Deck 8: Plane Electromagnetic Waves

Question 1

Define uniform plane wave.

Answer 1

The particular solution of Maxwell's equations with electric field E assuming the same phase, same direction and same magnitude in infinite planes perpendicular to the direction of propagation is called uniform plane wave. The uniform plane wave is similar to the magnetic field H also. The uniform plane wave is also called the transverse electromagnetic wave (TEM) as it contains the transverse plane containing both the electric field E and magnetic field H. A uniform plane wave does not exist in practice because a source infinite in extent would be required to create it.

Question 2

Write the phasor expressions for the electric and magnetic field intensity vectors of an x-polarized uniform plane wave propagating in the + z-direction.

Answer 2

Consider the equation for phasor electric field intensity for a uniform plane wave propagating in the +z-direction.
Where,
is a constant vector.
The phasor magnetic field intensity for a uniform plane wave propagating in the +z-direction is,
Where,
Thus, the required phasor form of is written.

Question 3

What is the significance of plasma frequency

Answer 3

The effective permittivity of the plasma on the wave propagation as follows.
Where,
Plasma angle frequency
The expression for plasma angle frequency as follows.
Where,
Number of electrons per unit volume
Electron charge
Mass of electrons
Permittivity of free space
The plasma frequency is proportional to Number of electrons per unit volume.
So using the plasma frequency we will decide frequency of communication because the number of electrons per unit volume will vary in each layer of ionosphere.

Question 4

What do we know about the magnitude of the tangential components of E and H at the interface when a wave impinges normally on a perfectly conducting plane boundary

Question 5

What is meant by the wave impedance of the total field. When is this impedance equal to the intrinsic impedance of the medium

Question 6

Under what conditions will the reflection and transmission coefficients for perpendicular polarization be the same as those for parallel polarization

Question 7

Prove that the electric field intensity in Eq. (8-22) satisfies the homogeneous Helmholtz's equation provided that the condition in Eq. (8-23) is satisfied.

Question 8

Determine and compare the intrinsic impedance, attentuation constant (in both Np/m and dB/m), and skin depth of copper [ cu = 5.80 x 10 7 (S/m)], silver [ ag = 6.15 x 10 7 (S/m)], and brass [ br = 1.59 x 10 7 (S/m)] at the following frequencies:
a) 60 (Hz),
(b) 1 (MHz), and
(c) 1 (GHz).

Question 9

Assuming that the radiation electric field intensity of an anten system is

find the expression for the average outward power flow per unit area.

Question 10

Determine the condition under which the magnitude of the reflection coefficient equals that of the transmission coefficient for a uniform plane wave at normal incidence on an interface between two lossless dielectric media. What is the standing-wave ratio in dB under this condition

Question 11

A uniform plane wave is incident on the ionosphere at an angle of incidence i = 60°. Assuming a constant electron density and a wave frequency equal to one-half of the plasma frequency of the ionosphere determine
a) and , and
b) and .
Interpret the significance of these complex quantities.

Question 12

An electromagnetic wave in dielectric medium 1 ( 1 , 0 ) impinges obliquely on a boundary lane with dielectric medium 2 ( 2 , 0 ). Let i , and t , denote the incident and refraction angles, respectively, and prove the following:
a) For perpendicular polarization:

b) For parallel polarization:

(These four relations are known as Fresnel formulas)

Question 13

What is a wavefront

Question 14

What is meant by the polarization of a wave When is a wave linearly polarized Circularly polarized

Question 15

When does the equivalent permittivity of the ionosphere become negative What is the significance of a negative permittivity in terms of wave propagation

Question 16

Define plane of incidence.

Question 17

A thin dielectric coating is sprayed on optical instruments to reduce glare. What factors determine the thickness of the coating

Question 18

Prove that, under the condition of no reflection at an interface, the sum of the Brewster angle and the angle of refraction is /2 for:
a) perpendicular polarization ( 1 2 ), and
b) parallel polarization ( 1 2 ).

Question 19

A Doppler radar is used to determine the speed of a moving vehicle by measuring the frequency shift of the wave reflected from the vehicle.
(a) Assuming that the reflecting surface of the vehicle can be represented by a perfectly conducting plane and that the transmitted signal is a time-harmonic uniform plane wave of a frequency f incident normally on the reflecting surface, find the relation between the frequency shift f and the speed u of the vehicle.
(b) Determine u both in (km/hr) and in (miles/hr) if f = 2.33 (kHz) with f = 10.5 (GHz).

Question 20

A 3 (GHz), y-polarized uniform plane wave propagates in the +x-direction in a nonmagnetic medium having a dielectric constant 2.5 and a loss tangent 10 - 2.
a) Determine the distance over which the amplitude of the propagating wave will be cut in half.
b) Determine the intrinsic impedance, the wavelength, the phase velocity, and the group velocity of the wave in the medium.
c) Assuming E = a y 50 sin (6 10 9 t + /3)(V/m) at x = 0, write the instantaneous ex­pression for H for all t and x.

