1. Air–perfect conductor interface. A uniform plane electromagnetic
wave traveling in air with its electric field given by...is normally
incident on a perfect conductor boundary located at x = 0. (a) Find the
phase constant β. (b) Find the corresponding magnetic field Hi(x, t).
(c) Find the electric and magnetic fields Ēr(x, t) and Hr(x, t) of the
reflected wave. (d) Find the nearest two positions in air away from the
boundary where the total electric field is always zero.
Get solution 2.
Air–perfect conductor interface. A uniform plane wave of time-average
power density 75 mW-cm−2 in air is normally incident on the surface of a
perfect conductor located at y = 0, as shown in Figure. The total
magnetic field phasor in air is given by...(a) What is H0? (b) What is
the frequency, f ? (c) Find the total electric field Ē1(y, t) at y =
−3.5 cm....Figure Normal incidence on a perfect conductor. Problem.
Get solution 3.
Air–perfect conductor interface. A uniform plane wave propagating in
air given by...is normally incident on a perfectly conducting plane
located at z = 0. (a) Find the frequency and the wavelength of this
wave. (b) Find the corresponding magnetic field Hi(z). (c) Find the
electric and magnetic field vectors of the reflected wave [i.e., Er(z)
and Hr(z)]. (d) Find the total electric field in air [i.e., E1(z)], and
plot the magnitude of each of its components as a function of z.
Get solution 4.
Air–perfect conductor interface. A uniform plane wave of frequency 12
GHz traveling in free space having a magnetic field given by...is
normally incident on a perfect conductor boundary located at y = 0. (a)
Find the real-time expression for the reflected wave, Hr(y, t). (b)
Compare the polarizations of the incident and the reflected waves. Is
there any difference? (c) Find the maximum value of the total magnetic
field at y = 0, −1.25 cm, −2.5 cm, −3.75 cm, and −5 cm, respectively.
Get solution 5.
Air-perfect conductor interface. A 10 mW-m−2, 3 GHz uniform plane wave
traveling in air is normally incident on a perfect conductor boundary
located at the x = 0 plane, as shown in Figure. (a) Find the phasor-form
electric and magnetic field vectors of the incident and reflected
waves: Ei(x), Hi(x), Er(x), and Hr(x). (b) Find the two closest
positions to the boundary in air where the total magnetic field is
always zero....Figure Normal incidence on a perfect conductor. Problem.
Get solution 6.
Air-dielectric interface. A 65 mW-m−2, 3 GHz uniform plane wave
traveling in the z direction in air is normally incident onto a planar,
lossless, nonmagnetic dielectric interface with ∈r = 6 located at z = 0,
as shown in Figure (a) Find the phasor-form electric and magnetic
fields of the incident, reflected, and transmitted waves. (b) Find the
two nearest positions in air with respect to the boundary where the
total magnetic field is at a local minimum....Figure Normal incidence on
a dielectric. Problem.
Get solution 7.
Air–GaAs interface. A uniform plane wave having a magnetic field given
by...is normally incident from air onto a plane air–gallium arsenide
(GaAs) interface located at y = 0. Assume GaAs to be a perfect
dielectric with ∈r ≃ 13 and μr = 1. (a) Find the reflected (Hr(y)) and
the transmitted (Ht(y)) magnetic fields. (b) Calculate the power density
of the incident, reflected, and transmitted waves independently and
verify the conservation of energy principle. (c) Find the expression for
the total magnetic field (H1(y)) in air and sketch its magnitude as a
function of y between y = −2 cm and y = 0.
Get solution 8.
Air–salty lake interface. Consider a uniform plane wave traveling in
air with its electric field given by...normally incident on the surface
(z = 0) of a salty lake (σ = 0.88 S-m−1, ∈r = 78.8, and μr = 1 at 1
GHz). (a) Assuming the lake to be perfectly flat and lossless (i.e.,
assume σ = 0), find the electric fields of the reflected and transmitted
waves (i.e., Ēr and Ēt). (b) Modify the Ēt expression found in part (a)
by introducing in it the exponential attenuation term due to the
nonzero conductivity of lake water, given in part (a), and justify this
approximation. (c) Using this expression, find the thickness of the lake
over which 90% of the power of the transmitted wave is dissipated. What
percentage of the power of the incident wave corresponds to this
amount?
Get solution 9.
Aircraft–submarine communication. A submarine submerged in the ocean is
trying to communicate with a Navy airplane, equipped with a VLF
transmitter operating at 20 kHz, approximately 10 km immediately
overhead from the location of the submarine. If the output power of the
VLF transmitter is 200 kW and the receiver sensitivity of the submarine
is 1 μV-m−1, calculate the maximum depth of the submarine from the
surface of the ocean for it to be able to communicate with the
transmitter. Assume the transmitter is radiating its power isotropically
and assume normal incidence at the air–ocean boundary. Use σ = 4 S-m−1,
∈r = 81, and μr = 1 for the properties of the ocean.
Get solution 10.
