1. GPS signal transmission through the ionosphere. The Global
Positioning System (GPS) uses frequencies of 1.228 and 1.575 GHz. The
highest electron density in the earth’s ionosphere occurs near the
so-called F-region (at altitudes of ∼300 km) of the ionosphere, reaching
values of Ne = 107 cm−3 during periods of high solar activity. By
modeling the ionosphere as a single layer of 50-km thickness with
uniform electron density Ne, assess the effect of the ionosphere on the
GPS signal. What fraction of a GPS signal vertically incident from space
onto the ionospheric layer is reflected? Get solution

2. Space vehicle reentry. The intense friction around a space vehicle reentering the atmosphere generates a plasma sheath which is 1 meter thick and is characterized by an electron density of Ne = 1013 cm−3, and a collision frequency of ν = 1011 s−1. What frequency is required in order to transmit plane waves through the plasma sheath with a minimum power loss of 10 dB? Get solution

3. Reflection from the ionosphere. Consider the propagation of waves by reflection from the ionosphere as shown in Figure (a) On a particular date and time of day, and above a particular geographic location, the highest electron density occurs at an altitude of ∼400 km, and the plasma frequency at this altitude is fp ≃ 5 MHz. Neglecting any ionization below this densest layer (i.e., assume electron density to be zero below 400 km altitude), determine the greatest possible angle of incidence θi at which an electromagnetic wave originating at a ground-based transmitter could possibly strike the layer. (b) How far away from the transmitter will the reflected radiation return to earth? (c) What is the highest frequency at which this obliquely incident radiation will be totally reflected?Figure Reflection from the ionosphere. Problem.... Get solution

4. Attenuation rate in the ionosphere. Show that the effective permittivity of a plasma with collisions can be expressed as...where ν is the collision frequency and ωp is the plasma frequency. Using this expression to find a simple approximation for the attenuation rate (in dB-m−1) for a 30 MHz wave passing through the lower ionosphere (assume ν ≃ 107 s−1, Ne ≃ 108 m−3). Get solution

5. Lossy LH Media. (a) In a lossy LH medium with Re{є}0 and Re{μ}0, show that Im{β2} = Im{ω2μє}0. (b) In a lossy RH medium with Re{є}0 and Re{μ}0, show that Im{β2} = Im{ω2μ_}0. Get solution

6. Lossy LH Media. In a lossy LH medium with Re{_}0 and Re{μ}0, use the result from part(a) of Problem to show that Re{β}Im{β}0.Lossy LH Media. (a) In a lossy LH medium with Re{є}0 and Re{μ}0, show that Im{β2} = Im{ω2μє} 0. (b) In a lossy RH medium with Re{є}0 and Re{μ}0, show that Im{β2} = Im{ω2μє}0 Get solution

7. Lorentz model of a metamaterial. In Section 11.2.1, we developed a model for the relative permittivity by considering the dynamics of a bound electron in the presence of an external electric field. The formula for єr given by (11.28) is called the Lorentz model. Using (11.5), we can rewrite (11.28a) as...where we added the subscript e to associate the characteristic frequencies with the bound electron model. A similar model may be used to approximate the permeability of artificially constructed metamaterials:28...(a) First consider the lossless case, where κe = κm = 0. Using ωpe = 2π × 8.0× 109 rad-s−1, ωpm = 2π × 7.0 × 109 rad-s−1, ω0e = 2π × 2.8 × 109 rad-s−1, and ω0m = 2π × 2.5 × 109 rad-s−1: (i) Plot the real and imaginary components of єr and μr in the frequency range 4 f11 GHz. (ii) Plot the real and imaginary components of the index of refraction n in the same frequency range. Use the convention that Re{β} 0 for backward propagation. (iii) In the frequency range 4 f 11 GHz, which frequencies allow for wave propagation? What is the frequency range of backward wave propagation? At what frequency is Re{n} = −1? (iv) Plot βc versus f over the domain 4 f 11 GHz, and comment on the sign of the phase and group velocities over frequencies where propagating solutions exist. (b) Repeat part (a) using κe = κm = 0.05ωpe . Get solution

2. Space vehicle reentry. The intense friction around a space vehicle reentering the atmosphere generates a plasma sheath which is 1 meter thick and is characterized by an electron density of Ne = 1013 cm−3, and a collision frequency of ν = 1011 s−1. What frequency is required in order to transmit plane waves through the plasma sheath with a minimum power loss of 10 dB? Get solution

3. Reflection from the ionosphere. Consider the propagation of waves by reflection from the ionosphere as shown in Figure (a) On a particular date and time of day, and above a particular geographic location, the highest electron density occurs at an altitude of ∼400 km, and the plasma frequency at this altitude is fp ≃ 5 MHz. Neglecting any ionization below this densest layer (i.e., assume electron density to be zero below 400 km altitude), determine the greatest possible angle of incidence θi at which an electromagnetic wave originating at a ground-based transmitter could possibly strike the layer. (b) How far away from the transmitter will the reflected radiation return to earth? (c) What is the highest frequency at which this obliquely incident radiation will be totally reflected?Figure Reflection from the ionosphere. Problem.... Get solution

4. Attenuation rate in the ionosphere. Show that the effective permittivity of a plasma with collisions can be expressed as...where ν is the collision frequency and ωp is the plasma frequency. Using this expression to find a simple approximation for the attenuation rate (in dB-m−1) for a 30 MHz wave passing through the lower ionosphere (assume ν ≃ 107 s−1, Ne ≃ 108 m−3). Get solution

5. Lossy LH Media. (a) In a lossy LH medium with Re{є}0 and Re{μ}0, show that Im{β2} = Im{ω2μє}0. (b) In a lossy RH medium with Re{є}0 and Re{μ}0, show that Im{β2} = Im{ω2μ_}0. Get solution

6. Lossy LH Media. In a lossy LH medium with Re{_}0 and Re{μ}0, use the result from part(a) of Problem to show that Re{β}Im{β}0.Lossy LH Media. (a) In a lossy LH medium with Re{є}0 and Re{μ}0, show that Im{β2} = Im{ω2μє} 0. (b) In a lossy RH medium with Re{є}0 and Re{μ}0, show that Im{β2} = Im{ω2μє}0 Get solution

7. Lorentz model of a metamaterial. In Section 11.2.1, we developed a model for the relative permittivity by considering the dynamics of a bound electron in the presence of an external electric field. The formula for єr given by (11.28) is called the Lorentz model. Using (11.5), we can rewrite (11.28a) as...where we added the subscript e to associate the characteristic frequencies with the bound electron model. A similar model may be used to approximate the permeability of artificially constructed metamaterials:28...(a) First consider the lossless case, where κe = κm = 0. Using ωpe = 2π × 8.0× 109 rad-s−1, ωpm = 2π × 7.0 × 109 rad-s−1, ω0e = 2π × 2.8 × 109 rad-s−1, and ω0m = 2π × 2.5 × 109 rad-s−1: (i) Plot the real and imaginary components of єr and μr in the frequency range 4 f11 GHz. (ii) Plot the real and imaginary components of the index of refraction n in the same frequency range. Use the convention that Re{β} 0 for backward propagation. (iii) In the frequency range 4 f 11 GHz, which frequencies allow for wave propagation? What is the frequency range of backward wave propagation? At what frequency is Re{n} = −1? (iv) Plot βc versus f over the domain 4 f 11 GHz, and comment on the sign of the phase and group velocities over frequencies where propagating solutions exist. (b) Repeat part (a) using κe = κm = 0.05ωpe . Get solution