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Chapter 8: Problem 9

The treatment in the text tacitly assumes that the interior of the waveguide has the electromagnetic properties of vacuum. Show that if the waveguide is filled with a material of relative permittivity \(\pi_{e}\) and permeability \(\pi_{m}\), all equations remain valid if \(c\) is replaced by \(c^{\prime}\) of \((8.2 .21)\) and \(\lambda_{9}\) in \((8.7 .20)\) is replaced by \(\lambda^{\prime}=2 \pi c^{\prime} / \omega\).

Short Answer

Expert verified
In conclusion, the waveguide equations remain valid when filled with a medium having relative permittivity \(\pi_{e}\) and permeability \(\pi_{m}\), as long as the speed of light in the medium (\(c^{\prime}\)) and the wavelength in the medium (\(\lambda^{\prime}\)) are substituted accordingly. The wave equation, which is the basis of these waveguide equations, holds true for any medium, provided the appropriate substitutions are made. Therefore, the given waveguide equations are still applicable in the case of a medium with different electromagnetic properties.

Step by step solution

01

Finding the speed of light in the medium (\(c^{\prime}\))

According to the given information, the relative permittivity is \(\pi_{e}\) and the relative permeability is \(\pi_{m}\). The speed of light in the medium, \(c^{\prime}\), can be found using the relationship: \(c^{\prime} = \frac{c}{\sqrt{\pi_{e}\pi_{m}}}\)
02

Expressing \(\lambda^{\prime}\) in terms of \(c^{\prime}\) and \(\omega\)

We are given that \(\lambda^{\prime} = 2 \pi c^{\prime} / \omega\). We can substitute the expression of \(c^{\prime}\) from Step 1 to obtain: \(\lambda^{\prime} = \frac{2 \pi c}{\omega\sqrt{\pi_{e}\pi_{m}}}\)
03

Checking the validity of the equations

Now that we have the expressions for \(c^{\prime}\) and \(\lambda^{\prime}\), we need to verify whether the given equations still hold when these substitutions are made. We can safely assume that if the waveguide equations are derived without any approximations, they should hold for any medium, as long as the appropriate substitutions are made. In general, the wave equation in a medium with permittivity \(\epsilon\) and permeability \(\mu\) is given by: \(\nabla^{2} \mathbf{E} - \frac{1}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0\) where \(\epsilon = \epsilon_0 \pi_{e}\) and \(\mu = \mu_0 \pi_{m}\), with \(\epsilon_0\) and \(\mu_0\) being the vacuum permittivity and permeability, respectively. We can rewrite the wave equation using \(c^{\prime}\) and \(\lambda^{\prime}\) as: \(\nabla^{2} \mathbf{E} - \frac{1}{c^{\prime 2}} \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0\) Notice how the equation still holds, as long as we substitute \(c\) with \(c^{\prime}\) and \(\lambda_{9}\) with \(\lambda^{\prime}\). This provides evidence that the equations remain valid, even when the medium inside the waveguide has electromagnetic properties that differ from a vacuum.

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Most popular questions from this chapter

Practical coaxial lines used for the distribution of high-frequency signals often consist of a thin copper wire in a polyethylene sleeve on which a copper braid is woven (usually there is also a protective plastic jacket over the braid). Commercial lines are made with nominal characteristic impedances of 50,75 , or 90 ohms. A common 50 -ohm variety has a center conductor of diameter 0035 in. The dielectric constant of polyethylene is \(2.3\) What is the nominal (inside) diameter of the copper braid? What are the capacitance and inductance per foot? What is the speed of propagation, expressed as a percent of the velocity of light? Arswer: \(0120 \mathrm{in} ; 30 \mathrm{pF} / \mathrm{ft} ; 0074 \mu \mathrm{H} / \mathrm{ft} ; 66\) percent.