Vertical incidence on plasma. An ionized gas (plasma) consists of free
electrons and ions, the motion of which can be represented in terms of a
frequency-dependent dielectric constant given by...where fp is known as
the plasma frequency. Consider a plasma with fp = 1 MHz onto which a
uniform plane wave at frequency f = 500 kHz is normally (θi = 0)
incident from air (∈1 = ∈0, μ1 = μ0). In this case, the dielectric
constant for the second medium is ∈2 = 3∈0. (a) Determine the reflection
coefficient Γ and the complete time-domain expression for the electric
field reflected wave. What fraction of the power carried by the incident
wave is reflected? (b) Determine the complete time-domain expression
for the electric field of the transmitted wave. (c) If the incident wave
is left hand circularly polarized (LHCP), what is the polarization of
the reflected and transmitted waves? Specify both the type (linear,
circular, elliptical) and sense (RH or LH) of the polarization.
Get solution 11.
Air–fat interface. Consider a planar interface between air and fat
tissue (assume it to be of semi-infinite extent). If a plane wave is
normally incident from air at this boundary, find the percentage of the
power absorbed by the fat tissue at (a) 100 MHz, (b) 300 MHz, (c) 915
MHz, and (d) 2.45 GHz, and compare your results with the results of
Example Use Table 9.1 for the parameters of the fat tissue and assume μr
= 1.ExampleAir–muscle interface. Consider a planar interface between
air and muscle tissue. If a plane wave is normally incident at this
boundary, find the percentage of incident power absorbed by the muscle
tissue at (a) 100 MHz, (b) 300 MHz, (c) 915 MHz, and (d) 2.45 GHz.
Assume μr = 1.TABLE ∈r AND σ FOR BIOLOGICAL TISSUES Muscle, Skin, and
Tissues with High Water ContentFat, Bone, and Tissues with Low Water
Contentf(MHz)(∈m)rσm(S-m−1)*(∈f)rσf(mS-m−1)
*10071.70.8897.4519.1–75.9300541.375.731.6–107750521.545.649.8–138915511.605.655.6–1471,500491.775.670.8–1712,450472.215.596.4–2135,000443.925.5162–30910,00039.910.34.5324–549
Get solution 12.
Air–concrete interface. A uniform plane wave operating at 1 GHz is
normally incident from air onto the air–concrete interface. At 1 GHz,
the complex relative dielectric constants of wet and dry concrete are
measured as ∈wr ≃ 14.8 − j 1.73 and ∈dr ≃ 4.5 − j 0.03, respectively.55
For each case, calculate (a) the percentage of the incident power
reflected and (b) the penetration depth in the concrete. Assume concrete
to be semi-infinite in extent with μr= 1.
Get solution 13.
Shielding with a copper foil. A 1-GHz, 1-kW-m−2 microwave beam is
incident upon a sheet of copper foil of 10 μm thickness (see Example for
the electromagnetic properties of copper). Consider neglecting multiple
reflections, if justified. (a) Find the power density of the reflected
wave. (b) Find the power density transmitted into the foil. (c) Find the
power density of the wave that emerges from the other side of the foil.
Comment on the shielding effectiveness of this thin copper
foil.Example: Air–copper interface. Consider a uniform plane wave
propagating in air incident normally on a large copper block. Find the
percentage time-average power absorbed by the copper block at 1, 10, and
100 MHz and at 1 GHz.
Get solution 14.
Absorbing material. Consider a commercial absorber slab56 made of
EHP-48 material of 1 m thickness backed by a perfectly conducting metal
plate. A 100-MHz uniform plane wave is normally incident from the air
side at the air–absorber–metal interface. Find the percentage of
incident power lost in the absorber material. For EHP-48, use ∈r = 6.93 −
j 8.29 and μr = 1 at 100 MHz.
Get solution 15.
Radome design. A common material in dielectric radomes for aeronautical
applications is fiberglass. For L-band (1–2 GHz), fiberglass has a
typical relative dielectric constant of approximately ∈r ≃ 4.6. (a)
Assuming a flat-plane radome, determine the minimum thickness of
fiberglass that causes no reflections at the center of the L-band. (b)
Using the thickness found in part (a), find the percentage of the
incident power which transmits to the other side of the radome at each
end of the L-band (i.e., 1 GHz and 2 GHz). Assume nonmagnetic case.
Get solution 16.
Radome design. A radome is to be designed for the nose of an aircraft
to protect an X-band weather radar operating between 8.5 and 10.3 GHz. A
new type of foam material with ∈r = 2 (assume lossless) is chosen for
the design. (a) Assuming a flat planar radome, determine the minimum
thickness of the foam that will give no reflections at the center
frequency of the band. Assume μr = 1. (b) Using the thickness found in
part (a), what percentage of the incident power is reflected at each end
of the operating frequency band? (c) A thin layer of a different
material (∈r = 4.1, tan δc = 0.04, thickness 0.25 mm) is added on one
side of the radome designed in part (a) to protect the radome from rain
erosion. What percent of the incident power is reflected at the center
frequency?
Get solution 17.
