522 CHAPTER 10 ELECTROMAGNETIC WAVE PROPAGATION

I 0.9 In a good conductor, E and H are in time phase.

(a) True (b) False

I 0.10 The Poynting vector physically denotes the power density leaving or entering a given volume in a time-varying field.

(a) True (b) False

Answers: 10.lb, 10.2f, 10.3a, 10.4b,c 10.Sb,e,f, 10.6c, 10.7c, 10.8b, 10.9b, 10.lOa.

Section 10.2-Waves in General

10.1 An EM wave propagating in a certain medium is described by

E = 25 sin(27T X 106t - 6x)az V/m

(a) Determine the direction of wave propagation. (b) Compute the period T, the wavelength A, and the velocity u. (c) Sketch the wave at t = 0, T/8, T/4, T/2.

I 0.2 Calculate the wavelength for plane waves in vacuum at the following frequencies:

(a) 60 Hz (power line) (b) 2 MHz (AM radio) (c) 120 MHz (FM radio) (d) 2.4 GHz (microwave oven)

I 0.3 An EM wave in free space is described by

H = 0.4cos(108t + /3y) AIM

Determine (a) the angular frequency w, (b) the wave number f3, (c) the wavelength A, ( d) the direction of wave propagation, ( e) the value of H(2, 3, 4, 10 ns).

10.4 (a) Show that E(x, t) = cos(x + wt) + cos(x - wt) satisfies the scalar wave equation. (b) Determine the velocity of wave propagation.

Section 10.3-Wave Propagation in Lossy Dielectrics

10.5 (a) Derive eqs. (10.23) and (10.24) from eqs. (10.18) and (10.20). (b) Using eq. (10.29) in conjunction with Maxweffs equations, show that

jwµ, 17=­

/'

(c) From part (b), derive eqs. (10.32) and (10.33).

Problems 523

10.6 At 50 MHz, a lossy dielectric material is characterized by s = 3.6s0 , µ = 2.1µ, 0 , and u = 0.08 Sim. If Es = 6e-yx az V/m, compute (a) y, (b) A, (c) u, (d) YJ, (e) H 5 •

10. 7 Determine the loss tangent for each of the following nonmagnetic media at 12 MHz.

(a) wet earth (s = lOs0

, u = 10-2 Sim) (b) dry earth (s = 4s

0, <T = 10- 4 Sim)

(c) seawater (s = 8ls0 , u = 4 Sim)

10.8 Alumina is a ceramic material used in making printed circuit boards. At 15 GHz, s = 9. 6s0 , µ, = µ, 0 , tan 8 = 3 X 10 - 4

• Calculate (a) the penetration depth, (b) the total attenuation over a thickness of 5 mm.

10.9 A medium is characterized bys = 4s0 , µ, = 2.5µ, 0 , u = 8 X 10- 3 Sim. Calculate the phase difference between E and H at 20 MHz.

10.10 Atf = 100 MHz, show that silver (u = 6.1 X 107 Sim, JLr = 1, er= 1) is a good con­ductor, while rubber (u = 10- 15 Sim, /Lr= 1, sr = 3.1) is a good insulator.

10.11 Seawater plays a vital role in the study of submarine communications. Assuming that for seawater, u = 4 Sim, sr = 80, µ,, = 1, and f = 100 kHz, calculate (a) the phase velocity, (b) the wavelength, ( c) the skin depth, ( d) the intrinsic impedance.

10.12 In a certain medium withµ = µ, 0 , s = 4s0

,

Find (a) the wave period T, (b) the wavelength A, (c) the electric field E, (d) the phase difference between E and H.

10.13 In a nonmagnetic medium,

H = 50e-rnox cos(21T X 109 t - 200x)ay mAlm

Find E.

10.14 A certain medium has u = 1 Sim, s = 4s0 , and µ, = 9 µ,0 at a frequency of 1 GHz. Determine the (a) attenuation constant, (b) phase constant, (c) intrinsic impedance, and ( d) wave velocity.

Sections 10.4 and 10.5-Waves in Lossless Dielectrics and Free Space

10.15 The electric field of a TV broadcast signal progagating in air is given by

E (z,t) = 0.2 cos( wt - 6.Sz)ax V /m

(a) Determine the wave frequency wand the wavelength A. (b) Sketch Ex as a function oft at z = 0 and z = A.12. (c) Find the corresponding H(z, t).

