Question

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

Short answer

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

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): \(\eta_{3}=\sqrt{\frac{\mu_{0}}{\epsilon_{0}}}\)

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: \(T_{2}=1+\Gamma_{2}\)

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)

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 \(z=0\).

Key concepts

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—some 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.