# Problems

# Section 7.2: Propagation in Lossless Media

1. 7.1 The magnetic field of a wave propagating through a certain nonmagnetic material is given by

   H=z ˆ 30 cos(10 8 t−0.5 y)(m A/m).

   Find:

   1. *a.the direction of wave propagation,

   2. b.the phase velocity,

   3. *c.the wavelength in the material,

   4. d.the relative permittivity of the material, and

   5. e.the electric field phasor.

2. 7.2 Write general expressions for the electric and magnetic fields of a 1 GHz sinusoidal plane wave traveling in the +y direction in a lossless nonmagnetic medium with relative permittivity ε r=9. The electric field is polarized along the x direction, its peak value is 6 V/m, and its intensity is 4 V/m at t=0 and y=2 cm.

3. 7.3 The electric field phasor of a uniform plane wave is given by E˜=y ˆ 10 e j 0.2 z (V/m). If the phase velocity of the wave is 1.5×10 8 m/s and the relative permeability of the medium is μ r=2.4, find:

   1. *a.the wavelength,

   2. b.the frequency f of the wave,

   3. c.the relative permittivity of the medium, and

   4. d.the magnetic field H(z,t).

4. 7.4 The electric field of a plane wave propagating in a nonmagnetic material is given by

   E=[y ˆ 3 sin(π×10 7 t−0.2 π x) +z ˆ 4 cos(π×10 7 t−0.2 π x)] (V/m)

   Determine:

   1. the wavelength

   2. ε r

   3. H

5. *7.5 A wave radiated by a source in air is incident upon a soil surface, where a part of the wave is transmitted into the soil medium. If the wavelength of the wave is 60 cm in air and 20 cm in the soil medium, what is the soil’s relative permittivity? Assume the soil to be a very low-loss medium.

6. 7.6 The electric field of a plane wave propagating in a lossless, nonmagnetic, dielectric material with ε r=2.56 is given by

   E=y ˆ 20 cos(6 π×10 9 t−k z)(V/m).

   Determine:

   1. f, u p, λ, k, and η.

   2. the magnetic field H.

7. 7.7 The magnetic field of a plane wave propagating in a nonmagnetic material is given by

   H=x ˆ 60 cos(2 π×10 7 t+0.1 π y) z ˆ 30 cos(2 π×10 7 t+0.1 π y)(mA/m).

   Determine:

   1. *a.the wavelength

   2. b.ε r

   3. c.E

8. 7.8 A 60 MHz plane wave traveling in the −x direction in dry soil with relative permittivity ε r=4 has an electric field polarized along the z direction. Assuming dry soil to be approximately lossless, and given that the magnetic field has a peak value of 10 (mA/m) and that its value was measured to be 7 (mA/m) at t=0 and x=−0.75 m, develop complete expressions for the wave’s electric and magnetic fields.

# Section 7.3: Wave Polarization

1. *7.9 An RHC-polarized wave with a modulus of 2 (V/m) is traveling in free space in the negative z direction. Write the expression for the wave’s electric field vector given that the wavelength is 6 cm.

2. 7.10 For a wave characterized by the electric field

   E(z,t)=x ˆ a x cos(ω t−k z)+y ˆ a y cos(ω t−k z+δ),

   identify the polarization state, determine the polarization angles (γ,χ), sketch the locus of E(0,t), and verify your solution using Module 7.3 for each of the following cases.

   1. a x=3 V/m, a y=4 V/m, and δ=0

   2. a x=3 V/m, a y=4 V/m, and δ=180∘

   3. a x=3 V/m, a y=3 V/m, and δ=45∘

   4. a x=3 V/m, a y=4 V/m, and δ=−135∘

3. 7.11 The electric field of a uniform plane wave propagating in free space is given by

   E˜=(x ˆ+j y ˆ)30 e−j π z/6(V/m).

   Specify the modulus and direction of the electric field intensity at the z=0 plane at t=0, 5, and 10 ns.

4. *7.12 The magnetic field of a uniform plane wave propagating in a dielectric medium with ε r=36 is given by

   H˜=30(y ˆ+j z ˆ)e−j π x/6(m A/m).

   Specify the modulus and direction of the electric field intensity at the x=0 plane at t=0 and 5 ns.

5. 7.13 A linearly polarized plane wave of the form E˜=x ˆ a x e−j k z can be expressed as the sum of an RHC-polarized wave with magnitude a R and an LHC-polarized wave with magnitude a L. Prove this statement by finding expressions for a R and a L in terms of a x.

