1.12: Exercises

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1. The plane wave electric field of a radar pulse propagating in free space in the $$z$$ direction is given by (with $$\omega_{0} ≫ π/(2\tau))$$
$\overline{\mathcal{E}}(t) =\left\{\begin{array}{cc}{A\cos\left(\frac{\pi t}{2\tau}\right)\cos(\omega_{0}t)\hat{\mathbf{x}}}&{-\tau\leq t\leq\tau}\\{0}&{otherwise}\end{array}\right.\nonumber$
1. Sketch $$\overline{\mathcal{E}}(t)$$ over the interval $$−1.5\tau ≤ t ≤ 1.5\tau$$.
2. What is, $$\overline{\mathcal{H}}(t)$$.
3. What is the Poynting vector, $$\overline{\mathcal{S}}(t)$$.
4. Determine the total energy density in the pulse (in units of $$\text{J/m}^{2}$$).
2. Consider a material with a relative permittivity of $$72$$, a relative permeability of $$1$$, and a static electric field ($$E$$) of $$1\text{ kV/m}$$. How much energy is stored in the $$E$$ field in a $$10\text{ cm}^{3}$$ volume of the material? [Parallels Example 1.5.1]
3. A time-varying electric field in the $$x$$ direction has a strength of $$1\text{ kV/m}$$ and a frequency of $$1\text{ GHz}$$. The medium has a relative permittivity of $$70$$. What is the time-domain polarization vector? [Parallels Example 1.5.2]
4. A medium has a dielectric constant of $$20$$. What is the index of refraction?
5. Consider a plane EM wave propagating in a medium with a permittivity $$\varepsilon$$ and permeability $$\mu$$. The complex permittivity measured at two frequencies is characterized by the relative permittivity $$(\varepsilon /\varepsilon_{0})$$ of the real $$\Re\{\varepsilon/\varepsilon_{0}\}$$ and imaginary parts $$ℑ \{\varepsilon/\varepsilon_{0}\}$$ as follows:
Measured relative permittivity:
Frequency Real Part Imaginary Part
$$1\text{ GHz}$$ $$3.8$$ $$-0.05$$
$$10\text{ GHz}$$ $$4.0$$ $$-0.03$$
Table $$\PageIndex{1}$$
Measured relative permeability:
Frequency Real Part Imaginary Part
$$1\text{ GHz}$$ $$0.999$$ $$-0.001$$
$$10\text{ GHz}$$ $$0.998$$ $$-0.001$$
Table $$\PageIndex{2}$$
Since there is an imaginary part of the dielectric constant there could be either dielectric damping or material conductivity, or both. [Parallels Example 1.9.1]
1. Determine the dielectric loss tangent at $$10\text{ GHz}$$.
2. Determine the relative dielectric damping factor at $$10\text{ GHz}$$ (the part of the permittivity due to dielectric damping).
3. What is the conductivity of the dielectric at $$10\text{ GHz}$$?
6. A medium has a relative permittivity of $$13$$ and supports a $$5.6\text{ GHz}$$ EM signal. By default, if not specified otherwise, a medium is lossless and will have a relative permeability of $$1$$.
1. Calculate the characteristic impedance of an EM plane wave.
2. Calculate the propagation constant of the medium.
7. A plane wave in free space is normally incident on a lossless medium occupying a half space with a dielectric constant of $$12$$. [Parallels Example 1.9.2]
1. Calculate the electric field reflection coefficient referred to the interface medium.
2. What is the magnetic field reflection coefficient?
8. Water, or more specifically tap water or sea water, has a complex dielectric constant resulting from two effects: conductivity resulting from dissolved ions in the water leading to charge carriers that can conduct current under the influence of an electric field, and dielectric loss resulting from rotation or bending of the water molecules themselves under the influence of an electric field. The rotation or bending of the water molecules results in motion of the water molecules and thus heat. The relative permittivity is
$\label{eq:1}\varepsilon_{wr}=\varepsilon_{wr}'-\jmath\varepsilon_{wr}''$
and the real and imaginary components are
\begin{align}\label{eq:2}\varepsilon_{wr}'&=\varepsilon_{w00}+\frac{\varepsilon_{w0}-\varepsilon_{w00}}{1+(2\pi f\tau_{w})^{2}} \\ \label{eq:3}\varepsilon_{wr}''&=\frac{2\pi f\tau_{w}(\varepsilon_{w0}-\varepsilon_{w00})}{1+(2\pi f\tau_{w})^{2}}+\frac{\sigma_{i}}{2\pi\varepsilon_{0}f}\end{align}