10.16 A 60 MHz plane wave travels in a lossless medium withe = 3s0 andµ = 4µ 0 • Find the wave velocity u, its wavelength A, and the intrinsic impedance 1J of the medium.

524 CHAPTER 10 ELECTROMAGNETIC WAVE PROPAGATION

10.17 The magnetic field component of an EM wave propagating through a nonmagnetic medium (µ., = µ

0) is

H = 25 sin(2 X 108 t + 6x)~ mA/m

Determine:

(a) The direction of wave propagation

(b) The permittivity of the medium

( c) The electric field intensity

10.18 A manufacturer produces a ferrite material withµ = 750µ,0 , e = Se0 , and a- = 10-6 Sim at 10 MHz.

(a) Would you classify the material as lossless, lossy, or conducting?

(b) Calculate f3 and A.

(c) Determine the phase difference between two points separated by 2 m.

(d) Find the intrinsic impedance.

10.19 The electric field intensity of a uniform plane wave in air is given by

E = 50sin(1087Tt - f3x)az m V Im

(a) Calculate ~·

(b) Determine the location(s) where E vanishes at t = 50 ns.

(c) Find H.

10.20 For a uniform plane wave at 4 GHz, the intrinsic impedance and phase velocity of an unknown material are measured as 105 fl and 7.6 X 107 m/s, respectively. Find Br and JLr

of the material.

10.21 In a lossless medium (sr = 4.5, /Lr = 1), a uniform plane wave

E = 8 cos( wt - {3z) ax - 6 sin( wt - {3z)ay V /m

propagates at 40 MHz. (a) Find H. (b) Determine {3, A, TJ, and u.

10.22 A uniform plane wave in a lossy medium has a phase constant of 1.6 rad/m at 107 Hz, and its magnitude is reduced by 60% for every 2 m traveled. Find the skin depth and the speed of the wave.

10.23 A uniform plane wave in a lossy nonmagnetic medium has

(a) Compute the magnitude of the wave at z = 4 m, t = T/8.

(b) Find the loss in decibels suffered by the wave in the interval 0 < z < 3 m.

( c) Calculate the intrinsic impedance.

Section 10.6-Plane Waves in Good Conductors

10.24 The magnet field intensity of a uniform plane wave in a good conductor (e = e0

, µ.,=/Lo) is

H = 20e- 12z cos(27T X 106t + 12z)~ mA/m

Find the conductivity and the corresponding E field.

10.25 Which of the following media may be treated as conducting at 8 MHz?

(a) Wet marshy soil (s = lSs0

, µ., = µ.,0

, a = 10-2 Sim)

(b) Intrinsic germanium (s = l6s0 , µ., = µ.,0

, a = 0.025 Sim)

(c) Seawater (E = 8ls0 , µ = µ 0 , a = 25 Sim)

Problems 525

10.26 Calculate the skin depth and the velocity of propagation for a uniform plane wave at frequency of 6 MHz traveling in polyvinyl chloride (PVC) (µr = 1, Er = 4,

a - = 7 x 10-2). wt

10.27 (a) Determine the de resistance of a round copper wire (a = 5.8 X 107 Sim, µr = 1, Er = 1) of radius 1.2 mm and length 600 m.

(b) Find the ac resistance at 100 MHz.

( c) Calculate the approximate frequency at which de and ac resistances are equal.

10.28 For aluminum (a = 3.5 X 107 Sim, s = E0

, µ, = µ,0

) at 150 MHz, find (a) the propaga­tion constant y, (b) the skin depth 8, (c) the wave velocity u.

10.29 For silver, a = 6.1 X 107 Sim, JLr = 1, Br = 1, determine the frequency at which the penetration depth is 2 mm.

10.30 Show that in a good conductor, the skin depth 8 is approximately given by B = 2'TTIA.

10.31 Brass waveguides are often silver plated to reduce losses. If the thickness of silver (µ = µ 0 , s = E 0 , a = 6.1 X 107 Sim) must be 58, find the minimum thickness required for a waveguide operating at 12 GHz.

10.32 How deep does a radar wave at 2 GHz travel in seawater before its amplitude is reduced to 10-5 of its amplitude just below the surface? Assume thatµ, = µ,

0, s = 24s

0, a = 4 S/m.

Section 10.7-Wave Polarization

10.33 The electric field intensity of a uniform plane wave in a medium (a = 0, µ, = µ

0, e = B

0Br) is

E = 12 sin(2'TT X 107 t - 3y)az V/m

(a) Determine the polarization of the wave.