6. *7.14 The electric field of an elliptically polarized plane wave is given by

   E(z,t)=[−x ˆ 10 sin(ω t−k z−60∘) +y ˆ 30 cos(ω t−k z)](V/m).

   Determine:

   1. the polarization angles (γ,χ), and

   2. the direction of rotation.

7. 7.15 Compare the polarization states of each of the following pairs of plane waves:

   1. Wave 1: E 1=x ˆ 2 cos(ω t−k z)+y ˆ 2 sin(ω t−k z)

      Wave 2: E 2=x ˆ 2 cos(ω t+k z)+y ˆ 2 sin(ω t+k z)

   2. Wave 1: E 1=x ˆ 2 cos(ω t−k z)−y ˆ 2 sin(ω t−k z)

      Wave 2: E 2=x ˆ 2 cos(ω t+k z)−y ˆ 2 sin(ω t+k z)

8. 7.16 Plot the locus of E(0,t) for a plane wave with

   E(z,t)=x ˆ sin(ω t+k z)+y ˆ 2 cos(ω t+k z).

   Determine the polarization state from your plot.

# Sections 7.4: Propagation in a Lossy Medium

1. 7.17 For each of the following combinations of parameters, determine if the material is a low-loss dielectric, a quasi-conductor, or a good conductor. Then calculate α, β, λ, u p, and η c.

   1. *a.Glass with μ r=1, ε r=5, and σ=10−12 S/m at 10 GHz.

   2. b.Animal tissue with μ r=1, ε r=12, and σ=0.3 S/m at 100 MHz.

   3. c.Wood with μ r=1, ε r=3, and σ=10−4 S/m at 1 kHz.

2. 7.18 Dry soil is characterized by ε r=2.5, μ r=1, and σ=10−4 (S/m). At each of the following frequencies, determine if dry soil may be considered a good conductor, a quasi-conductor, or a low-loss dielectric. Then calculate α, β, λ, μ p, and η c:

   1. 60 Hz

   2. 1 kHz

   3. 1 MHz

   4. 1 GHz

3. *7.19 In a medium characterized by ε r=9, μ r=1, and σ=0.1 S/m, determine the phase angle by which the magnetic field leads the electric field at 100 MHz.

4. 7.20 Generate a plot for the skin depth δ s versus frequency for seawater for the range from 1 kHz to 10 GHz (use log-log scales). The constitutive parameters of seawater are μ r=1, ε r=80, and σ=4 S/m.

5. *7.21 Ignoring reflection at the air–soil boundary, if the amplitude of a 3 GHz incident wave is 10 V/m at the surface of a wet soil medium, at what depth will it be down to 1 mV/m? Wet soil is characterized by μ r=1, ε r=9, and σ=5×10−4 S/m.

6. 7.22 Ignoring reflection at the air–water boundary, if the amplitude of a 1 GHz incident wave in air is 20 V/m at the water surface, at what depth will it be down to 1 μV/m? Water has μ r=1 and at 1 GHz, ε r=80, and σ=1 S/m.

7. *7.23 The skin depth of a certain nonmagnetic conducting material is 3 μm at 5 GHz. Determine the phase velocity in the material.

8. 7.24 Based on wave attenuation and reflection measurements conducted at 1 MHz, it was determined that the intrinsic impedance of a certain medium is 28.1 45°(Ω) and the skin depth is 2 m. Determine:

   1. the conductivity of the material,

   2. the wavelength in the medium, and

   3. the phase velocity.

9. *7.25 The electric field of a plane wave propagating in a nonmagnetic medium is given by

   E=z ˆ 25 e−30 x cos(2 π×10 9 t−40 x)(V/m).

   Obtain the corresponding expression for H.

10. 7.26 The magnetic field of a plane wave propagating in a nonmagnetic medium is given by

    H=y ˆ 60 e−10 z cos(2 π×10 8 t−12 z)(mA/m).

    Obtain the corresponding expression for E.

11. 7.27 At 2 GHz, the conductivity of meat is on the order of 1 (S/m). When a material is placed inside a microwave oven and the field is activated, the presence of the electromagnetic fields in the conducting material causes energy dissipation in the material in the form of heat.

    1. Develop an expression for the time-average power per m m 3 dissipated in a material of conductivity σ if the peak electric field in the material is E 0.

    2. Evaluate the result for an electric field E 0=4×10 4 (V/m).

# Section 7.5: Current Flow in Conductors

1. 7.28 In a nonmagnetic, lossy, dielectric medium, a 300 MHz plane wave is characterized by the magnetic field phasor

   H˜=(x ˆ−j 4 z ˆ)e−2 y e−j 9 y(A/m).