Transmission through a multilayered dielectric. (a) Find the three
lowest frequencies at which all of the incident power would be
transmitted through the three-layer structure shown in Figure. The
permeability of all three media is μ0. (b) If complete transmission is
required for any thickness of the center medium, what is the lowest
usable frequency? (c) Find the bandwidth of the transmission, defined as
the range between the two lowest percentage values adjacent to and on
either side of the frequency found in (b). Also find the lowest
percentage values of transmission. (d) Why does the reflection from
multiply coated optical lenses tend to be purple in color?...Figure
Multilayered dielectric. Problem.
Get solution 18.
Glass slab. Consider a 1-cm thick slab of crown glass, with index of
refraction n = 1.52. (a) If a beam of visible light at 500 nm is
normally incident from one side of the slab, what percentage of the
incident power transmits to the other side? (b) Repeat for 400 and 600
nm.
Get solution 19.
Refractive index of a liquid. To measure the refractive index of a
liquid, a container is designed as shown in Figure to hold the liquid
sample.57 Consider a container made of Teflon (∈r ≃ 2.08 and μr = 1 at
10 GHz) with wall thickness of L1 ≃ 1.04 cm on each side. A liquid with
refractive index n is poured inside the container’s compartment with
thickness L2 ≃ 1.49 cm. When a 10-GHz plane wave is normally incident
from one side of the container, the effective reflection coefficient at
that side is measured to be Γeff ≃ −0.39. (a) Find the refractive index
of the liquid (assume lossless case). (b) Recalculate Γeff at 20 GHz
(assume the same material properties apply). (c) Repeat part (b) at 5
GHz....Figure Refractive index of a liquid. Liquid with unknown index of
refraction n in a container of known dielectric constant ∈r. Problem.
Get solution 20.
Antireflection (AR) coating on a glass slab. A beam of light is
normally incident on one side of a 1-cm thick slab of flint glass
(assume n = 1.86) at 550 nm. (a) What percentage of the incident power
reflects back? (b) To minimize reflections, the glass is coated with a
thin layer of antireflection coating material on both sides. The
material chosen is magnesium fluoride (MgF2), which has a refractive
index around 1.38 at 550 nm. Find the approximate thickness of each
coating layer of MgF2 needed. (c) If a beam of light at 400 nm is
normally incident on the coated glass with the thickness of the coating
layers found in part (b), what percentage of the incident power reflects
back? (d) Repeat part (c) for a light beam at 700 nm.
Get solution 21.
AR coating on a glass slab. A 1-cm thick slab of flint glass (n = 1.86)
is to be coated only on one side so that when a beam of light is
incident on the uncoated side, a sample of the light beam that reflects
back from that side can be used to monitor the power of the incident
beam. Assuming the light beam to be normally incident at 550 nm and the
coating material used on the other side to be MgF2, calculate the
percentage of the incident power that reflects back.
Get solution 22.
Infrared antireflection coating. To minimize reflections at the
air–germanium interface in the infrared frequency spectrum, a coating
material with an index of refraction of 2.04 is introduced as shown in
Figure. (a) If the thickness of the coating material is adjusted to be a
quarter-wavelength in the coating material for a free-space wavelength
of 4 μm, find the effective reflection coefficient at free-space
wavelengths of 4 and 8 μm. (b) Repeat part (a) if the thickness of the
coating material is adjusted to be a quarter-wavelength in the coating
material at 8 μm....Figure Infrared antireflection coating. Problem.
Get solution 23.
Reflection from ferrite-titanate slab. A 30-GHz uniform plane wave is
normally incident from air onto an interface consisting of a second
section (medium 2) of electrical length d having a complex permittivity
of ∈c = ∈0(3 − j 4) and a complex permeability of μc = μ0(3 − j 4)
followed by a third section (medium 3) consisting of air, as shown in
Figure (a) Calculate the values of the real and imaginary parts of the
propagation constant γ2 (i.e., α2 and β2) in medium 2 and determine all
of the value(s) of d for which the effective reflection coefficient Γeff
is zero. (b) For d = 0.25 cm, determine the fraction of the incident
wave power that is transmitted into medium 3 (air)....Figure Reflection
from ferritetitanate slab. Problem.
Get solution 24.
Superwide infrared antireflection coating. A wideband antireflection
coating system as shown in Figure is designed to be used between air and
germanium at infrared frequencies. If the thicknesses of the coating
layers are each quarter wavelength for operation at λ1 = 3.5 μm, find
and sketch the magnitude of the effective reflection coefficient over
the range from 3 to 12 μm....Figure Infrared antireflection coating.
Problem.
Get solution 25.
A snow-covered glacier. Glaciers are huge masses of ice formed in the
cold polar regions and in high mountains. Most glaciers range in
thickness from about 100 m to 3000 m. In Antarctica, the deepest ice on
the polar plateau is 4.7 km. Consider a large glacier in Alaska covered
with a layer of snow of 1 m thickness during late winter. A radar signal
operating at 56 MHz is normally incident from air onto the air–snow
interface. Assume both the snow and the ice to be lossless and
nonmagnetic; for ice, ∈r = 3.2, and for snow, ∈r can vary between 1.2
and 1.8. Assuming both the snow and the ice to be homogeneous and the
ice to be semi-infinite in extent, calculate the reflection coefficient
at the air–snow interface for three different permittivity values of
snow: ∈r = 1.2, 1.5, and 1.8. For which case is the snow layer most
transparent (invisible) to the radar signal? Why?
