1. EE 542 Antennas & Propagation for Wireless Communications
Topic 3 - Basic EM Theory and Plane Waves

2. Outline EM Theory Concepts Maxwell’s Equations
Notation
Differential Form
Integral Form
Phasor Form
Wave Equation and Solution (lossless, unbounded, homogeneous medium)
Derivation of Wave Equation
Solution to the Wave Equation – Separation of Variables
Plane waves
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3. EM Theory Concept
The fundamental concept of em theory is that a current at a point in space is capable of inducing potential and hence currents at another point far away. J
E, H
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4. Introduction to EM Theory
The existence of propagating em waves can be predicted as a direct consequence of Maxwell’s equations.
These equations satisfy the relationship between the vector electric field, E and vector magnetic field, H in time and space in a given medium.
Both E and H are vector functions of space and time; i.e. E (x,y,z;t), H (x,y,z;t.)
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5. What is an Electromagnetic Field?
The electric and magnetic fields were originally introduced by means of the force equation.
In Coulomb’s experiments forces acting between localized charges were observed.
There, it is found useful to introduce E as the force per unit charge.
Similarly, in Ampere’s experiments the mutual forces of current carrying loops were studied.
B is defined as force per unit current.
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6. Why not use just force?
Although E and B appear as convenient replacements for forces produced by distributions of charge and current, they have other important aspects.
First, their introduction decouples conceptually the sources from the test bodies experiencing em forces.
If the fields E and B from two source distributions are the same at a given point in space, the force acting on a test charge will be the same regardless of how different the sources are.
This gives E and B meaning in their own right.
Also, em fields can exist in regions of space where there are no sources.
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7. Maxwell’s Equations
Maxwell's equations give expressions for electric and magnetic fields everywhere in space provided that all charge and current sources are defined.
They represent one of the most elegant and concise ways to state the fundamentals of electricity and magnetism.
These set of equations describe the relationship between the electric and magnetic fields and sources in the medium.
Because of their concise statement, they embody a high level of mathematical sophistication.
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8. Notation: (Time and Position Dependent Field Vectors)
E (x,y,z;t)
Electric field intensity (Volts/m)
H (x,y,z;t)
Magnetic field intensity (Amperes/m)
D (x,y,z;t)
Electric flux density (Coulombs/m2)
B (x,y,z;t)
Magnetic flux density (Webers/m2, Tesla)
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9. Notation: Sources and Medium
J (x,y,z;t)
Electric current density (Amperes/m2)
Jd (x,y,z;t)
Displacement current density (Amperes/m2) re
Electric charge density (Coulombs/m3) e er
Permittivity of the medium (Farad/m)
Relative permittivity (with respect to free space eo) m
Permeability of the medium (Henry/m)
Relative permittivity (with respect to free space mo) s
Conductivity of the medium (Siemens/m)
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10. Maxwell’s Equations – Physical Laws
Faraday’s Law  Changes in magnetic field induce voltage.
Ampere’s Law  Allows us to write all the possible ways that electric currents can make magnetic field. Magnetic field in space around an electric current is proportional to the current source.
Gauss’ Law for Electricity The electric flux out of any closed surface is proportional to the total charge enclosed within the surface.
Gauss’ Law for Magnetism The net magnetic flux out of any closed surface is zero.
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11. Differential Form of Maxwell’s Equations
Faraday’s Law:
(1)
Ampere’s Law:
(2)
Gauss’ Law:
(3)
(4)
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12. Constitutive Relations
Constitutive relations provide information about the environment in which electromagnetic fields occur; e.g. free space, water, etc.
permittivity
(5)
permeability
(6)
Free space values.
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13. Time Harmonic Representation - Phasor Form
In a source free ( ) and lossless ( ) medium characterized by permeability m and permittivity e, Maxwell’s equations can be written as:
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14. Examples of del Operations
The following examples will show how to take divergence and curl of vector functions
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15. Example 1
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16. Solution 1
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17. Example 2
Calculate the magnetic field for the electric field given below. Is this electric field realizable?
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18. Solution
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19. Solution continued
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20. Solution continued
To be realizable, the fields must satisfy Maxwell’s equations!