(a) From \((8.7 .10)\) and \((8.7 .13)\), show that the guided ave impedance \(\left(E_{x}{ }^{2}+E_{y}{ }^{2}\right)^{1 / 2} /\) \(\left(H_{x}{ }^{2}+H_{z}{ }^{2}\right)^{1 / 2}\) is \(Z_{\mathrm{TE}}=\frac{Z_{0}}{\left[1-\left(\lambda_{0} / \lambda_{e}\right)^{2}\right]^{1 / 2}} \quad\) TE modes \(Z_{\text {T? }}=Z_{0}\left[1-\left(\lambda_{0} / \lambda_{c}\right)^{2}\right]^{1 / 2} \quad\) TM modes, where \(Z_{0}\) is the unbounded wave impedance \((8.3 .10)\) or, more generally, \((8.3 .12) .(b)\) For the TE o dominant mode in rectangular waveguide, show that the peak potential difference between opposite points in the cross section is $$ V_{0}=\left[\int_{0}^{b} E_{y}\left(x=\frac{1}{2} a\right) d y\right]_{p e s k}=b E_{0} $$ and that the peak axial current flowing in the top wall is $$ I_{0} \equiv\left[\frac{1}{\mu_{0}} \int_{0}^{a} B_{x}(y=b) d x\right]_{\text {pak }}=\frac{2 a E_{0}}{\pi Z_{\mathrm{TE}}} $$ Since the result of Prob. 8.7.10b can be written $$ \bar{P}=\frac{a b}{4} \frac{E_{0}{ }^{2}}{Z_{\mathrm{TB}}} $$ we can define three other (mode-dependent) waveguide impedances as follows: $$ \begin{aligned} &Z_{V, I}=\frac{V_{0}}{I_{0}}=\frac{\pi}{2}\left(\frac{b}{a} Z_{\mathrm{TE}}\right) \\ &Z_{P, V}=\frac{V_{0}{ }^{2}}{2 P}=2\left(\frac{b}{a} Z_{\mathrm{TE}}\right) \\\ &Z_{P, I}=\frac{2 \bar{P}}{I_{0}{ }^{2}}=\frac{\pi^{2}}{8}\left(\frac{b}{a} Z_{\mathrm{TE}}\right) \end{aligned} $$ which differ by small numerical factors. Only systems supporting a TEM mode (e.g., Sec. 8.1), have a unique impedance.

A long straight wire of radius a carries a current \(I\) and has resistance \(R_{1}\) per unit length. Compute \(\mathbf{E}\) and \(\mathbf{B}\) at its surface and show that the rate of energy flow into the wire via the Poynting flux is \(I^{2} R_{1}\) per unit length.

Show that the resistive and reactive parts of an unknown load impedance \(\breve{Z}_{i}=\) \(R_{l}+j X_{1}\) are given by $$ \begin{aligned} &R_{l}=Z_{9} \frac{1-|\not{R}|^{2}}{1-2|\not{R}| \cos \phi+|\vec{R}|^{2}} \\ &X_{1}=Z_{0} \frac{2|\not{R}| \sin \phi}{1-2|\vec{R}| \cos \phi+|\vec{R}|^{2}} \end{aligned} $$ where \(|\not{R}|\) and \(\phi\) specify the complex reflection coeflicient \(R\) and \(Z_{0}\) is the characteristic impedance. Note: See Prob. 1.4.3.

Show that $$ \begin{aligned} &\mathbf{E}=(1+j j) E_{1} e^{j(\omega t-\alpha o)} \\ &\mathbf{B}=(-1 j+j) \frac{E_{1}}{c} e^{j(\omega t-\alpha t)} \end{aligned} $$ represent a circularly polarised plane wave. (Note that \(j=\sqrt{-1}\), while \(\mathbf{1}, \mathbf{j}\) are the cartesian unit vectors in the \(x\) and \(y\) directions!) If you watch the time variation of the electric field at a fixed position, will the direction of the field rotate in the right-or left-handed sense with respect to the direction of travel \((+z) ?\) If you could take a snapshot of the electric field over space, in which sense would the direction rotate? Repeat these questions for the magnetic field. How would you represent a circularly polarized wave of the opposite handedness? Answer: Left-handed; right-handed; magnetic same as electric; reverse sign of \(j\) in coefficients.
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