Question 21

From the point of view of electromagnetics, the power transmitted by a lossless coaxial cable can be considered in terms of the Poynting vector inside the dielectric medium between the inner conductor and the outer sheath. Assuming that a d-c voltage V 0 applied between the inner conductor (of radius a ) and the outer sheath (of inner radius b ) causes a current I to flow to a load resistance, verify that the integration of the Poynting vector over the cross-sectional area of the dielectric medium equals the power V 0 I that is transmitted to the load.

Question 22

A uniform plane wave in air with E i (z) = a x 10e -J6z (V/m) is incident normally on an interface at z = 0 with a lossy medium having a dielectric constant 2.5 and a loss tangent 0.5. Find the following:
a) The instantaneous expressions for E r (z, t), H r (z, f), E t (z, t), and H r (z, t), using a cosine reference.
b) The expressions for time-average Poynting vectors in air and in the lossy medium.

Question 23

A 10(kHz) parallelly polarized electromagnetic wave in air is incident obliquely on an ocean surface at a near-grazing angle i = 88°. Using r = 81, r - 1, and = 4 (S/m) for seawater, find
(a) the angle of refraction t ,
(b) the transmission coefficient T||,
(c) ( P av ) r / ( P av ) i and
(d) the distance below the ocean surface where the field intensity has been diminished by 30 (dB).

Question 24

For an incident wave with parallel polarization:
Find the relation between the critical angle c and the Brewster angle bll for nonmagnetic media.
Plot c and Bll versus the ratio 1/ 2.

Question 25

Write the homogeneous vector Helmholtz's equation for E in free space.

Question 26

Two orthogonal linearly polarized waves of the same frequency are combined. State the conditions under which the resultant will be (a) another linearly polarized wave, (b) a circularly polarized wave, and (c) an elliptically polarized wave.

Question 27

What is meant by the dispersion of a signal Give an example of a dispersive medium.

Question 28

What do we mean when we say that an incident wave has
(a) perpendicular polar­ization, and
(b) parallel polarization

Question 29

How should the thickness of the radome in a radar installation be chosen

Question 30

By using Snell's law of refraction,
(a) express T and in terms of r1 , r2 , and i ; and
(b) plot T and versus c , for r1/ r2 = 2.25 for both perpendicular and parallel polarizations.

Question 31

For a harmonic uniform plane wave propagating in a simple medium, both E and Hvary in accordance with the factor exp (-jk • R ) as indicated in Eq. (8-26). Show that the four Maxwell's equations for uniform plane wave in a source-free region reduce to the following:
k × E = µH,
k × H = - µE,
k • E = 0,
k • H = 0.

Question 32

The magnetic field intensity of a linearly polarized uniform plane wave propagating in the + y-direction in seawater [ r = 80, r = 1, = 4 (S/m)] is
H = a x 0.1 sin (10 10 t - /3) (A/m) at y = 0.
a) Determine the attenuation constant, the phase constant, the intrinsic impedance,the phase velocity, the wavelength, and the skin depth.
b) Find the location at which the amplitude of H is 0.01 (A/m).
c) Write the expressions for E(y, t) and H(y, t) at y = 0.5 (m) as functions of t.

Question 33

A uniform plane electromagnetic wave propagates in the +z- (downward) direction and impinges normally at z = 0 on an ocean surface. Let the magnetic field at z = 0 be H(0, t) = a y H 0 cos 10 4 t (A/m).
a) Determine the skin depth. (For the ocean: Conductivity = , permeability = 0.)
b) Find the expressions for H(z, t) and E (z, t).
c) Find the power loss per unit area (in terms of H 0 ) into the ocean.

Question 34

A uniform plane wave in air with E i (z) = a x E 0 exp (-j 0 z) impinges normally onto the surface at z = 0 of a highly conducting medium having constitutive parameters 0 , , and ( / 0 1)
a) Find the reflection coefficient.
b) Derive the expression for the fraction of the incident power absorbed by the conducting medium.
c) Obtain the fraction of the power absorbed at 1 (MHz) if the medium is iron.

Question 35

A light ray is incident from air obliquely on a transparent sheet of thickness d with an index of refraction n , as shown in Fig.. The angle of incidence is i. Find (a) i , (b) the distance 1 at the point of exit, and (c) the amount of the lateral displacement 2 of the emerging ray.

Question 36

A perpendicularly polarized uniform plane wave in air of frequency f is incidentobliquely at an angle of incidence i; on a plane boundary with a lossy dielectric mediumthat is characterized by a complex permittivity 2 = ' - j ". Let the incident electric field be

a)Find the expressions of the transmitted electric and magnetic field intensity phasorsin terms of the given parameters.
b) Show that the angle of refraction is complex and that H t is elliptically polarized.