where $$f$$ is frequency, $$\varepsilon_{0} = 8.854\times 10^{−12}\text{ F/m}$$ is the permittivity magnitude, and $$\varepsilon_{w00} = 4.9$$ is the high-frequency limit of $$\varepsilon_{r}$$ and is known to be independent of salinity and is also that of pure deionized water. The other quantities are τw, the relaxation time of water; $$\varepsilon_{w0}$$, the static relative dielectric constant of water (which for pure water at $$0^{\circ}\text{C}$$ is equal to $$87.13$$); and $$\sigma_{i}$$, the ionic conductivity of the water. At $$5^{\circ}\text{C}$$ and for tap water with a salinity of $$0.03$$ (in parts per thousand by weight) [17, 18], $$\varepsilon_{w0} = 85.9,\: \tau_{w} = 14.6\text{ ps}$$, and $$\sigma_{i} = 0.00548\text{ S/m}$$.
1. Calculate and then plot the real and imaginary dielectric constant of water over the frequency range from DC to $$40\text{ GHz}$$ for tap water at $$5^{\circ}\text{C}$$. You should see two distinct regions. Identify the part of the imaginary dielectric constant graph that results from ionic conductivity and the part of the graph that relates to dielectric relaxation.
2. Consider a plane wave propagating in water. Determine the attenuation constant and the propagation constant of the wave at $$500\text{ MHz}$$.
3. Consider a plane wave propagating in water. Determine the attenuation constant and the propagation constant of the wave at $$2.45\text{ GHz}$$.
4. Consider a plane wave propagating in water. Determine the attenuation constant and the propagation constant of the wave at $$22\text{ GHz}$$.
5. Again considering the plane wave in water, calculate the loss factor over a distance of $$1\text{ cm}$$ at $$2.45\text{ GHz}$$. Express your answer in terms of decibels.
6. Consider a plane wave making normal incidence from above into a bucket of tap water that is $$1\text{ m}$$ deep so that you do not need to consider the reflection at the bottom of the bucket. Calculate the reflection coefficient of the incident plane wave at $$2.45\text{ GHz}$$ referred to the air immediately above the surface of the water.
7. Again consider the situation in (f), but this time with the plane wave incident on the air–water interface from the water side, calculate the reflection coefficient of the incident wave at $$2.45\text{ GHz}$$ referred to the water immediately below the surface of the water.
8. If the bucket in (f) and (g) is $$0.5\text{ cm}$$ deep you will need to consider the reflection at the bottom of the bucket. Consider that the bottom of the bucket is a perfect conductor. Draw the bounce diagram for calculating the reflection of the plane wave in air. Determine the total reflection coefficient of the plane wave (referred to the air immediately above the surface of the water) by using the bounce diagram.
9. Repeat (h) using the formula for multiple reflections.
9. A $$4\text{ GHz}$$ time-varying EM field is traveling in the $$+z$$ direction in Region 1 and is incident on another material in Region 2. The permittivity of Region 1 is $$\varepsilon_{1} =\varepsilon_{0}$$ and that of Region 2 is $$\varepsilon_{2} = (4−\jmath 0.04)\varepsilon_{0}$$. For both regions $$\mu_{1} =\mu_{2} =\mu_{0}$$. [Parallels Example 1.9.4]
1. What is the characteristic impedance (or wave impedance) in Region 2?
2. What is the propagation constant in Region 1?
10. A $$4\text{ GHz}$$ time-varying EM field is traveling in the $$+z$$ direction in Region 1 and is normally incident on another material in Region 2. The boundary between the two regions is in the $$z = 0$$ plane. The permittivity of Region 1 is $$\varepsilon_{1} =\varepsilon_{0}$$, and that of Region 2 is $$\varepsilon_{2} = 4\varepsilon_{0}$$. For both regions, $$\mu_{1} =\mu_{2} =\mu_{0}$$. The phasor of the forward-traveling electric field (i.e., the incident field) is $$\overline{\text{ffl}}$$ $$E+ = 100\hat{\mathbf{y}}\text{ V/m}$$ and the phase is normalized with respect to $$z = 0$$. $$Q_{0} = 0$$. [Parallels Example 1.9.4]
1. What is the wave impedance of Region 1?
2. What is the wave impedance of Region 2?
3. What is the electric field reflection coefficient at the boundary?
4. What is the magnetic field reflection coefficient at the boundary?
5. What is the electric field transmission coefficient at the boundary?
6. What is the power transmitted into Region 2?
7. What is the power reflected from the boundary back into Region 1?

1.12.1 Exercises By Section

$$†$$challenging, $$‡$$very challenging

$$§1.5\: 1†, 2†, 3†$$

$$§1.7\: 4, 5†$$

$$§1.9\: 6, 7†, 8‡, 9†, 10†$$

1. $$4.47$$
1. (c) $$\gamma=49+\jmath 468\text{ m}^{-1}$$
2. $$0.84+\jmath 168\text{ m}^{-1}$$