# Problems

# Section 8.1: Reflection and Transmission at Normal Incidence

1. *8.1 A plane wave in air with an electric field amplitude of 20 V/m is incident normally upon the surface of a lossless, nonmagnetic medium with ε r=25. Determine:

   1. the reflection and transmission coefficients,

   2. the standing-wave ratio in the air medium, and

   3. the average power densities of the incident, reflected, and transmitted waves.

2. 8.2 A plane wave traveling in medium 1 with ε r 1=2.25 is normally incident upon medium 2 with ε r 2=4. Both media are made of nonmagnetic, nonconducting materials. The electric field of the incident wave is given by

   E i=y ˆ 8 cos(6 π×10 9 t−30 π x)(V/m).

   1. Obtain time-domain expressions for the electric and magnetic fields in each of the two media.

   2. Determine the average power densities of the incident, reflected and transmitted waves.

3. 8.3 A plane wave traveling in a medium with ε r 1=9 is normally incident upon a second medium with ε r 2=4. Both media are made of nonmagnetic, nonconducting materials. The magnetic field of the incident plane wave is given by

   H i=z ˆ 2 cos(2 π×10 9 t−k y)(A/m).

   1. a.Obtain time-domain expressions for the electric and magnetic fields in each of the two media.

   2. *b.Determine the average power densities of the incident, reflected, and transmitted waves.

4. 8.4 A 200 MHz, left-hand circularly polarized plane wave with an electric field modulus of 5 V/m is normally incident in air upon a dielectric medium with ε r=4 and occupies the region defined by z≥0.

   1. Write an expression for the electric field phasor of the incident wave given that the field is a positive maximum at z=0 and t=0.

   2. Calculate the reflection and transmission coefficients.

   3. Write expressions for the electric field phasors of the reflected wave, the transmitted wave, and the total field in the region z≤0.

   4. Determine the percentages of the incident average power reflected by the boundary and transmitted into the second medium.

5. 8.5 Repeat Problem 8.4, but replace the dielectric medium with a poor conductor characterized by ε r=2.25, μ r=1, and σ=10−4 S/m.

6. 8.6 A 50 MHz plane wave with electric field amplitude of 50 V/m is normally incident in air onto a semi-infinite, perfect dielectric medium with ε r=36. Determine:

   1. *a.Γ,

   2. b.the average power densities of the incident and reflected waves, and

   3. c.the distance in the air medium from the boundary to the nearest minimum of the electric field intensity, |E|.

7. *8.7 What is the maximum amplitude of the total electric field in the air medium of Problem 8.6, and at what nearest distance from the boundary does it occur?

8. 8.8 Repeat Problem 8.6, but replace the dielectric medium with a conductor with ε r=1, μ r=1, and σ=2.78×10−3 S/m.

9. *8.9 The three regions shown in Fig. P8.9 contain perfect dielectrics. For a wave in medium 1, incident normally upon the boundary at z=−d, what combination of ε r 2 and d produces no reflection? Express your answers in terms of ε r 1, ε r 3 and the oscillation frequency of the wave, f.

   # Figure P8.9 Dielectric layers for Problems 8.9 to 8.11.

   Figure P8.9 Full Alternative Text

10. 8.10 For the configuration shown in Fig. P8.9, use transmission-line equations (or the Smith chart) to calculate the input impedance at z=−d for ε r 1=1, ε r 2=9, ε r 3=4, d=1.2 m, and f=50 MHz. Also determine the fraction of the incident average power density reflected by the structure. Assume all media are lossless and nonmagnetic.

11. *8.11 Repeat Problem 8.10, but interchange ε r 1 and ε r 3.

12. 8.12 Orange light of wavelength 0.61 μ m in air enters a block of glass with ε r=1.44. What color would it appear to a sensor embedded in the glass? The wavelength ranges of colors are violet (0.39 to 0.45 μ m), blue (0.45 to 0.49 μ m), green (0.49 to 0.58 μ m), yellow (0.58 to 0.60 μ m), orange (0.60 to 0.62 μ m), and red (0.62 to 0.78 μ m).

13. *8.13 A plane wave of unknown frequency is normally incident in air upon the surface of a perfect conductor. Using an electric-field meter, it was determined that the total electric field in the air medium is always zero when measured at a distance of 2 m from the conductor surface. Moreover, no such nulls were observed at distances closer to the conductor. What is the frequency of the incident wave?

