Region 2 (\(01\) m): \(\epsilon_{3}=\epsilon_{0}\), \(\mu_{3}=\mu_{0}\)

Step by step solution

01

Compute the intrinsic impedances in each region

Given that \(\epsilon_{1}=\epsilon_{0},\mu_{1}=\mu_{0}\) for \(z<0\), \(\epsilon_{2}=4\epsilon_{0},\mu_{2}=\mu_{0}\) for \(01 \mathrm{~m}\). To compute the intrinsic impedance in each region, we'll use the formula: \(\eta=\sqrt{\frac{\mu}{\epsilon}}\). For Region 1 (\(z<0\)): \(\eta_{1}=\sqrt{\frac{\mu_{0}}{\epsilon_{0}}}\) For Region 2 (\(01\) m): \(\eta_{3}=\sqrt{\frac{\mu_{0}}{\epsilon_{0}}}\)
02

Calculate reflection coefficients and transmission coefficients

To compute the reflection coefficients, we'll use the formula: \(\Gamma=\frac{\eta_{2}-\eta_{1}}{\eta_{1}+\eta_{2}}\). At the interface of Region 1 and Region 2, the reflection coefficient is: \(\Gamma_{1}=\frac{\eta_{2}-\eta_{1}}{\eta_{1}+\eta_{2}}\) At the interface of Region 2 and Region 3, the reflection coefficient is: \(\Gamma_{2}=\frac{\eta_{3}-\eta_{2}}{\eta_{2}+\eta_{3}}\) Now, we calculate the transmission coefficients using the formula: \(T=1+\Gamma\) .Transmission coefficient at the interface of Region 1 and Region 2: \(T_{1}=1+\Gamma_{1}\) And for the interface of Region 2 and Region 3: \(T_{2}=1+\Gamma_{2}\)
03

Compute Standing Wave Ratios for each region

Using the reflection coefficient, we can determine the standing wave ratio (SWR) in each region by: \(SWR=\frac{1+|\Gamma|}{1-|\Gamma|}\) For Region 1: \(SWR_{1}=\frac{1+|\Gamma_{1}|}{1-|\Gamma_{1}|}\) For Region 2: \(SWR_{2}=\frac{1+|\Gamma_{3}|}{1-|\Gamma_{3}|}\) For Region 3: \(SWR_{3}=1\) (Since no additional reflections are present for \(z>1\) m)
04

Calculate the location of maximum \(|\mathbf{E}|\) for \(z

To find the location of maximum electric field magnitude for \(z<0\) nearest to the \(z=0\) interface, we'll use the formula: \(z_{max}=\frac{\lambda}{4}(1+SWR_{1})\) First, we calculate the wavelength in Region 1 using the formula: \(\lambda=\frac{2\pi c}{\omega\sqrt{\epsilon_{1}\mu_{1}}}\) Now, we can find the location of the maximum electric field magnitude: \(z_{max}=\frac{\lambda}{4}(1+SWR_{1})\) By following these steps, we can calculate the required SWR for each region and the location of maximum electric field magnitude for the region \(z<0\) near the interface at \(z=0\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electromagnetic Wave Propagation
Understanding the behavior of electromagnetic waves as they move through different media is fundamental in diverse fields such as communication, antenna design, and physics. Electromagnetic wave propagation involves how electric and magnetic fields spread through space. The waves can be reflected, refracted, or absorbed, depending on the medium's properties.

In the given problem, a uniform plane wave travels toward an interface, and we see examples of reflection and transmission because of differences in medium properties. This behavior is influenced significantly by changes in the permittivity \(\epsilon\) and permeability \(\mu\) of the media the wave encounters, which in turn affect the wave's speed and direction.
Reflection Coefficients
When it comes to electromagnetic waves meeting an interface between two media, not all of the wave's energy passes through鈥攕ome of it invariably reflects back into the originating medium. This is where reflection coefficients come into play. They are a measure of the wave's power reflected by an interface relative to the incident power.

