Problem 8.2 A plane wave traveling in medium 1 with r1 = 2.25 is normally
incident upon medium 2 with r2 = 4. Both media are made of nonmagnetic, nonconducting materials. If the electric field of the incident wave is given by
Ei = y 8 cos(6 109t 30 x) (V/
Problem 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 r = 25.
Determine the following:
(a) The reflection and transmission coefficients.
(b) The standing-wav
Problem 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 r = 36.
Determine the following:
(a)
(b) The average power densities of the incident and reflected wa
Problem 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 r2 and d produces no reflection? Express your answers in terms of r1 , r3 and the
oscil
Problem 7.21 Based on wave attenuation and reflection measurements conducted
at 1 MHz, it was determined that the intrinsic impedance of a certain medium is
28.145 () and the skin depth is 2 m. Determine the following:
(a) The conductivity of the material
Problem 7.22 The electric field of a plane wave propagating in a nonmagnetic
medium is given by
E = z 25e30x cos(2 109t 40x) (V/m)
Obtain the corresponding expression for H.
Solution: From the given expression for E,
= 2 109 (rad/s),
= 30 (Np/m),
= 40
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Problem 7.9 For a wave characterized by the electric field
E(z,t) = x ax cos( t kz) + y ay cos( t kz + )
identify the polarization state, determine the polarization angles ( , ), and sketch
the locus of E(0,t) for each of the following
Problem 7.15 For each of the following combinations of parameters, determine if
the material is a low-loss dielectric, a quasi-conductor, or a good conductor, and then
calculate , , , up , and c :
(a) Glass with r = 1, r = 5, and = 1012 S/m at 10 GHz.
(b)
Problem 7.32 At microwave frequencies, the power density considered safe for
human exposure is 1 (mW/cm2 ). 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.
Problem 7.16 Dry soil is characterized by r = 2.5, r = 1, and = 104 (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, and then calculate , , , p ,
and c :
(a
Problem 7.29 The electric-field phasor of a uniform plane wave traveling
downward in water is given by
e = x 5e0.2z e j0.2z
E
(V/m)
where z is the downward direction and z = 0 is the water surface. If = 4 S/m,
(a) Obtain an expression for the average powe
Problem 7.25 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
Problem 7.26 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 r = 1,
r = 1, and = 5.8 107 S/m, and the outer conductor is 0.5 mm thick. At 10 MHz:
(a) Are the conductors
Problem 7.27 The magnetic field of a plane wave traveling in air is given by
H = x 50 sin(2 107t ky) (mA/m). Determine the average power density carried
by the wave.
Solution:
H = x 50 sin(2 107t ky) (mA/m),
E = 0 y H = z 0 50 sin(2 107t ky) (mV/m),
Sav =
Problem 7.34
Repeat Problem 7.33 for a wave traveling in a lossy medium in which
E = x 100e20y cos(2 109t 40y) (V/m)
H = z 0.64e20y cos(2 109t 40y 36.85 )
(A/m)
The box has dimensions a = 1 cm, b = 2 cm, and c = 0.5 cm.
Solution:
(a)
H
S(t) = E
= x 100e20
Problem 7.14 Plot the locus of E(0,t) for a plane wave with
E(z,t) = x sin( t + kz) + y 2 cos( t + kz)
Determine the polarization state from your plot.
Solution:
y
t=/2
2
1
z
1
x
t=0
2
Figure P7.14: Locus of E versus time.
E = x sin( t + kz) + y 2 cos( t
Problem 7.19 Ignoring reflection at the airsoil 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 r = 1, r = 9, and = 5 104
S/m.
Solutio
Problem 7.20 The skin depth of a certain nonmagnetic conducting material is 3 m
at 2 GHz. Determine the phase velocity in the material.
Solution: For a good conductor, = , and for any material s = 1/ . Hence,
up =
2 f
=
= 2 f s = 2 5 109 3 106 = 9.42 104
Problem 7.9 For a wave characterized by the electric field
E(z,t) = x ax cos( t kz) + y ay cos( t kz + )
identify the polarization state, determine the polarization angles ( , ), and sketch
the locus of E(0,t) for each of the following cases:
(a) ax = 3 V
Problem 7.5 A wave radiated by a source in air is incident upon a soil surface,
whereupon 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 soils relative
permit
Problem 7.1 The magnetic field of a wave propagating through a certain
nonmagnetic material is given by
H = z 30 cos(108t 0.5y) (mA/m)
Find the following:
(a) The direction of wave propagation.
(b) The phase velocity.
(c) The wavelength in the material.
(
Problem 7.12 The electric field of an elliptically polarized plane wave is given by
E(z,t) = [x 10 sin( t kz 60 )
+ y 30 cos( t kz)] (V/m)
Determine the following:
(a) The polarization angles ( , ).
(b) The direction of rotation.
Solution:
(a)
E(z,t) = [x
Problem 7.8 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 waves electric
field vector, given that the wavelength is 6 cm.
Solution:
y
t=0
z
x
t=/2
Figure P7.8: Locus of
Problem 7.3 The electric field phasor of a uniform plane wave is given by
e = y 10e j0.2z (V/m). If the phase velocity of the wave is 1.5 108 m/s and the
E
relative permeability of the medium is r = 2.4, find the following:
(a) The wavelength.
(b) The fre
Problem 7.4 The electric field of a plane wave propagating in a nonmagnetic
material is given by
E = [y 3 sin( 107t 0.2 x)
+ z 4 cos( 107t 0.2 x)] (V/m)
Determine
(a) The wavelength.
(b) r .
(c) H.
Solution:
(a) Since k = 0.2 ,
=
(b)
up =
2
2
= 10 m.
=
k
Problem 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 r = 9. The electric field is polarized along the
x-direct
Problem 7.7 A 60-MHz plane wave traveling in the x-direction in dry soil
with relative permittivity 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 v
Problem 7.6 The electric field of a plane wave propagating in a lossless,
nonmagnetic, dielectric material with r = 2.56 is given by
E = y 20 cos(6 109t kz) (V/m)
Determine:
(a) f , up , , k, and .
(b) The magnetic field H.
Solution:
(a)
= 2 f = 6 109 ra
Problem 7.10 The electric field of a uniform plane wave propagating in free space
is given by
e = (x + jy )30e j z/6 (V/m)
E
Specify the modulus and direction of the electric field intensity at the z = 0 plane at
t = 0, 5, and 10 ns.
Solution:
e j t ]
E(z
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