(b) Find the frequency.

( c ) Calculate Er-

( d) Obtain the magnetic field intensity H.

10.34 Let E = 2 sin(wt -{3x)ay - 5 sin(wt -{3x)az Vim. What is the wave polarization?

10.35 Determine the wave polarization of each of the following waves:

(a) E0 cos(wt + {3y)ax + E0 sin( wt+ {3y)az Vim

(b) E0 cos( wt - {3y)ax - 3E0 sin( wt+ {3y)az V /m

10.36 Determine the polarization of the following waves:

(a) Es = 40ej10zax + 60e j10za V Im (b) Es= 12ej'7T/3e -jlOxay + s:-j'TT/3e-jIOxaz Vim

526 CHAPTER 10 ELECTROMAGNETIC WAVE PROPAGATION

10.37 The electric field intensity of a uniform plane wave in free space is given by

E = 40 cos(wt -{3z)ax + 60 sin(wt -{3z)a1 V/m

(a) What is the wave polarization?

(b) Determine the magnetic field intensity.

I 0.38 Show that a linearly polarized plane wave of the form Es = E0 e - j f3z ax can be expressed as the sum of two circularly polarized waves.

10.39 Suppose E(y,t) = E01 cos(wt - /3y)ax + E02 cos(wt - /3y + </J)az V/m. Determine the polarization when (a) ¢ = 0, (b) ¢ = 7T/2, ( c) ¢ = 'TT.

Section 10.8-Power and the Poynting Vector

10.40 Show that eqs. (10.77) and (10.78) are equivalent.

10.41 The electric field intensity in a dielectric medium (µ, = µ,0 , e = e 0er) is given by

E = 150 cos(109t + 8x)a2 V/m

Calculate

(a) The dielectric constant er

(b) The intrinsic impedance

( c) The velocity of propagation

(d) The magnetic field intensity

( e) The Poynting vector <!P

10.42 The composite fields resulting from the superposition of two uniform plane waves are given by

E = E01 cos ax cos(wt -{3z)~ + E02 sin ax sin(wt -{3z)az

H = H0 cos ax cos(wt - {3z)eiy

Determine the time-average Poynting vector.

I 0.43 The electric field due a short dipole antenna located in free space is

10 ·3 E = - sin Oe- 1 ra V im

s r o

Find (a) H5

, (b) the average power crossing the surfacer = 2, 0 < (} < 7T/6, 0 < <P < 'TT.

10.44 The electric field component of a uniform plane wave traveling in seawater (u = 4 Sim, e = 81 e0 , µ, = µ, 0 ) is

E = 8e-o.iz cos( wt - 0.3z)ax V im

Problems 527

(a) Determine the average power density. (b) Find the depth at which the power density is reduced by 20 dB.

10.45 In a coaxial transmission line filled with a lossless dielectric ( c; = 4.Sc:0 , µ, = µ.0 ),

40 E = - sin( wt - 2z)aP V/m

p

Find (a) w and H, (b) the Poynting vector, ( c) the total time-average power crossing the surface z = 1 m, 2 mm < p < 3 mm, 0 < <f> < 2'1T.

10.46 An antenna is located at the origin of a spherical coordinate system. The fields produced by the antenna in free space are

E E = ~sin 0 sin w(t - rlc)a8 r

E H = ~ sin 0 sin w(t - r/c)a</>

1JT

where c = ~and T/ = ;µ;;,.Determine the time-average power radiated by the JLoBo \j ~

antenna.

10.47 A plane wave in free space has

H(x, t) = (10~ - 20az) sin( wt - 40x) Alm

Find w, E, and P ave·

10.48 Human exposure to the electromagnetic radiation in air is regarded as safe if the power density is less than 10 m W/m2

. What is the corresponding electric field intensity?

10.49 Given that E = cos(wt - f3z)ax +sin( wt - /3Z)3.y V/m, show that the Poyntingvector is constant everywhere.

10.50 At the bottom of a microwave oven, E = 2.4 kV Im. If this value is found uniformly over the entire area of the oven, which is 450 cm2

, determine the power delivered by the oven. Assumeµ. = µ.

0, s = s

0•

10.51 A coaxial cable consists of two conducting cylinders of radii a and b. The electric and magnetic fields in the cable are

E = Vo sin(wt - {3z)ap, a< p < b p In(b!a)

H = ~ sin(wt - f3z)a</>, a < p < b 27Tp

where V0 andJ0 areconstants. (a) Determinethetime-averagePoyntingvector. (b) Find the time-average power flowing through the cable.