Mathematically, the reflection coefficient, represented by \(\Gamma\), can be calculated using the intrinsic impedances of the respective media. The solution to our problem showed that by knowing the intrinsic impedances, \(\eta\), of the two regions involved, one could determine how much of the wave is reflected when it hits the interface. A coefficient closer to 1 means higher reflection, indicating a significant mismatch in media properties, while a value close to 0 indicates minimal reflection.
Transmission Coefficients
In contrast to reflection coefficients, transmission coefficients quantify the part of the electromagnetic wave's power that is successfully transmitted across an interface into the second medium. The transmission coefficient, often denoted by \(T\), complements the reflection coefficient because together they account for the total incident power on an interface.

If we have the reflection coefficient \(\Gamma\), the transmission coefficient can simply be calculated as \(T = 1 + \Gamma\). High transmission implies that the wave continues through the new medium with little power loss, which is ideal in scenarios where signal strength preservation across mediums is critical.
Intrinsic Impedance
A cornerstone concept in understanding electromagnetic waves' interaction with materials is the intrinsic impedance, symbolized as \(\eta\). It represents the material's resistance to electromagnetic wave propagation and is a complex quantity that relates the electric and magnetic fields within a medium.

Intrinsic impedance is not just a property of the material itself but also a function of the frequency of the electromagnetic wave. It's calculated as \(\eta = \sqrt{\frac{\mu}{\epsilon}}\), involving both the permeability (\(\mu\)) and permittivity (\(\epsilon\)) of the medium. As the exercise demonstrates, calculating the intrinsic impedances of different regions allows us to understand how the electromagnetic waves behave at those boundaries, impacting how we define both the reflection and transmission coefficients.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A left-circularly polarized plane wave is normally incident onto the surface of a perfect conductor. (a) Construct the superposition of the incident and reflected waves in phasor form. (b) Determine the real instantaneous form of the result of part ( \(a\) ). \((c)\) Describe the wave that is formed.

A \(T=5\) ps transform-limited pulse propagates in a dispersive medium for which \(\beta_{2}=10 \mathrm{ps}^{2} / \mathrm{km}\). Over what distance will the pulse spread to twice its initial width?

A uniform plane wave is normally incident onto a slab of glass \((n=1.45)\) whose back surface is in contact with a perfect conductor. Determine the reflective phase shift at the front surface of the glass if the glass thickness is (a) \(\lambda / 2 ;(b) \lambda / 4 ;(c) \lambda / 8\).

The plane \(z=0\) defines the boundary between two dielectrics. For \(z<0\), \(\epsilon_{r 1}=9, \epsilon_{r 1}^{\prime \prime}=0\), and \(\mu_{1}=\mu_{0} .\) For \(z>0, \epsilon_{r 2}^{\prime}=3, \epsilon_{r 2}^{\prime \prime}=0\), and \(\mu_{2}=\mu_{0}\) Let \(E_{x 1}^{+}=10 \cos (\omega t-15 z) \mathrm{V} / \mathrm{m}\) and find \((a) \omega ;(b)\left\langle\mathbf{S}_{1}^{+}\right\rangle ;(c)\left\langle\mathbf{S}_{1}^{-}\right\rangle\); (d) \(\left\langle\mathbf{S}_{2}^{+}\right\rangle\).

A wave starts at point \(a\), propagates \(1 \mathrm{~m}\) through a lossy dielectric rated at \(0.1 \mathrm{~dB} / \mathrm{cm}\), reflects at normal incidence at a boundary at which \(\Gamma=0.3+j 0.4\), and then returns to point \(a .\) Calculate the ratio of the final power to the incident power after this round trip, and specify the overall loss in decibels.
See all solutions

Recommended explanations on Physics Textbooks

Radiation

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Mechanics and Materials

Read Explanation

Quantum Physics

Read Explanation

Dynamics

Read Explanation

Medical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.