Get solution 26.
Minimum ice thickness. Consider a 500-MHz uniform plane wave radiated
by an aircraft radar normally incident on a freshwater (∈r = 88, μr = 1)
lake covered with a layer of ice (∈r = 3.2, μr = 1), as shown in
Figure. (a) Find the minimum thickness of the ice such that the
reflected wave has maximum strength. Assume the lake water to be very
deep. (b) What is the ratio of the amplitudes of the reflected and
incident electric fields?...Figure Aircraft radar signal incident on an
icy lake surface. Problem.
Get solution 27.
A snow–ice-covered lake. An interior lake in Alaska can be 30–100 m
deep and is covered with ice and snow on the top, each of which can be
about a meter deep in late winter.58 Consider a 5-GHz C-band radar
signal normally incident from air onto the surface of a lake that is 50 m
deep, covered with a layer of snow (assume ∈r = 1.5) of 60 cm thickness
over a layer of ice (∈r = 3.2) of 1.35 m. Assume both the snow and the
ice to be lossless and nonmagnetic. Also assume the lake water, with ∈cr
= 68 − j 35 at 5 GHz at 0◦C, to be slightly brackish (salty), with an
approximate conductivity of σ = 0.01 S-m−1, and the bottom of the lake
to consist of thick silt with ∈r ≃ 50. (a) Calculate the reflection
coefficient at the air interface with and without the snow layer. (b)
Repeat part (a) at an X-band radar frequency of 10 GHz. Assume all the
other parameters to be the same except for the lake water, ∈cr = 42 − j
41 at 10 GHz and 0◦C. Use any approximations possible, with the
condition that sufficient justifications are provided.
Get solution 28.
Incidence on an air gap between two dielectrics. A 1 GHz uniform plane
wave having an incident electric field amplitude of and propagating in a
lossless dielectric medium (∈1 = 4∈0, μ1 = μ0) is normally incident on
an airgap (i.e., ∈2 = ∈0, μ2 = μ0) of length d between itself and an
identical dielectric (∈3 = 4∈0, μ3 = μ0). (a) Determine the magnitude of
the reflected wave and the complete timedomain expression for the
electric field Er(z , t) of the reflected wave for d = λ2/8 and d =
λ2/4. (b) Determine the time-average electromagnetic power density (in
W-m−2) transmitted through the airgap and the complete time-domain
expression for the transmitted electric field ∈3(z , t) for d = λ2/8 and
d = λ2/4. (c) Determine the the complete time-domain expression for the
electric field of the electromagnetic wave within the airgap (i.e.,
E2(z , t)) for d = λ2/8 and d = λ2/4....
Get solution 29.
Air–water–air interface. A uniform plane wave operating at 2.45 GHz in
air is normally incident onto a planar water boundary at 20◦C (∈r = ∈ʹr −
j∈ʺr = 79 − j 11).59 (a) Calculate the percentage of the incident power
that is transmitted into the water (assume the water region to be
nonmagnetic and semi-infinite in extent). (b) What percentage of the
incident power is absorbed in the first 1-cm thick layer of water? (c)
If the water layer has a finite thickness of 1 cm with air on the other
side (i.e., air–water–air interface), calculate the percentage of the
incident power that is absorbed in the water layer and compare it with
the result of part (b).
Get solution 30.
Incidence on a coated perfect conductor. A 1.8 GHz uniform plane wave
having an incident electric field amplitude of Ei0 = 1ej0 V-m−1 and
propagating in a lossless dielectric medium (∈1, μ0) is incident on a
perfect conductor coated with a lossy dielectric as shown in Figure. (a)
Determine the total amount of power absorbed in the lossy dielectric.
(b) Is there a nonzero surface current induced on the surface of the
perfect conductor (at z = 0)? If so, determine its magnitude, phase, and
direction and write the real time dependent expression, that is, Īs (z ,
t)....Figure Normal incidence on a coated perfect conductor. Problem.
Get solution 31.
Air–concrete wall–air. A 900-MHz wireless communication signal is
normally incident from one side onto a reinforced concrete wall of
thickness d having air on both sides. (a) Find the percentages of the
incident power that is reflected back and that is transmitted to the
other side of the wall for three different wall thicknesses: 10, 20, and
30 cm. (See Problem for data on the properties of reinforced concrete
wall.) (b) Repeat part (a) at 1.8 GHz.ProblemConcrete wall. The
effective complex dielectric constant of walls in buildings are
investigated for wireless communication applications.72 The relative
dielectric constant of the reinforced concrete wall of a building is
found to be ∈r = 6.7 − j 1.2 at 900 MHz and ∈r = 6.2 − j 0.69 at 1.8
GHz, respectively. (a) Find the appropriate thickness of the concrete
wall to cause a 10 dB attenuation in the field strength of the 900 MHz
signal traveling over its thickness. Assume μr = 1 and neglect the
reflections from the surfaces of the wall. (b) Repeat the same
calculations at 1.8 GHz.