14. 8.14 Consider a thin film of soap in air under illumination by yellow light with λ=0.6 μ m in vacuum. The film is treated as a planar dielectric slab with ε r=1.72 and is surrounded on both sides by air. What film thickness would produce strong reflection of the yellow light at normal incidence?

15. *8.15 A 5 MHz plane wave with electric field amplitude of 10 (V/m) is normally incident in air onto the plane surface of a semi-infinite conducting material with ε r=4, μ r=1, and σ=100 (S/m). Determine the average power dissipated (lost) per unit cross-sectional area in a 2 mm penetration of the conducting medium.

16. 8.16 A 0.5 MHz antenna carried by an airplane flying over the ocean surface generates a wave that approaches the water surface in the form of a normally incident plane wave with an electric-field amplitude of 3,000 (V/m). Seawater is characterized by ε r=72, μ r=1, and σ=4 (S/m). The plane is trying to communicate a message to a submarine submerged at a depth d below the water surface. If the submarine’s receiver requires a minimum signal amplitude of 0.01 (μ V/m), what is the maximum depth d where successful communication is still possible?

# Sections 8.2 and 8.3: Snell’s Laws and Fiber Optics

1. *8. 17A light ray is incident on a prism in air at an angle θ, as shown in Fig. P8.17. The ray is refracted at the first surface and again at the second surface. In terms of the apex angle ϕ of the prism and its index of refraction n, determine the smallest value of θ for which the ray will emerge from the other side. Find this minimum θ for n=1.4 and ϕ=60°.

   # Figure P8.17 Prism of Problem 8.17.

   Figure P8.17 Full Alternative Text

2. 8.18 For some types of glass, the index of refraction varies with wavelength. A prism is made of a material with

   n=1.71−4 30 λ 0(λ 0 in μ m),

   where λ 0 is the wavelength in vacuum, and is used to disperse white light, as shown in Fig. P8.18. The white light is incident at an angle of 50°, the wavelength λ 0 of red light is 0.7 μ m, and that of violet light is 0.4 μ m. Determine the angular dispersion in degrees.

   # Figure P8.18 Prism of Problem 8.18.

   Figure P8.18 Full Alternative Text

3. *8.19 The two prisms in Fig. P8.19 are made of glass with n=1.5. What fraction of the power density carried by the ray incident upon the top prism emerges from the bottom prism? Neglect multiple internal reflections.

   # Figure P8.19 Periscope prisms of Problem 8.19.

   Figure P8.19 Full Alternative Text

4. 8.20 A parallel-polarized plane wave is incident from air at an angle θ i=30° onto a pair of dielectric layers, as shown in Fig. P8.20.

   1. Determine the angles of transmission θ 2, θ 3, and θ 4.

   2. Determine the lateral distance d.

   # Figure P8.20 Problem P8.20.

   Figure P8.20 Full Alternative Text

5. 8.21 A light ray incident at 45° passes through two dielectric materials with the indices of refraction and thicknesses given in Fig. P8.21. If the ray strikes the surface of the first dielectric at a height of 2 cm, at what height will it strike the screen?

   # Figure P8.21 Light incident on a screen through a multilayered dielectric (Problem 8.21).

   Figure P8.21 Full Alternative Text

6. *8.22 Figure P8.22 depicts a beaker containing a block of glass on the bottom and water over it. The glass block contains a small air bubble at an unknown depth below the water surface. When viewed from above at an angle of 60°, the air bubble appears at a depth of 6.81 cm. What is the true depth of the air bubble?

   # Figure P8.22 Apparent position of the air bubble in Problem 8.22.

   Figure P8.22 Full Alternative Text

7. 8.23 A glass semicylinder with n=1.5 is positioned such that its flat face is horizontal, as shown in Fig. P8.23, and its horizontal surface supports a drop of oil, as also shown. When light is directed radially toward the oil, total internal reflection occurs if θ exceeds 53°. What is the index of refraction of the oil?

   # Figure P8.23 Oil drop on the flat surface of a glass semicylinder (Problem 8.23).

8. *8.24 A penny lies at the bottom of a water fountain at a depth of 30 cm. Determine the diameter of a piece of paper that when placed to float on the surface of the water directly above the penny, would totally obscure the penny from view. Treat the penny as a point and assume that n=1.33 for water.

9. 8.25 Suppose that the optical fiber of Example 8-5 is submerged in water (with n=1.33) instead of air. Determine θ a and f p in that case.