Question 37

Define wavenumber. How is wavenumber related to wavelength

Question 38

How is the E-field from AM broadcast stations polarized From television stations From FM broadcast stations

Question 39

Define group velocity. In what ways is group velocity different from phase velocity

Question 40

Define reflection coefficient and transmission coefficient. What is the relationship between them

Question 41

State Snell's law of reflection in words.

Question 42

In some books the reflection and transmission coefficients for parallel polarization are defined as the ratios of the amplitude of the tangential components of the reflected andtransmitted E fields, respectively, to the amplitude of the tangential component of theincident E field. Let the coefficients defined in this manner be designated
respectively.
a) Find
in terms of n 1 , n 2 , i and i t and compare them with and t m in Eqs. (8-221) and (8-222).
b) Find the relation between
, and compare it with Eq. (8-223).

Question 43

The instantaneous expression for the magnetic field intensity of a uniform plane wave propagating in the + y direction in air is given by

a) Determine k 0 and the location where H z vanishes at t = 3 (ms).
b) Write the instantaneous expression for E.

Question 44

Given that the skin depth for graphite at 100 (MHz) is 0.16 (mm), determine (a) the conductivity of graphite, and (b) the distance that a l-(GHz) wave travels in graphite such that its field intensity is reduced by 30 (dB).

Question 45

A right-hand circularly polarized plane wave represented by the phasor

impinges normally on a perfectly conducting wall at z = 0.
a) Determine the polarization of the reflected wave.
b) Find the induced current on the conducting wall.
c) Obtain the instantaneous expression of the total electric intensity based on a cosine time reference.

Question 46

Consider the situation of normal incidence at a lossless dielectric slab of thickness d. in air, as shown in Fig. 8-15 with
1 = 3 = 0 and 1 = 3 = 0
a) Find E r0 , E 2 + , E 2 - , and E t0 in terms of E i0 , d, 2 , and 2.
b) Will there be reflection at interface z = 0 if d = 2 /4 If d = 2 /2 Explain.

Question 47

A uniform plane wave with perpendicular polarization represented by Eqs. (8-196) and (8-197) is incident on a plane interface at z = 0, as shown in below Fig. Assuming 2 t and i C ,
(a) obtain the phasor expressions for the transmitted field (E t , H t ) , and
(b) verify that the average power transmitted into medium 2 vanishes.

Question 48

Define phase velocity.

Question 49

Define
(a) propagation constant,
(b) attenuation constant, and
(c) phase constant.

Question 50

Define Poynting vector. What is the SI unit for this vector

Question 51

Under what conditions will reflection and transmission coefficients be real

Question 52

State Snell's law of refraction.

Question 53

The E-field of a uniform plane wave propagating in a dielectric medium is given by

a) Determine the frequency and wavelength of the wave.
b) What is the dielectric constant of the medium
c) Describe the polarization of the wave.
d) Find the corresponding H -field.

Question 54

Assume the ionosphere to be modeled by a plasma region with an electron density that increases with altitude from a low value at the lower boundary toward a value N max and decreases again as the altitude gets higher. A plane electromagnetic wave impinges on the lower boundary at an angle i with the normal. Determine the highest frequency of the wave that will be turned back toward the earth. (Hint: Imagine the ionosphere to be strat­ified into layers of successively decreasing constant permittivities until the layer containing N max. The frequency to be determined corresponds to that for an emerging angle of /2.)

Question 55

A uniform sinusoidal plane wave in air with the following phasor expression for electric intensity
E i (x, z) = ay10e- J{6x +8z) (V/m)
is incident on a perfectly conducting plane at z = 0.
a) Find the frequency and wavelength of the wave.
b) Write the instantaneous expressions for E ; (x, z; t) and H,(x, z; t), using a cosine reference.
c) Determine the angle of incidence.
d) Find E r (x, z) and H,.(x, z) of the reflected wave.
e) Find E 1 (x, z) and H 1 (x, z) of the total field.

Question 56

A transparent dielectric coating is applied to glass ( r = 4, r = 1) to eliminate the reflection of red light [ 0 = 0.75 ( m)].
a) Determine the required dielectric constant and thickness of the coating.
b) If violet light [ 0 = 0.42 (( m)] is shone normally on the coated glass, what percent­age of the incident power will be reflected

Question 57

A uniform plane wave of angular frequency in medium 1 having a refractive index n 1 is incident on a plane interface at z = 0 with medium 2 having a refractive index n 2 ( n 1 ) at the critical angle. Let E i 0 and E t 0 denote the amplitudes of the incident and refracted electric field intensities, respectively.
a) Find the ratio E t 0 / E i 0 for perpendicular polarization.
b) Find the ratio E t 0 / E i 0 for parallel polarization.
c) Write the instantaneous expressions of E i ( x , z ; t ) and E t ( x , z ; t ) for perpendicular polarization in terms of the parameters , n 1 n 2 , i , and E i 0.