528 CHAPTER 10 ELECTROMAGNETIC WAVE PROPAGATION

Section 10.9-Reflection at Normal Incidence

10.52 (a) For a normal incidence upon the dielectric- dielectric interface for which /Li - JL2 - JL 0 , we define R and T as the reflection and transmission coefficients for average powers, that is, P r,ave = RP;,ave and P t,ave = TP;,ave· Prove that

(n - n )

2 R _ 1 2

n1 + n1 and

4n1n2 T= ----(n1 + n1)2

where n1 and n2 are the refractive indices of the media.

(b) Determine the ratio n1/n2 so that the reflected and the transmitted waves have the same average power.

10.53 The plane wave E = 30 cos(wt - z)ax V/m in air normally hits a lossless medium (JL = JLo, e = 4eJ at z = 0. (a) Find r , T , ands. (b) Calculate the reflected electric and magnetic fields.

10.54 A uniform plane wave E; = 50 sin(27T X 108t - f3 1x)az Vim is incident normally from air to a perfect conductor. Determine Er and Er.

10.55 A uniform plane wave in air with

H = 4 sin( wt - 5x)a1

A /m

is normally incident on a plastic region ( x > 0) with the parameters IL = JL 0 , s = 4s0 ,

and u = 0. (a) Obtain the total electric field in air. (b) Calculate the time-average power density in the plastic region. ( c) Find the standing wave ratio.

10.56 Region 1 is a lossless medium for which y > 0, IL = JL 0 , e = 4e0 , whereas region 2 is free space, y < 0. If a plane wave E; = 5 cos( 108t + f3y )a2 V /m exists in region 1, find (a) the total electric field component of the wave in region 1, (b) the time­average Poynting vector in region 1, (c) the time-average Poynting vector in region 2.

10.57 A plane wave in free space (z < 0) is incident normally on a large block of material with ( e, = 12, JLr = 3, u = 0) that occupies z > O. If the incident electric field is

E = 30 cos( wt - z)a1

V/m

Find (a) w, (b) the standing wave ratio, (c) the reflected magnetic field, (d) the average power density of the transmitted wave.

10.58 A uniform plane wave in air is normally incident on an infinite lossless dielectric material occupying z > 0 and having e = 3e0 and IL = /Lo· If the incident wave is E; = 10 cos(wt - z)ay V/m, find

(a) .A and w of the wave in air and the transmitted wave in the dielectric medium

(b) The incident H ; field

(c) rand 'T

(d) The total electric field and the time-average power in both regions

Problems 529

10.59 A 100 MHz plane wave is normally incident from air to the sea surface, which may be assumed to be calm and smooth. If <T = 4 Sim, /Lr= 1, and Br = 81 for seawater, calculate the fractions of the incident power that are transmitted and reflected.

10.60 A uniform plane wave in a certain medium (µ, = µ, 0 , e = 4e0 ) is given by

E = 12 cos(wt - 40'1Tx)~ Vim

(a) Find w.

(b) If the wave is normally incident on a dielectric (µ, = µ,0 , e Er and Ei-

*10.61 A signal in air (z > 0) with the electric field component

E = 10 sin( wt+ 3z)ax Vim

3.2e0), determine

hits normally the ocean surface at z = 0 as in Figure 10.24. Assuming that the ocean surface is smooth and that e = 80e0 , J.L = µ,0 , u = 4 Sim in ocean, determine

(a) w

(b) The wavelength of the signal in air

( c) The loss tangent and intrinsic impedance of the ocean

( d) The reflected and transmitted E field

10.62 Sketch the standing wave in eq. (10.97) at t = 0, T/8, T/4, 3T/8, T/2, and so on, where T = 2Trlw.

10.63 A uniform plane wave is incident at an angle O; = 45° on a pair of dielectric slabs joined together as shown in Figure 10.25. Determine the angles of transmission Ot1 and Ot2 in the slabs.

FIGURE 10.24 For Problem 10.61.

z

Ocean

e = 80eo, µ. = µ o, <T = 4

Free space Free space FIGURE 10.25 For Problem 10.63.

