1. Electromagnetic Waves
Chapter 1
Electromagnetic Waves

2. Chapter Outlines Chapter 1 Electromagnetic Waves Faraday’s Law
Transformer and Motional EMFs
Displacement Current
Maxwell’s Equations
Lossless TEM Waves
EM Wave Fundamental and Equations
EM Wave Propagation in Different Media
EM Wave Reflection and Transmission at Normal or Oblique Incidence

3. Introduction
In your previous experience in studying electromagnetic, you have learned about and experimented with electrostatics and magnetostatics … concentrating on static, or time invariant electromagnetic fields (EM Fields).
Henceforth, we shall examine situations where electric and magnetic fields are dynamic or time varying !!

4. Introduction (Cont’d..)
Where :
In static EM Fields, electric and magnetic fields are independent each other, but in dynamic field both are interdependent.
Time varying EM Fields, represented by E(x,y,z,t) and H(x,y,z,t) are of more practical value than static EM Fields.
In time varying fields, it usually due to accelerated charges or time varying currents.

5. Introduction (Cont’d..)
In summary:
Stationary charges  electrostatic fields
Steady currents  magnetostatic fields
Time varying currents  electromagnetic fields or waves

6. 1.1 Faraday’s Law
According to faraday’s experiment, a static magnetic field produces no current flow, but a time varying field produces an induced voltage called electromotive force or emf in a closed circuit, which causes a flow of current.
Faraday’s Law – the induced emf, Vemf in volts, in any closed circuit is equal to the time rate of change of the magnetic flux linkage by the circuit.

7. Faraday’s Law (Cont’d..)
Where,
The negative sign is a consequence of Len’z Law. If we consider a single loop, Faraday’s Law can be written as:

8. Faraday’s Law (Cont’d..)
An increasing magnetic field out of the page induces a current in (a) or an emf in (b). (c) The distributed resistance in a continuous conductive loop can be modeled as lumped resistor Rdist in series with a perfectly conductive loop.

9. Faraday’s Law (Cont’d..)
Generating emf requires a time varying magnetic flux linking the circuit. This occurs if the magnetic field changes with time ‘transformer emf’’ or if the surface containing the flux changes with time ‘motional emf’.
The emf is measured around the closed path enclosing the area through which the flux is passing, can be written as:
It is clear that in time varying situation, both E and B are present and interrelated.

10. Example 1
Consider the rectangular loop moving with velocity u=uyay in the field from an infinite length line current on the z axis. Assume the loop has a distributed resistance Rdist. Find an expression for the current in the loop including its direction.

11. Solution to Example 1 Remember ?
First calculate the flux through the loop at an instant time,
Remember ?
Where,
unit vector along the line current
unit vector perpendicular from the line current to the field point
So,

12. Solution to Example 1 (Cont’d..)
Arbitrarily choose dS in the +ax direction,
So the flux can be easily calculated as:

13. Solution to Example 1 (Cont’d..)
Then, we want to find how this flux changes with time,
By chain rule,
By considering uy=dy/dt,

14. Solution to Example 1 (Cont’d..)
Our emf is negative of this, where:
Since we considered dS in the +ax direction, our emf is taken counterclockwise circulation. But since the emf is negative, our induced current is apparently going in the clockwise direction with value of:

15. 1.2 Transformer and motional emf
The variation of flux with time as in previous equation maybe caused in three ways:
By having a stationary loop in a time varying B field.
(transformer emf)
By having a time varying loop area in a static B field.
(motional emf)
By having a time varying loop area in a time varying
B field.

16. Transformer and motional emf (Cont’d..)
Stationary Loop in Time Varying B Field
This is the case where a stationary conducting loop is in a time varying magnetic B field. The equation becomes:
By applying Stokes Theorem in the middle term, we obtain:

17. Transformer and motional emf (Cont’d..)
This leads us to the point or differential form of Faraday’s Law,
Based on this equation, the time varying electric field is not conservative, or not equal to zero. The work done in taking a charge about a closed path in a time varying electric field, for example, is due to the energy from the time varying magnetic field.

18. Transformer and motional emf (Cont’d..)
Moving Loop in static B Field
When a conducting loop is moving in a static B field, an emf is induced in the loop. Recall that the force on a charge moving with uniform velocity in magnetic field,
So then, we define the motional electric field Em,

19. Transformer and motional emf (Cont’d..)
If we consider a conducting loop moving with uniform velocity u as consisting of a large number of free electrons, the emf induced in the loop is:
Moving Loop in Time Varying Field
The total emf would be:

20. Example 2
The loop shown is inside a uniform magnetic field B = 50 ax mWb/m2 . If side DC of the loop cuts the flux lines at the frequency of 50Hz and the loop lies in the yz plane at time t = 0, find the induced emf at t = 1 ms.