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21. Solution Continued
These fields are NOT realizable. They do not form em fields.
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22. Time Harmonic Fields
We will now assume time harmonic fields; i.e. fields at a single frequency.
We will assume that all field vectors vary sinusoidally with time, at an angular frequency w; i.e.
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23. Time Harmonics and Phasor Notation
Using Euler’s identity
The time harmonic fields can be written as
Phasor notation
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24. Phasor Form Information on amplitude, direction and phase
Note that the E and H vectors are now complex and are known as phasors
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25. Time Harmonic Fields in Maxwell’s Equations
With the phasor notation, the time derivative in Maxwell’s equations becomes a factor of jw:
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26. Maxwell’s Equations in Phasor Form (1)
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27. Maxwell’s Equations in Phasor Form (2)
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28. Phasor Form of Maxwell’s Equations (3)
Maxwell’s equations can thus be written in phasor form as:
Phasor form is dependent on position only. Time dependence is removed.
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29. Examples on Phasor Form
Determine the phasor form of the following sinusoidal functions:
f(x,t)=(5x+3) cos(wt + 30)
g(x,z,t) = (3x+z) sin(wt)
h(y,z,t) = (2y+5)4z sin(wt + 45)
V(t) = 0.5 cos(kz-wt)
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30. Solutions a)
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31. Solutions b)
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32. Solution c)
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33. Solution d)
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34. Example Find the phasor notation of the following vector:
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35. Solution
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36. Example
Show that the following electric field satisfies Maxwell’s equations.
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37. Solution
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38. The Wave Equation (1) If we take the curl of Maxwell’s first equation:
Using the vector identity:
And assuming a source free, i.e.
and lossless;
i.e.
medium:
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39. The Wave Equation (2) Define k, which will be known as wave number:
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40. Wave Equation in Cartesian Coordinates
where
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41. Laplacian
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42. Scalar Form of Maxwell’s Equations
Let the electric field vary with x only.
and consider only one component of the field; i.e. f(x).
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43. Possible Solutions to the Scalar Wave Equation
Energy is transported from one point to the other
Standing wave solutions are appropriate for bounded propagation such as wave guides.
When waves travel in unbounded medium, traveling wave solution is more appropriate.
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44. The Traveling Wave
The phasor form of the fields is a mathematical representation.
The measurable fields are represented in the time domain.
Let the solution to the a-component of the electric field be:
Then
Traveling in +x direction
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45. Traveling Wave As time increases, the wave moves along +x direction
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46. Standing Wave
Then, in time domain:
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47. Standing Wave Stationary nulls and peaks in space as time passes.
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48. To summarize
We have shown that Maxwell’s equations describe how electromagnetic energy travels in a medium
The E and H fields satisfy the “wave equation”.
The solution to the wave equation can be in various forms, depending on the medium characteristics
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49. The Plane Wave Concept
Plane waves constitute a special set of E and H field components such that E and H are always perpendicular to each other and to the direction of propagation.
A special case of plane waves is uniform plane waves where E and H have a constant magnitude in the plane that contains them.
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50. Plane Wave Characteristics
amplitude
Frequency (rad/sec)
phase
polarization
Wave number, depends on the medium characteristics
Direction of propagation
amplitude
phase
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51. Plane Waves in Phasor Form
polarization
Complex amplitude
Position dependence
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52. Example 1
Assume that the E field lies along the x-axis (i.e. x-polarized) and is traveling along the z-direction.
wave number
We derive the solution for the H field from the E field using Maxwell’s equation #1: H E
Intrinsic impedance;
377 W for free space
Note the I = V/R analogy in circuit theory.
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53. Example 1 (2 of 4) direction of propagation z y E, H plane x
E and H fields are not functions of x and y, because they lie on x-y plane
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54. Example 1 (3 of 4) In time domain: phase term
*** The constant phase term j is the angle of the complex number Eo
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55. Example 1 (4 of 4)
Wavelength: period in space
kl = 2p
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56. Velocity of Propagation (1/3)
We observe that the fields progress with time.
Imagine that we ride along with the wave.
At what velocity shall we move in order to keep up with the wave???