   Obtain time-domain expressions for the electric and magnetic field vectors.

2. *7.29 A rectangular copper block is 30 cm in height (along z). In response to a wave incident upon the block from above, a current is induced in the block in the positive x direction. Determine the ratio of the ac resistance of the block to its dc resistance at 1 kHz. The relevant properties of copper are given in Appendix B.

3. 7.30 Repeat Problem 7.29 at 10 MHz.

4. 7.31 The inner and outer conductors of a coaxial cable have radii of 0.5 cm and 1 cm, respectively. The conductors are made of copper with ε r=1, μ r=1, and σ=5.8×10 7 S/m. The outer conductor is 0.5 mm thick. At 10 MHz:

   1. Are the conductors thick enough to be considered infinitely thick as far as the flow of current through them is concerned?

   2. Determine the surface resistance R s.

   3. Determine the ac resistance per unit length of the cable.

5. 7.32 Repeat Problem 7.31 at 1 GHz.

# Section 7.6: EM Power Density

1. *7.33 The magnetic field of a plane wave traveling in air is given by H=x ˆ 50 sin(2 π×10 7 t−k y) (mA/m). Determine the average power density carried by the wave.

2. 7.34 A wave traveling in a nonmagnetic medium with ε r=9 is characterized by an electric field given by

   E=[y ˆ 3 cos(π×10 7 t+k x) −z ˆ 2 cos(π×10 7 t+k x)](V/m).

   Determine the direction of wave travel and average power density carried by the wave.

3. 7.35 The electric-field phasor of a uniform plane wave traveling downward in water is given by

   E˜=x ˆ 5 e−0.2 z e−j 0.2 z(V/m),

   where z ˆ is the downward direction and z=0 is the water surface. If σ=4 S/m,

   1. a.Obtain an expression for the average power density.

   2. b.Determine the attenuation rate.

   3. *c.Determine the depth at which the power density has been reduced by 40 dB.

4. 7.36 The amplitudes of an elliptically polarized plane wave traveling in a lossless, nonmagnetic medium with ε r=4 are H y 0=3 (mA/m) and H z 0=4 (mA/m). Determine the average power flowing through an aperture in the y–z plane if its area is 20 m 2.

5. *7.37 A wave traveling in a lossless, nonmagnetic medium has an electric field amplitude of 24.56 V/m and an average power density of 2.4 W/m 2. Determine the phase velocity of the wave.

6. 7.38 At microwave frequencies, the power density considered safe for human exposure is 1 (mW/c m 2). A radar radiates a wave with an electric field amplitude E that decays with distance as E(R)=(3,000/R) (V/m), where R is the distance in meters. What is the radius of the unsafe region?

7. 7.39 Consider the imaginary rectangular box shown in Fig. P7.39.

   1. a.Determine the net power flux P(t) entering the box due to a plane wave in air given by

      E=x ˆ E 0 cos(ω t−k y)(V/m).

   2. *b.Determine the net time-average power entering the box.

   # Figure P7.39 Imaginary rectangular box of Problems 7.39 and 7.40.

8. 7.40 Repeat Problem 7.39 for a wave traveling in a lossy medium in which

   E=x ˆ 100 e−20 y cos(2 π×10 9 t−40 y) (V/m),H=−z ˆ 0.64 e−20 y cos(2 π×10 9 t−40 y−36.85∘) (A/m).

   The box has dimensions a=1 cm, b=2 cm, and c=0.5 cm.

9. 7.41 Given a wave with E=x ˆ E 0 cos(ω t−k z):

   1. *a.Calculate the time-average electric energy density

      (w e)av=1 T∫0 T w e d t=1 2 T∫0 T ε E 2 d t.

   2. b.Calculate the time-average magnetic energy density

      (w m)av=1 T∫0 T w m d t=1 2 T∫0 T μ H 2 d t.

   3. c.Show that (w e)av=(w m)av.

10. 7.42 A team of scientists is designing a radar as a probe for measuring the depth of the ice layer over the antarctic land mass. In order to measure a detectable echo due to the reflection by the ice-rock boundary, the thickness of the ice sheet should not exceed three skin depths. If ε r′=3 and ε r′′=10−2 for ice and if the maximum anticipated ice thickness in the area under exploration is 1.2 km, what frequency range is useable with the radar?

Test Your Understanding of Chapter 7:
http://em8e.eecs.umich.edu/all_qs/chapter7.phtml