10. *8.26 Equation (8.45) was derived for the case where the light incident upon the sending end of the optical fiber extends over the entire acceptance cone shown in Fig. 8-12(b). Suppose the incident light is constrained to a narrower range extending between normal incidence and θ′, where θ′<θ a.

    1. Obtain an expression for the maximum data rate f p in terms of θ′.

    2. Evaluate f p for the fiber of Example 8-5 when θ′=5°.

# Sections 8-4 and 8-5: Reflection and Transmission at Oblique Incidence

1. 8.27 A 2-km long optical fiber uses a fiber core with n f=1.6 and a cladding with n c=1.57. Compute the maximum data rate and confirm your result using Module 8.2. The operating frequency is 100 THz.

2. 8.28 A 3-km long optical fiber uses a fiber core with n f=1.51 and a cladding with n c=1.48. Compute the maximum data rate and confirm your result using Module 8.2. The operating frequency is 300 THz.

3. 8.29 A plane wave in air with

   E˜i=y ˆ 20 e−j(3 x+4 z)(V/m)

   is incident upon the planar surface of a dielectric material with ε r=4, and it occupies the half-space z≥0. Determine:

   1. a.the polarization of the incident wave,

   2. *b.the angle of incidence,

   3. c.the time-domain expressions for the reflected electric and magnetic fields,

   4. d.the time-domain expressions for the transmitted electric and magnetic fields, and

   5. e.the average power density carried by the wave in the dielectric medium.

4. 8.30 Repeat Problem 8.29 for a wave in air with

   H˜i=y ˆ 2×10−2 e−j(8 x+6 z)(A/m)

   incident upon the planar boundary of a dielectric medium (z≥0) with ε r=9.

5. 8.31 A plane wave in air with

   E˜i=(x ˆ 9−y ˆ 4−z ˆ 6)e−j(2 x+3 z)(V/m)

   is incident upon the planar surface of a dielectric material, with ε r=2.25, and it occupies the half-space z≥0. Determine:

   1. *a.the incidence angle θ i,

   2. b.the frequency of the wave,

   3. c.the field E˜r of the reflected wave,

   4. d.the field E˜t of the wave transmitted into the dielectric medium, and

   5. e.the average power density carried by the wave into the dielectric medium.

6. 8.32 Natural light is randomly polarized, which means that, on average, half the light energy is polarized along any given direction (in the plane orthogonal to the direction of propagation) and the other half of the energy is polarized along the direction orthogonal to the first polarization direction. Hence, when treating natural light incident upon a planar boundary, we can consider half of its energy to be in the form of parallel-polarized waves and the other half as perpendicularly polarized waves. Determine the fraction of the incident power reflected by the planar surface of a piece of glass with n=1.5 when illuminated by natural light at 70°.

7. *8.33 A parallel-polarized plane wave is incident from air onto a dielectric medium with ε r=9 at the Brewster angle. What is the refraction angle?

8. 8.34 Repeat Problem 8.33 using Module 8.3.

9. 8.35 A perpendicularly polarized wave in air is obliquely incident upon a planar glass–air interface at an incidence angle of 30°. The wave frequency is 600 THz (1 THz=10 12 THz), which corresponds to green light, and the index of refraction of the glass is 1.6. If the electric field amplitude of the incident wave is 50 V/m, determine:

   1. the reflection and transmission coefficients, and

   2. the instantaneous expressions for E and H in the glass medium.

10. 8.36 Repeat part (a) of Problem 8.35 using Module 8.3.

11. 8.37 Show that the reflection coefficient Γ⊥ can be written in the following form:

    Γ⊥=sin(θ t−θ i)sin(θ t+θ i)

12. 8.38 Show that for nonmagnetic media, the reflection coefficient Γ∥ can be written in the following form:

    Γ∥=tan(θ t−θ i)tan(θ t+θ i).

13. *8.39 A parallel-polarized beam of light with an electric field amplitude of 10 (V/m) is incident in air on polystyrene with μ r=1 and ε r=2.6. If the incidence angle at the air–polystyrene planar boundary is 50°, determine:

    1. the reflectivity and transmissivity, and

    2. the power carried by the incident, reflected, and transmitted beams if the spot on the boundary illuminated by the incident beam is 1 m 2 in area.

14. 8.40 Repeat part (a) of Problem 8.39 using Module 8.3.

15. 8.41 A 50-MHz right-hand circularly polarized plane wave with an electric field modulus of 30 V/m is normally incident in air upon a dielectric medium with ε r=9 and occupying the region defined by z≥0.