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57. Velocity of Propagation (2/3)
E field as a function of different times
Constant phase points kz
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58. Velocity of Propagation (3/3)
In free space:
Note that the velocity is independent of the frequency of the wave, but a function of the medium properties.
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59. Example 2
A uniform em wave is traveling at an angle q with respect to the z-axis as shown below. The E field is in the y-direction. What is the direction of the H field? x z y k q E
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60. Solution: Example 2 The E field is along y xz y k q
The direction of propagation is the unit vector
The E field is along y E
Because E, H and the direction of propagation are perpendicular to each other, H lies on x-z plane. It should be in the direction parallel to:
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61. Example 3
Write the expression for an x-polarized electric field that propagates in +z direction at a frequency of 3 GHz in free space with unit amplitude and 60o phase.
+ z-direction
W = 2pf = 2p*3*109 =1 x
60o
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62. Solution 3
+ z-direction
W = 2pf = 2p*3*109 =1 x
60o
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63. Example 4
If the electric field intensity of a uniform plane wave in a dielectric medium where e = eoer and m = mo is given by:
Determine:
The direction of propagation and frequency
The velocity
The dielectric constant (i.e. permittivity)
The wavelength
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64. Solution: Example 4 (1/2) +y direction; w = 2pf = 109 Velocity:
Permittivity:
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65. Solution: Example 4 (2/2)
4. Wavelength: m
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66. Example 5
Assume that a plane wave propagates along +z-direction in a boundless and a source free, dielectric medium. If the electric field is given by:
Calculate the magnetic field, H.
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67. Example 5 - observations
Note that the phasor form is being used in the notation; i.e. time dependence is suppressed.
We observe that the direction of propagation is along +z-axis.
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68. Solution: Example 5 (1/2) E k Intrinsic impedance, I = V/R
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69. Solution: Example 5 (2/2)
E, H and the direction of propagation are orthogonal to each other.
Amplitudes of E and H are related to each other through the intrinsic impedance of the medium.
Note that the free space intrinsic impedance is 377W.
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70. Example 6
Sketch the motion of the tip of the vector A(t) as a function of time.
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71. Solution: Example 6 (1/2)
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72. Solution: Example 6 (2/2) y wt = 90o wt = 180o x wt = 0 wt = 270o
The vector A(t) rotates clockwise wrt z-axis. The tip traces a circle of radius equal to unity with angular frequency w.
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73. Polarization
The alignment of the electric field vector of a plane wave relative to the direction of propagation defines the polarization.
Three types:
Linear
Circular
Elliptical (most general form)
Polarization is the locus of the tip of the electric field at a given point as a function of time.
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74. Linear Polarization y
Electric field oscillates along a straight line as a function of time
Example: wire antennas E x y E x
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75. Example 7
For z = 0 (any position value is fine) y
- Eo Eo x
t = p
t = 0
Linear Polarization: The tip of the E field always stays on x-axis. It oscillates between ±Eo
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76. Example 8 Linear Polarization Exo=1 Eyo=2
Let z = 0 (any position is fine) y 2
t = 0
Linear Polarization x 1
t = p/2
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77. Circular Polarizationy
RHCP
Electric field traces a circle as a function of time.
Generated by two linear components that are 90o out of phase.
Most satellite antennas are circularly polarized. x y
LHCP x
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78. Example 7 Exo=1 Eyo=1 Let z= 0 y RHCP t=p/2w t=p/w x t=0 t=3p/2w
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79. Elliptical Polarizationy x RH
This is the most general form
Linear and circular cases are special forms of elliptical polarization
Example: log spiral antennas y x LH
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80. Example 8 Ey Ex Linear when Circular when
Elliptical if no special condition is met.
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81. Example 9
Determine the polarization of this wave.
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82. Solution: Example 9 (1/2)
Note that the field is given in phasor form. We would like to see the trace of the tip of the E field as a function of time. Therefore we need to convert the phasor form to time domain.
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83. Solution: Example 9 (2/2) Elliptical polarization Let z=0
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84. Example 10 Find the polarization of the following fields: a) b) c)
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85. Solution: Example 10 (1/4) a)
Observe that orthogonal components have same amplitude but 90o phase difference.