CD 0 µ = µ o µ = µ o

e. = 4.5e0 e == 2. 25e0

530 CHAPTER 10 ELECTROMAGNETIC WAVE PROPAGATION

FIGURE 10.26 For Problem 10.65. Air Tissue

: z

10.64 Show that the field

where k; + ~ = w 2µ,0 e0 , can be represented as the superposition of four propagating plane waves. Find the corresponding H 5•

10.65 Electromagnetic radiation can be used to heat cancerous tumor. If a plane wave is nor­mally incident on the tissue surface at 1.2 GHz as shown in Figure. 10.26, determine the refection coefficient. At 1.2 GHz, the electrical properties of the tissue are er = 50, µ r =

1, a =4 Sim.

10.66 An EM plane wave in a lossless medium impinges normally on a lossy medium.

(a) Determine the ratio of transmitted to incident power in terms of the standing wave ratios.

(b) Express the ratio of reflected to incident power in terms of s.

Section 10.10-Reflection at Oblique Incidence

*10.67 By assuming the time-dependent fields E = E0e J(k · r-wt) and H = H

0eJ(k-r-wt) where

k = kx3-x + k1<ly + k2az is the wave number vector and r = xax + ya1 + zaz is the radius vector, show that V X E = -aB/at can be expressed as k X E = µwH and deduce

ak X aE = aH.

10.68 A plane wave in free space has a propagation vector

k = 124ax + 1243y +263a2

Find the wavelength, frequency, and angles k makes with the x-, y- , and z-axes.

Problems 531

10.69 In free space,

Determine E0 , H 5, and frequency.

10.70 Assume the same fields as in Problem 10.67 and show that Maxwell's equations in a source-free region can be written as

From these equations deduce

k · E = 0

k·H = 0

k XE= wµH

k X H = -wsE

10.71 Show that for nonmagnetic dielectric media, the reflection and transmission coefficients for oblique incidence become

tan ( (} t - (};) r =-----11 tan ( (} t + (}; )'

sin ( (} t - (} J r -'- =----­sin ( (} t + (} J '

10.72 If region 1 is in free space, while region 2 is a nonmagnetic dielectric medium (a-2 = 0, sr2 = 6.4) , compute ErJEio and Et0 /Ei0 for oblique incidence at Oi = 12°. Assume parallel polarization.

10. 73 A parallel-polarized wave in air with

Air

E = (8a1

- 6az) sin( wt - 4y - 3z) V/m

i1npinges a dielectric half-space as shown in Figure 10.27. Find (a) the incidence angle O;, (b) the time-average power in air(µ, = µ 0 , s = s 0 ), (c) the reflected and transmitted E fields.

z FIGURE 10.27 For Pro blem 10.73.

Dielectric

( e = 4co, µ = µo)

--~--------------- y ei

k l

532 CHAPTER 10 ELECTROMAGNETIC WAVE PROPAGATION

10.74 In a dielectric medium (s = 9s0

, µ, = µ,J, a plane wave with

H = 0.2 cos(109 t - kx - kVsz)ay Alm

is incident on an air boundary at z = 0. Find

(a) Or and (}t

(b) k

( c) The wavelength in the dielectric and in air

(d) The incident E

( e) The transmitted and reflected E (f) The Brewster angle

10.75 Determine the Brewster angle for an air-seawater (e = 8le0 ) interface for the following cases: (a) EM plane wave passing from air to seawater, (b) EM wave passing from seawa­ter to air.

10. 76 If u is the phase velocity of an EM wave in a given medium, the index of refraction of the medium is n = c/u, where c is the speed of light in vacuum.

(a) Paraffin has /Lr= 1, e r = 2.1. Determine n for unbounded medium of paraffin.

(b) Distilled water has /Lr= 1, er= 81. Find n.

(c) Polystyrene hasµ,= 1, er= 2.7. Calculate n.

Section 10.11 -Application Note-Microwaves

10.77 Discuss briefly some applications of microwaves other than those discussed in the text.

10.78 A useful set of parameters, known as the scattering transfer parameters, is related to the incident and reflected waves as

[::] = [~:: (a) Express the T-parameters in terms of the S-parameters.

(b) Find T when

S= [0.2

0.4 0.4] 0.2

10.79 The S-parameters of a two-port network are:

S11 = 0.33 - j0.16, S12 = S21 = 0.56, S22 = 0.44 - j0.62

Find the input and output reflection coefficients when Zr = Z0 = 50 il and Zg = 2Z0 •

Problems 533

10.80 Why can't regular lumped circuit components such as resistors, inductors, and capacitors be used at microwave frequencies?

10.81 In free space, a microwave signal has a frequency of 8.4 GHz. Calculate the wavelength of the signal.