Get solution 32.
Reflection from multiple interfaces. A uniform plane wave is normally
incident on a multiple dielectric interface consisting of two sections
(mediums 2 and 3) of the same electrical length (l1 = d and l2 = 2d) as
shown in Figure. Determine the value of d (if any) such that the all of
the incident wave power is transmitted to medium 4 (i.e., such that the
reflection coefficient in medium 1 is zero)....Figure Multiple
dielectric interfaces. Problem.
Get solution 33.
Oblique incidence on a perfect conductor. A 30W-m−2 uniform plane wave
in air is obliquely incident on a perfect conductor boundary located at
the y = 0 plane. The electric field of the incident wave is given
by...(a) Find E0, f , and θi. (b) Find Er. (c) Find the total electric
field E1 and the nearest positions (with respect to the conductor
surface) of the minima and maxima of its magnitude.
Get solution 34.
Oblique incidence on a perfect conductor. A parallel-polarized (with
respect to the plane of incidence) 100 μW-(cm)−2, 4-GHz wireless
communication signal in air is incident on a perfect conductor surface
located at y = 0 at an incidence angle of θi = 30◦ as shown in Figure.
The signal can be approximated as a uniform plane wave. (a) Write the
instantaneous expressions for Ēi(y, z , t) and Hi(y, z , t). (b) Find
Er(y, z) and Hr(y, z) of the reflected wave. (c) Find the magnitude of
the total magnetic field phasor H1(y, z) and sketch it as a function of
y....Figure Oblique incidence on a perfect conductor. Problem.
Get solution 35.
Plane wave at 45◦ angle. A plane wave in air is incident at 45◦ upon a
perfectly conducting surface located at x = 0. The plane wave consists
of two components as follows:...(a) Write the perpendicular and parallel
polarization components of the electric fields of the reflected wave,
and show that the tangential electric fields satisfy the boundary
conditions.(b) What are the polarizations of the incident and the
reflected waves?
Get solution 36.
Oblique incidence on a dielectric medium. Consider oblique incidence of
a uniform plane wave on the interface between two nonmagnetic
dielectric media with permittivities ∈A and ∈B. When a perpendicularly
polarized plane wave is incident from medium A (i.e., ∈1= ∈A) onto
medium B (i.e., ∈2= ∈B) at an incidence angle θi = θ1, the transmitted
angle is θt = θ2, and the reflection coefficient is.... If the
propagation direction is reversed, that is, if a new incident wave with
the same polarization is now incident from medium B (i.e., ∈1= ∈B) onto
medium A (i.e., ∈2= ∈A) at an incidence angle θi = θ2, what is the
numerical value of the reflection coefficient...?
Get solution 37.
Oblique incidence. A uniform plane wave is obliquely incident at an
angle θi at the interface between two nonmagnetic (μ1 = μ2 = μ0)
dielectric media as shown in Figure. The relative permittivity of the
second medium is known to be ∈2r = 3, and the electric field of the
incident wave is given by...(a) Calculate the relative dielectric
constant ∈1r and the angle of incidence θi. (b) Write the corresponding
expression for the magnetic field of the incident wave (i.e., Hi(x, z ,
t)).(c) Determine the percentage of the incident power that will be
transmitted across the interface....Figure Oblique incidence. Problem.
Get solution 38.
Reflection from ground. A 8-GHz and 200-μW-m−2 microwave communication
signal in air is obliquely incident at θi = 60◦ onto the ground (assume
lossless and ∈r = 15) located at z = 0. (a) If the incident wave is
perpendicularly polarized, write the complete expressions for Ei, Er,
and Et. (b) Repeat part (a) for a parallel-polarized wave.
Get solution 39.
Air–dielectric interface. A uniform plane wave propagating in air has
an electric field given by...where E0 is a real constant. The wave is
incident on the planar interface (located at y = 0) of a dielectric with
μr = 1, ∈r = 3, as shown in Figure. (a) What are the values of the wave
frequency and the angle of incidence? (b) What is the polarization of
the incident wave (i.e., linear, circular, elliptical, right-handed or
left-handed)? (c) Write the complete expression for the electric field
of the reflected wave in a simplified form. (d) What is the polarization
of the reflected wave?...Figure Air–dielectric interface. Figure for
Problem.
Get solution 40.
Oblique incidence on a dielectric. An elliptically polarized uniform
plane wave is incident from free space (∈0, μ0) onto on a lossless
nonmagnetic dielectric (∈2, μ0) at an incidence angle θi = 70◦. Assuming
the coordinate system and the placement of the interface to be as in
Section 9.6 (e.g., Figure), the electric field vector of the incident
wave can be written as:...(a) Determine the dielectric constant ∈2 for
which the reflected wave is right-hand (RH) circularly polarized. (b)
For the value of ∈2 found in part (a), determine the fraction of the
incident wave power that is transmitted into the second medium. (c)
Specify the polarization (the type, i.e., linear/circular/elliptical,
and the sense, i.e., RH or LH) of the wave transmitted into the second
medium....Figure Oblique incidence at a dielectric boundary. Uniform
plane wave obliquely incident at a dielectric boundary located at the z =
0 plane. Dashed lines AC, BD, and EB are the wavefronts (planar
surfaces of constant phase).