Circular Polarization y
t=0
Let kz=0 x z
t=p/2w
t=3p/2w
RHCP
t=p/w
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86. Solution: Example 10 (2/4) b)
Observe that orthogonal components have same amplitude but 90o phase difference.
Circular Polarization z
t=+p/4w
Let kx=0
t=3p/4w y x
t=-p/4w
RHCP
t=5p/4w
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87. Solution: Example 10 (3/4) c)
Observe that orthogonal components have different amplitudes and are out of phase.
Elliptical Polarization x
t=-a/w
Left Hand
Let ky=0 z y
t=+b/w
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88. Solution: Example 10 (4/4) d)
Observe that orthogonal components are in phase.
Linear Polarization y x z
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89. Coherence and Polarization
In the definition of linear, circular and elliptical polarization, we considered only completely polarized plane waves.
Natural radiation received by an anatenna operating at a frequency w, with a narrow bandwidth, Dw would be quasi-monochromatic plane wave.
The received signal can be treated as a single frequency plane wave whose amplitude and phase are slowly varying functions of time.
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90. Quasi-Monochromatic Waves
amplitude and phase are slowly varying functions of time
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91. Degree of Coherence where <….> denotes the time average.
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92. Degree of Coherence – Plane Waves
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93. Unpolarized Waves
An em wave can be unpolarized. For example sunlight or lamp light. Other terminology: randomly polarized, incoherent. A wave containing many linearly polarized waves with the polarization randomly oriented in space.
A wave can also be partially polarized; such as sky light or light reflected from the surface of an object; i.e. glare.
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94. Poynting Vector
As we have seen, a uniform plane wave carries em power.
The power density is obtained from the Poynting vector.
The direction of the Poynting vector is in the direction of wave propagation.
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95. Poynting Vector
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96. Example 11
Calculate the time average power density for the em wave if the electric field is given by:
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97. Solution: Example 11 (1/2)
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98. Solution: Example 11 (2/2)
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99. Plane Waves in Lossy Media
Finite conductivity, s results in loss
Ohm’s Law applies:
Conduction current
Conductivity, Siemens/m
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100. Complex Permittivity
From Ampere’s Law in phasor form:
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101. Wave Equation for Lossy Media
Wave number:
Loss tangent, d
Attenuation constant
Phase constant
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102. Example 12 (1/2) Plane wave propagation in lossy media: complex number
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103. Example 12 (2/2) attenuation propagation
Plane wave is traveling along +z-direction and dissipating as it moves.
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104. Field Attenuation in Lossy Medium
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105. Attenuation and Skin Depth
Attenuation coefficient, a, depends on the conductivity, permittivity and frequency.
Skin depth, d is a measure of how far em wave can penetrate a lossy medium
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106. Lossy Media
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107. Example 13
Calculate the attenuation rate and skin depth of earth for a uniform plane wave of 10 MHz. Assume the following properties for earth:
m = mo
e = 4eo
s = 10-4
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108. Solution: Example 13
First we check if we can use approximate relations.
Slightly conducting
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109. References
Applied Electromagnetism, Liang Chi Shen, Jin Au Kong, PWS
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110. Homework Assignments
Due 9/25/08
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111. Homework 3.1
The magnetic field of a uniform plane wave traveling in free space is given by
What is the direction of propagation?
What is the wave number, k in terms of permittivity, eo and permeability, mo?
Determine the electric field, E.
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112. Homework 3.2 Find the polarization state of the following plane wave:
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113. Homework 3.3
How far must a plane wave of frequency 60 GHz propagate in order for the phase of the wave to be retarded by 180o in a lossless medium with mr =1 and er = 3.5?
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114. Solution Homework 3.1 What is the direction of propagation? Ans: -z
What is the wave number, k in terms of permittivity, eo and permeability, mo?
Ans: free space 
Determine the electric field, E. E H k
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115. Solution: Homework 3.2
Observe that orthogonal components are in phase.
Linear Polarization y x z
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116. Solution 3.3 (1/2)
Wavelength: period in space
kl = 2p
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117. Solution 3.3 (2/2)
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