Question 58

Define intrinsic impedance of a medium. What is the value of the intrinsic impedance of free space

Question 59

What is meant by the skin depth of a conductor How is it related to the attenuation constant How does it depend on On f

Question 60

State Poynting's theorem.

Question 61

What are the values of the reflection and transmission coefficients at an interface with a perfectly conducting boundary

Question 62

Define critical angle. When does it exist at an interface of two nonmagnetic media

Question 63

Show that a plane wave with an instantaneous expression for the electric field
E(z, t) = a x E 10 sin ( t - kz) + a, E 20 sin ( t - kz + )
is elliptically polarized. Find the polarization ellipse.

Question 64

Prove the following relations between group velocity u g and phase velocity u p in a dispersive medium:

Question 65

Repeat below Problem for E i (y, z) = 5(a y + a z 3)e j6( 3y-z) (V/m).
PROBLEM:
A uniform sinusoidal plane wave in air with the following phasor expression for electric intensity
E i (x, z) = ay10e- J{6x +8z) (V/m)
is incident on a perfectly conducting plane at z = 0.
a) Find the frequency and wavelength of the wave.
b) Write the instantaneous expressions for E ; (x, z; t) and H,(x, z; t), using a cosine reference.
c) Determine the angle of incidence.
d) Find E r (x, z) and H,.(x, z) of the reflected wave.
e) Find E 1 (x, z) and H 1 (x, z) of the total field.

Question 66

Refer to Fig. 8-15, which depicts three different dielectric media with two parallel interfaces. A uniform plane wave in medium 1 propagates in the + z-direction. Let T 12 and T 23 denote the reflection coefficients between media 1 and 2 and between media 2 and 3, respectively. Express the effective reflection coefficient, r 0 , at z = 0 for the incident wave in terms of T 12 , T 23 , and 2 d.

Question 67

An electromagnetic wave from an underwater source with perpendicular polarization is incident on a water-air interface at i = 20°. Using r = 81 and r = 1 for fresh water, find (a) critical angle c , (b) reflection coefficient , (c) transmission coefficient , and (d) attenuation in dB for each wavelength into the air.

Question 68

What is Doppler effect

Question 69

What is the constitution of the ionosphere

Question 70

For a time-harmonic electromagnetic field, write the expressions in terms of electric and magnetic field intensity vectors for (a) instantaneous Poynting vector, and (b) time-average Poynting vector.

Question 71

A plane wave originating in medium 1 ( 1 , µ 1 = µ 0 , 1 = 0) is incident normally on a plane interface with medium 2 ( 2 1 , µ 2 = µ 0 , 2 = 0)- Under what condition will the electric field at the interface be a maximum A minimum

Question 72

Define Brewster angle. When does it exist at an interface of two nonmagnetic media

Question 73

Prove the following:
a) An elliptically polarized plane wave can be resolved into right-hand and left-hand circularly polarized waves.
b) A circularly polarized plane wave can be obtained from a superposition of two op­positely directed elliptically polarized waves.

Question 74

There is a continuing discussion on radiation hazards to human health. The following calculations provide a rough comparison.
a) The U.S. standard for personal safety in a microwave environment is that the power density be less than 10(mW/cm 2 ). Calculate the corresponding standard in terms of electric field intensity. In terms of magnetic field intensity.
b) It is estimated that the earth receives radiant energy from the sun at a rate of about 1.3 (kW/m 2 ) on a sunny day. Assuming a monochromatic plane wave (which it is not), calculate the equivalent amplitudes of the electric and magnetic field intensity vectors.

Question 75

For the case of oblique incidence of a uniform plane wave with perpendicular polarization on a perfectly conducting plane boundary as shown in Fig. 8-11, write
(a) the instantaneous expressions
E 1 (x, z; t) and H 1 (x, z; t)
for the total field in medium 1, using a cosine reference, and
(b) the time-average Poynting vector.

Question 76

A uniform plane wave with

in medium 1 ( 1 , 1 ) is incident normally onto a lossless dielectric slab ( 2 , 2 ) of a thickness d backed by a perfectly conducting plane, as shown in below Fig. Find
a) E r (z, t)
b)E 1 (z, t)
c) E 2 (z, t)
d) ( P av ) 1
e) ( P av ) 2
Determine the thickness d that makes Ej(z, t) the same as if the dielectric slab were absent.

Question 77

Glass isosceles triangular prisms shown in Fig. are used in optical instruments. Assuming r = 4 for glass, calculate the percentage of the incident light power reflected back by the prism.

Question 78

What is a TEM wave

Question 79

What is a plasma

Question 80

What is a standing wave