Chapter 34, Guided Waves Ii Video Solutions, Electromagnetic Fields and Waves: Including Electric Circuits

Problem 1

An electromagnetic wave propagating in the $\mathrm{TE}_{1}$ mode in a $34.0 \times$ $72.1$ millimeter rectangular waveguide has a wavelength $\lambda_{z}$ of 138 millimeters. Calculate its frequency.

Prachita Kush

Numerade Educator

00:43

Problem 2

Figure $34-10$ shows three sides of a rectangular guide that is split open and flattened. Draw a figure like this and show, on face $B$, lines of $\boldsymbol{H}$, electric charges, and vectors $\boldsymbol{E} \times \boldsymbol{H}$ at a given instant. Then add lines of current on all three faces.

Mayukh Banik

Numerade Educator

02:31

Problem 3

According to Table $34-1$, the recommended range of operating frequencies for a rectangular waveguide with a cross section of $34.0 \times$ $72.1$ millimeters extends from $2.61$ to $3.95$ gigahertz. Calculate the values of $\theta$ at both ends of this range.

Suhas Katkar

Numerade Educator

05:08

Problem 4

If a load is not properly matched to a waveguide, part of the incident wave turns back and there is a standing wave along the guide. Then only a fraction of the power available at the source reaches the load. It is therefore useful to be able to move a small probe along a longitudinal slot to sample the field inside the guide. The voltage standing wave ratio (VSWR) is the ratio of the maximum to the minimum time-averaged rms voltage measured at the probe. Under ideal conditions there is no reflected wave, and the VSWR is equal to unity.

The probe can be a short length of wire that responds to the electric field or a small loop coupled to the magnetic field. The probe projects into the field by about 1 millimeter.
(a) With the $\mathrm{TE}_{1}$ mode, where should the slot be cut to disturb the wave as little as possible?
(b) If the probe is a loop, how should it be oriented?
(c) How would you proceed to measure the wavelength of the guided wave?
(d) If the VSWR is equal to 2 , what is the value of the ratio $E_{\text {reftected }} / E_{\text {incident }} ?$

Vishal Gupta

Numerade Educator

03:40

Problem 5

Figure $34-11$ shows a set of parallel conducting plates uniformly spaced by a distance $s$. If $s / \lambda$ has the correct value and if the incident wave is correctly polarized, this medium acts as an artificial dielectric whose index of refraction is less than unity.
(a) What must be the orientation of $\boldsymbol{E} ?$
(b) Find the index of refraction as a function of the ratio $s / \lambda_{0}$.
(c) Show rays deflected by (i) a prism and (ii) a converging cylindrical lens made in this way.

Keshav Singh

Numerade Educator

05:39

Problem 6

A rectangular metallic waveguide $A$ is air-filled, and its cross section is $a, b$. Another rectangular metallic waveguide $D$ is filled with a dielectric $\epsilon_{r}$, and its cross section is $a / \epsilon_{r}^{1 / 2}, b / \epsilon_{r}^{1 / 2}$.
(a) Show that waveguides $A$ and $D$ have the same cutoff frequency and that, at a given frequency, $\lambda_{z D}=\lambda_{z A} / \epsilon_{r}^{1 / 2}$. Thus, for a given operating frequency, a dielectric-filled guide is smaller than an air-filled one. With Teflon $\left(\epsilon_{r}=2.1\right)$, both dimensions $a$ and $b$ are smaller by a factor of $1.45$. Also, the phase and signal velocities are smaller by a factor of $1.45$.
(b) Compare the power ratings of $A$ and $D$ at a given frequency.
The dielectric strength of a material is the maximum permissible value of $E$ before breakdown. The dielectric strength of a good dielectric such as Teflon is of the order of 10 times that of air. Set this ratio equal to $R$. The value of $R$ increases as the thickness decreases.

Keshav Singh

Numerade Educator

01:23

Problem 7

(a) Find $\lambda_{0} / \lambda_{z}$ as a function of $\lambda_{0} / \lambda_{c}$ for a rectangular waveguide, where $\lambda_{c}$ is the cutoff wavelength.
(b) Find $\omega \lambda_{c} / c$ as a function of $k_{z} \lambda_{c} .$ For a given frequency,
$$
\frac{v_{p}}{c}=\frac{\omega}{k_{z}}=\frac{\omega \lambda_{c}}{c} k_{z} \lambda_{c}
$$
The group, or signal, velocity is given by
$$
\frac{v_{R}}{c}=\frac{d\left(\omega \lambda_{c} / c\right)}{d\left(k_{z} \lambda_{c}\right)}
$$

Ankur S

Numerade Educator

01:14

Problem 8

Show that, in a hollow rectangular waveguide,
$$
v_{g}=\frac{1}{d k_{z} / d \omega}
$$

Urvashi Arora

Numerade Educator

02:14

Problem 9

One can measure the power transmitted down a rectangular metallic waveguide by reading the voltage induced in a tiny loop projecting into the guide, as in Fig. 34-12. The loop is situated at $x=a / 2, y=0$, and it lies in the $y z$-plane.
Show that
$$
P_{T, \mathrm{av}}=8.40 \times 10^{-6} \frac{\lambda_{0}^{2} a b}{\left[1-\lambda_{0}^{2} /\left(4 a^{2}\right)\right]^{1 / 2}} \frac{\mathcal{V}^{2}}{\mathscr{A}^{2}}
$$
where $\mathcal{V}$ is the rms voltage induced in the loop and $\mathscr{A}$ is the area of the loop. The effective value of $\mathscr{A}$ is unknown, but the method is satisfactory for measuring relative values of $P_{T}$.

Dominador Tan

Numerade Educator

01:43

Problem 10

(a) Show that, upon reflection from a good nonmagnetic conductor, in air, the amplitude of an electromagnetic wave decreases by a factor of approximately $1-\left(2 \epsilon_{0} \omega / \sigma\right)^{1 / 2} \cos \theta_{I}$ if $\boldsymbol{E}$ is normal to the plane of incidence.
(b) Show that this is in agreement with the attenuation calculated in Sec. $34.8$.

Ajay Singhal

Numerade Educator

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Problem 11

(a) Calculate the maximum power that can be carried by a coaxial line and by a rectangular metallic waveguide at $3.00$ gigahertz. The coaxial line has a diameter $\rho_{2}$ of $25.0$ millimeters, and the guide has an inside cross section of $37.5 \times 75.0$ millimeters. The coaxial line satisfies the condition for maximum power transfer, namely, $\rho_{2} / \rho_{1}=1.65$, and its outside radius is small enough to ensure the attenuation of higher-order modes. Both lines are air-filled, and the current-carrying surfaces are silver-plated. The maximum allowed $E$ is $1.5$ megavolts/meter. (Under ideal conditions the breakdown field at 3 gigahertz is about $10^{8}$ volts/meter.) There is no reflected wave.
(b) Calculate the power dissipation per meter in both cases. Clearly, these lines can operate at these power levels only during short pulses.
(c) Calculate the rms voltage and current at the input end of the coaxial line.

Victor Salazar

Numerade Educator

01:42

Problem 12

Attenuation on a transmission line is expressed in decibels per meter. The number of decibels per meter is 20 times the logarithm to the base 10 of the ratio of the $E$ 's (or the H's, or the voltages, or the currents) at the two ends of a line 1 meter long. The degree of attenuation is also expressed in nepers per meter, and this is simply the value of $\beta$.
Show that 1 neper/meter is equivalent to $8.686$ decibels/meter.

Abhishek Kumar

Numerade Educator