Get solution 41.
Brewster angle at the air–water interface. A perpendicularly polarized
uniform plane wave is obliquely incident from air onto the surface of a
smooth freshwater lake (assume lossless and nonmagnetic with ∈r ≃ 81) at
the Brewster angle (i.e., θi = θiB). Calculate the reflection and
transmission coefficients.
Get solution 42.
Air–ice interface. A 1-W-m−2, 1-GHz radar signal is obliquely incident
at an angle θi = 30◦ from air onto an air–ice interface. (a) Assuming
the ice to be lossless, nonmagnetic and semiinfinite in extent, with ∈r ≃
3.17, calculate the reflection and the transmission coefficients and
the average power densities of the reflected and the transmitted waves
if the incident wave is perpendicularly polarized. (b) Repeat part (a)
for an incident wave that is parallelpolarized. (c) Find the Brewster
angle and repeat parts (a) and (b) for a wave incident at the Brewster
angle.
Get solution 43.
Communication over a lake. Consider a ground-to-air communication
system as shown in Figure. The receiver antenna is on an aircraft over a
huge lake circling at a horizontal distance of ∼10 km from the
transmitter antenna as it waits for a landing time. The transmitter
antenna is located right at the shore mounted on top of a 100-m tower
above the lake surface overlooking the lake and transmits a parallel
polarized (with respect to the plane of incidence) signal. The
transmitter operates in the VHF band. The pilot of the aircraft
experiences noise (sometimes called ghosting effect) in his receiver due
to the destructive interference between the direct wave and the
ground-reflected wave and needs to adjust his altitude to minimize this
interference. Assuming the lake to be flat and lossless with ∈r ≃ 80,
calculate the critical height of the aircraft in order to achieve clear
transmission between the transmitter and the receiver....Figure
Communication over a lake. Problem.
Get solution 44.
Air-to-air communication. Consider two helicopters flying in air
separated by a horizontal distance of 2 km over a flat terrain as shown
in Figure. The pilot of one of the helicopters, located at an altitude
of 100 m, transmits a parallel-polarized (with respect to the plane of
incidence) VHF-band signal (assume 200 MHz) to communicate with the
other helicopter. The pilot of the other helicopter needs to adjust her
altitude to eliminate the noise on her receiver due to the interference
of the ground-reflected wave with the direct wave. (a) Assuming the
ground to be homogeneous, nonmagnetic and lossless with ∈r ≃ 16, find
the critical altitude of the receiver helicopter in order to minimize
this interference. (b) Consider another scenario when both helicopters
are at 250 m altitude. In this case, what should be the horizontal
separation distance between the helicopters in order to achieve clear
signal transmission? (c) Repeat both (a) and (b) for the case in which
the helicopters try to land at a remote site in Alaska where the ground
is covered with permafrost (assume ∈r ≃ 4)....Figure Air-to-air
communication. Problem.
Get solution 45.
Signal to Interference Ratio (SIR). You have just launched your
start-up company, with your headquarters located at a distance of d =
140 m from a cellular tower of height h1 = 50 m. As the CEO you choose
to have your office on the top floor of your building, at a height of h2
= 20 m, as shown in Figure. (a) Determine the signal-to-interference
ratio (SIR):...assuming that the permittivity of the ground is ∈ =
5.55∈0. Provide separate results for both parallel and perpendicular
polarization. (b) After a few dropped calls, you feel that you are not
satisfied with the SIR you have and decide that you can move your office
to a lower floor to maximize SIR. Calculate the height h2 that results
in maximum SIR and the value of maximum SIR. Provide separate results
for both parallel and perpendicular polarization....Figure Cell phone
reception. Problem.
Get solution 46.
Dry asphalt roads acting as mayfly traps? Adult mayflies have only a
few hours in which to find a mate and reproduce. Direct sunlight
reflected off the surface of water is strongly polarized in the
horizontal plane (i.e., parallel polarized), and many water-dwelling
insects, including mayflies, use this reflected polarized light to
identify open stretches of water where they can lay their eggs during
their brief mating period. However, researchers discovered that light
reflected from dry asphalt roads is also horizontally polarized and
visually deceives mayflies into laying their eggs on roads instead of in
rivers.60 The higher the degree of polarization of the reflected light,
the more attractive it is for mayflies. Hence, mayflies swarming,
mating, and egg-laying on asphalt roads are predominantly deceived by
and attracted to the asphalt surface because the largely horizontally
polarized reflected light imitates a water surface. Note that although
sunlight has mixed polarization and is incident on the asphalt over a
range of angles, the polarization of the reflected light is
predominantly horizontal because of the deep minimum for ρ|| in the
vicinity of the Brewster angle (see Figure). (a) If the reflected light
from asphalt is almost 100% horizontally polarized when the light is
incident on the asphalt surface at an incidence angle of about 57.5◦,
calculate the effective refractive index of asphalt. (b) Assuming the
refractive index of water to be nw ≃ 1.33, find the angle at which the
reflected light from the water surface is 100% horizontally
polarized....Figure Reflection coefficient versus the angle of
incidence. (a) Magnitude and phase of reflection coefficient for
perpendicular (Γ⊥) and parallel (Γ||) polarization versus angle of
incidence θi for distilled water (∈2r = 81), flint glass (∈2r = 10), and
paraffin (∈2r = 2), all assumed to be lossless. For clarity, the
complement of the phase angle φ|| (i.e., π − φ||) is sketched, rather
than φ|| itself. In all cases, φ|| = π for θiθiB and φ|| = 0 for θi >
θiB, while φ⊥ = π for all θi.