    1. Write an expression for the electric field phasor of the incident wave given that the field is a positive maximum at z=0 and t=0.

    2. Calculate the reflection and transmission coefficients.

    3. Write expressions for the electric field phasors of the reflected wave, the transmitted wave, and the total field in the region z≤0.

    4. Determine the percentages of the incident average power reflected by the boundary and transmitted into the second medium.

16. 8.42 Consider a flat 5-mm thick slab of glass with ε r=2.56.

    1. *a.If a beam of green light (λ 0=0.52 μ m) is normally incident upon one of the sides of the slab, what percentage of the incident power is reflected back by the glass?

    2. b.To eliminate reflections, it is desired to add a thin layer of antireflection coating material on each side of the glass. If you are at liberty to specify the thickness of the antireflection material as well as its relative permittivity, what would these specifications be?

17. 8.43 A perpendicular polarization wave in air is incident upon the surface of a perfect conductor at an incidence angle of 30°. The frequency of the wave is 2 GHz. Use Module 8.1 to determine the location of the nearest electric field maximum from the conductor surface.

18. 8.44 A 2 GHz plane wave in air is incident upon a conducting medium with ε r=4 and σ=2 S/m at an incidence angle of 60°. The wave is parallel-polarized. Use Module 8.4 to compute:

    1. the reflectivity and transmissivity, and

    2. the effective refraction angle.

19. 8.45 A 2 GHz plane wave in air is incident upon a conducting medium with ε r=4 and σ=2 S/m at an incidence angle of 60°. The wave is perpendicularly polarized. Use Module 8.4 to compute:

    1. the reflectivity and transmissivity, and

    2. the effective refraction angle.

# Sections 8.6 to 8.11: Waveguides and Resonators

1. 8.46 Derive Eq. (8.89b).

2. *8.47 A hollow, rectangular waveguide is to be used to transmit signals at a carrier frequency of 6 GHz. Choose its dimensions so that the cutoff frequency of the dominant TE mode is lower than the carrier by 25% and that of the next mode is at least 25% higher than the carrier.

3. 8.48 A TE wave propagating in a dielectric-filled waveguide of unknown permittivity has dimensions a=5 cm and b=3 cm. If the x component of its electric field is given by

   E x=−36 cos(40 π x) sin (100 π y)⋅sin(2.4 π×10 10 t−52.9 π z),(V/m)

   determine:

   1. the mode number,

   2. ε r of the material in the guide,

   3. the cutoff frequency, and

   4. the expression for H y.

4. *8.49 A waveguide filled with a material whose ε r=2.25 has dimensions a=2 cm and b=1.4 cm. If the guide is to transmit 10.5 GHz signals, what possible modes can be used for the transmission?

5. 8.50 For a rectangular waveguide operating in the TE 10 mode, obtain expressions for the surface charge density ρ˜s and surface current density J˜s on each of the four walls of the guide.

6. *8.51 A waveguide with dimensions a=1 cm and b=0.7 cm is to be used at 20 GHz. Determine the wave impedance for the dominant mode when

   1. the guide is empty, and

   2. the guide is filled with polyethylene (whose ε r=2.25).

7. 8.52 A narrow rectangular pulse superimposed on a carrier with a frequency of 9.5 GHz was used to excite all possible modes in a hollow guide with a=3 cm and b=2.0 cm. If the guide is 100 m in length, how long will it take each of the excited modes to arrive at the receiving end?

8. *8.53 If the zigzag angle θ′ is 25° for the TE 10 mode, what would it be for the TE 20 mode?

9. 8.54 Measurement of the TE 101 frequency response of an air-filled cubic cavity revealed that its Q is 4802. If its volume is 64 mm 3, what material are its sides made of? [Hint: See Appendix B.]

10. 8.55 A hollow cavity made of aluminum has dimensions a=4 cm and d=3 cm. Calculate Q of the TE 101 mode for

    1. *a.b=2 cm, and

    2. b.b=3 cm.

11. 8.56 A hollow rectangular waveguide with dimensions a=2 cm and b=1 cm is used to propagate signals at 20 GHz. Use Module 8.5 to determine:

    1. the cutoff frequency of TE 10 mode, and

    2. all possible transmission modes.

12. 8.57 A hollow rectangular waveguide with dimensions a=4 cm and b=2 cm is used to propagate signals at 12 GHz. Use Module 8.5 to determine:

    1. the cutoff frequency of TE 10 mode, and

    2. all possible transmission modes.

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