Get solution 47.
Total internal reflection. A uniform plane wave with a magnetic field
given by...is obliquely incident at an interface at z = 0 separating two
nonmagnetic lossless media as shown in Figure. (a) Calculate the
relative dielectric constant ∈1r of medium 1 and the angle of incidence
θi. (b) Find the maximum value of the relative dielectric constant ∈2r
of medium 2 for total internal reflection to occur. (c) Is it possible
to achieve total transmission by adjusting the incidence angle? If yes,
use the maximum value of ∈2r found in part (b) to determine the
incidence angle at which total transmission would occur....Figure Total
internal reflection. Problem.
Get solution 48.
Reflection from prisms. Consider the various right-angled prisms shown
in Figure. (a) What is the minimum index of refraction n1 of the prism
necessary in each case for there to be no time-average power transmitted
across the hypotenuse when the prisms are (i) in free space, (ii) in
water (assume n2 ≃ 1.33). (b) At these refractive index values (found in
(a)), what are the exit angles θte?...Figure Reflection from prisms.
Problem.
Get solution 49.
MgF2 prism. A 45◦–90◦–45◦ prism is constructed from MgF2 (n = 1.38) to
be used to turn a light beam around by 90◦ by internal reflection at its
hypotenuse. Does the light beam exit at the hypotenuse, and if so, what
is its exit angle?
Get solution 50.
Refractive index of a prism. An experiment is designed to measure the
refractive index of a prism using the principle of total internal
reflection.61 In this experiment, a plane-polarized, collimated,
monochromatic beam of light is obliquely incident from one side of the
prism at an incidence angle of θi as shown in Figure. The incidence
angle θi of the beam is adjusted to a critical value ψc such that the
incidence angle on the other side of the prism is equal to the critical
angle of incidence θic. Thus, by measuring the refracting angle A of the
prism and the critical incidence angle ψc, the refractive index np of
the prism can be calculated. (a) Show that...(b) For a prism under test,
the refracting angle of the prism and the critical incidence angle
adjusted are measured to be A = 60◦ and ψc = 42◦, respectively.
Calculate the refractive index np of this prism....
Get solution 51.
Right-angle prism. A new technique is proposed to measure small angles
in optical systems using right-angle prisms.62 Consider a 45°–90°–45°
prism as shown in Figure made of glass having a refractive index of n ≃
1.515 to be used in an experiment. A light beam that is obliquely
incident at an incidence angle θ1 on the entrance face undergoes
reflection and refraction at the entrance face, the hypotenuse face, and
the exit face of the prism. (a) For an incidence angle of θ1 = 30◦,
find the exit angles θ4 and θ6. (b) Find the critical incidence angle θ1
of the incident beam on the entrance face that results in no
transmission at the hypotenuse face. (c) What happens to the critical
angle found in part (b) when the hypotenuse face of the prism is coated
with an antireflection coating (single or multiple layers)? (d) Repeat
parts (a) and (b) if the prism is made of a different type of glass with
n ≃ 1.845....Figure Right-angle prism. Problem.
Get solution 52.
A V-shaped prism. A V-shaped right-angled prism is designed to measure
accurately the refractive index of liquids63 as shown in Figure. A laser
beam normally incident from one side is refracted at the inclined
faces, between which the liquid sample under test is placed, and leaves
the prism on the other side at a deflection angle ψ. (a) A liquid of
known refractive index n ≃ 1.512 is placed in the top compartment of the
prism with np ≃ 1.628. Find the deflection angle ψd. (b) Using the same
prism, the deflection angle for a different liquid with unknown
refractive index is approximately measured to be ψ≃ 55.5°. Calculate the
refractive index n of this liquid....Figure A V-shaped prism. Problem.
Get solution 53.
Turning a perfect corner. An interesting optical phenomenon is the
invariance of the angle between the incoming and outgoing light rays
passing through a right-angle isosceles prism with a silver-coated
hypotenuse.64 For the isosceles right-angle prism shown in Figure, the
incident ray enters at side AB and exits at side AC after being
reflected twice and refracted twice. Show that the total deviation angle
ψd between the incident and the exiting rays is exactly 90°....Figure
Turning a perfect corner. Problem.
Get solution 54.
Air–oil–water. Consider a layer of oil (assume n ≃ 1.6) about 5 mm
thick floating over a body of water (n ≃ 1.33). (a) If a light ray is
obliquely incident from air onto the oil surface, find the range of
incidence angles (if any) that results in total internal reflection at
the oil–water interface. (b) If a light ray is obliquely incident from
water onto the oil surface, find the range of incidence angles (if any)
that results in total internal reflection at the oil–air interface.
Get solution 55.
An in-line Brewster angle prism. Brewster angle prisms are optical
elements that use light at the polarizing angle to obtain perfect
transmission of parallel polarized light. An inline Brewster angle
polarizing prism is designed65 as shown in Figure, to polarize a light
beam without changing its direction. Consider an unpolarized light beam
incident on this prism at point A at an incidence angle of θi = 31.639◦.
Given the prism angles to be α = 63.278° and β = 103.939° and the prism
refractive index to be n = 1.623, (a) determine whether light exits the
prism at points B, C, and/or D, (b) find the exit angles θB, θC, and/or
θD at these exit points, (c) specify the polarization of each exiting
beam, and (d) find the angle between each exiting beam and the incident
beam. Discuss your results in terms of the stated purpose of this
particular prism design....Figure In-line polarizing prism. Problem.
Get solution 56.
V-shaped prism polarizer. A symmetrical three-reflection silicon (Si)
polarizer prism is designed as shown in Figure, based on the
Brewster-angle internal reflection that occurs at the base of the
prism.66 (a) In Figure, the prism angle A is adjusted for use at 1.3 μm
light wave communication wavelength (the refractive index of silicon at
1.30 μm is nSi ≃ 3.5053), to A ≃ 52.9613°. Assuming the incident light
beam entering the prism on the left side to be unpolarized, find the
polarization of the beam exiting on the right side. (b) Find the new
value of A for the prism to be used as a polarizer at 1.55-μm wavelength
(the refractive index of silicon at 1.55 μm is nSi ≃ 3.4777). (c)
Another interesting design is to coat the base of the silicon prism with
silicon dioxide (SiO2) as shown in Figure. At 1.3 μm, the prism angle
is adjusted to A ≃ 56.217◦. Find the refractive index of SiO2 at 1.3 μm.
(d) Repeat part (c) at 1.55 μm when the prism angle is adjusted to A ≃
56.277°....Figure V-shaped prism polarizer. Problem.
Get solution 57.
Limestone wall versus brick wall. Consider two buildings, one with
limestone (∈r = 7.51, σ = 0.03 S-m−1) exterior walls and the other with
brick (∈r = 4.44, σ = 0.01 S-m−1) exterior walls, with the material
properties cited measured67 at 4 GHz. These walls represent some of the
typical building surfaces that affect the propagation of mobile radio
signals. Assuming the walls to be lossless, nonmagnetic, semi-infinite
in extent, and neglecting the roughness of their surfaces, calculate the
reflection coefficients at the surface of each building for both
perpendicular and parallel polarizations at three different angles of
incidence of θi = 30°, 45°, and 60° and compare the results.
Get solution 58.
Refractive index of concrete. Knowledge of the dielectric properties of
construction material is important because the reflection and
transmission characteristics of buildings and rooms are governed by
these properties. The complex refractive index of a plain concrete plate
mixed from Portland cement, gravel, sand, and water was measured at
57.5 GHz for use in designing and testing millimeter-wave communication
systems.68 The measured refractive index of the concrete 14 months after
concreting was n = 2.55 − j 0.084. (a) Using the measured values for
concrete, calculate and sketch the magnitude of the reflection
coefficient at the air–concrete interface at 57.5 GHz as a function of
the incidence angle varying between 0◦ and 90◦ for both perpendicular
and parallel polarization cases. Assume the concrete to be semi-infinite
in extent. (b) For a 5-cm thick concrete wall having air on both sides,
calculate the magnitude of the normal incidence reflection coefficient
at 57.5 GHz and compare it with the result of part (a). (c) Repeat part
(b) for a thickness of 10 cm.
Get solution 59.
Oblique incidence on a multiple dielectric interface. Suppose that a
parallel polarized uniform plane wave is incident, in air, on a
nonmagnetic glass slab (assume ∈r = 2) of thickness d, as shown in
Figure. The angle of incidence θi is chosen to be the air-glass Brewster
angle so that there is no reflection at the first interface (z = 0).
Find the complete expression for the electric field phasor of the wave
transmitted into air (z > d), that is, Et(x, z). Hint: First find the
reflection coefficient at the z = d interface....Figure Oblique
incidence on a multiple interface. Problem.
Get solution 60.
Oblique incidence on a multiple dielectric interface. A perpendicularly
polarized uniform plane wave propagating in air is incident obliquely
(at an angle θi) on a structure consisting of two lossless and
nonmagnetic dielectrics: a coating layer with permittivity ∈1r and
thickness d coated on another dielectric with permittivity ∈2r and of
infinite thickness, as shown in Figure. Derive expressions for the
effective reflection and transmission coefficients (Γeff and Jeff) in
terms of parameters of the media (∈1r , ∈2r), the angle of incidence θi,
and the slab thickness d....Figure Oblique incidence on a multiple
interface. Problem.
Get solution 