21. Solution to Example 2
Since the B field is time invariant, the induced emf is motional, that is:
Where,

22. Solution to Example 2 (Cont’d..)
As u and dL is in cylindrical coordinates, transform B field into cylindrical coordinate (Chapter 1 in Electromagnetic Theory !! ):
Where B0 = 0.05 , therefore:

23. Solution to Example 2 (Cont’d..)
And
To determine recall that,
C is constant
at because the loop is in the yz plane!

24. Solution to Example 2 (Cont’d..)
Hence,
Therefore,
So that at t = 1 ms,

25. 1.3 Displacement Current
We recall from Ampere’s Circuital Law for static field,
‘c’ subscript is used to identify it as a conduction current density, which related to electric field Ohm’s Law by:
But divergence of curl of a vector is identically zero,

26. Displacement Current (Cont’d..)
The current continuity equation,
We see that the static form of Ampere’s Law is clearly invalid for time varying fields since it violates the law of current continuity, and it was resolved by Maxwell introduction which what we called displacement current density,
Where Jd is the rate of change of the electric flux density,

27. Displacement Current (Cont’d..)
The insertion of Jd was one of the major contribution of Maxwell. Without Jd term, electromagnetic wave propagation (e.g. radio or TV waves) would be impossible. At low frequencies, Jd is usually neglected compared with Jc. But at radio frequencies, the two terms are comparable.
Therefore,

28. Displacement Current (Cont’d..)
By applying the divergence of curl, rearrange, integrate and apply Stoke’s Theorem, we can get the integral form of Ampere’s circuital Law:
Do you really understand this displacement current??
Only formula and formula…????

29. Displacement Current (Cont’d..)
To have clear understanding of displacement current, consider the simple capacitor circuit of figure below.
A sinusoidal voltage source is applied to the capacitor, and from circuit theory we know the voltage is related to the current by the capacitance. i(t) here is the conduction current.

30. Displacement Current (Cont’d..)
Consider the loop surrounding the plane surface S1. By static form of Ampere’s Law, the circulation of H must be equal to the current that cuts through the surface. But, the same current must pass through S2 that passes between the plates of capacitor.

31. Displacement Current (Cont’d..)
But, there is no conduction current passes through an ideal capacitor, (where J=0, due to σ=0 for an ideal dielectric ) flows through S2. This is contradictory in view of the fact that the same closed path as S1 is used.
But to resolve this conflict, the current passing through S2 must be entirely a displacement current, where it needs to be included in Ampere’s Circuital Law.
So we obtain the same current for either surface though it is conduction current in S1 and displacement current in S2.

32. Displacement Current (Cont’d..)
The ratio of conduction current magnitude to the displacement current magnitude is called loss tangent, where it is used to measure the quality of the dielectric good dielectric will have very low loss tangent.
Other example for physical meaning:
Ji = current source
Jc= conducted current through resistor
Jd=displacement current through dielectric material
mi= magnetic current source
md=displacement magnetic current

33. 1.4 Maxwell Equations
Below is the generalized forms of Maxwell Equations:
Maxwell Equations
Point or
Differential Form
Integral Form
Gauss’s Law
Gauss’s Law for
Magnetic Field
Faraday’s Law
Ampere’s Circuital
Law

34. Maxwell Equations (Cont’d..)
It is worthwhile to mention other equations that go hand in hand with Maxwell’s equations.
Lorent’z Force Equation
Constitutive Relations
Current Continuity Equation

35. Circuit - Field relations:
Maxwell Equations (Cont’d..)
Circuit - Field relations:
Field Relation Circuit Relation

36. 1.5 Lossless TEM Waves
Let’s use Maxwell’s equations to study the relationship between the electric and magnetic field components of an electromagnetic wave.
Consider an x-polarized wave propagating in the +z direction in some ideal medium characterized by µ and ε, with σ = 0.
Electromagnetic wave as x-polarized means that the E field vector is always pointing in the x or –x direction. Choose σ = 0 to make medium lossless for simplicity, as given by:

37. Lossless TEM Waves (Cont’d..)
A plot of the equation E(z,0) = E0cos(z)ax at 10 MHz in free space with E0 = 1 V/m.

38. Lossless TEM Waves (Cont’d..)
Upon application of Maxwell equations, we would also find the magnetic field propagates in the +z direction, but the field is always normal or perpendicular to the electric field vector  the wave is said to propagate in a transverse electromagnetic wave mode, or TEM.
TEM Waves has no E field or H field components along the direction of propagation.
We can apply Faraday’s Law, to the propagating electric field equation previously.
Solve the right hand side and the left hand side to get H !!

39. Lossless TEM Waves (Cont’d..)
Where,

40. Lossless TEM Waves (Cont’d..)
We could see that the time varying E is the only source of H, if no conduction current given that can also generate H  Thus C must be zero.
The amplitudes of E and H are related by Maxwell equations.
Plot of the equation H(z,0) = (E0/) cos(–z)ay at 10 MHz in free space with E0 = 1 V/m along with the lighter plot of E(z,0).

41. Lossless TEM Waves (Cont’d..)
We can apply Ampere’s Circuital Law, to the propagating magnetic field equation previously.
By considering lossless characteristic, taking the curl of H, equating the equation and then integrate it, we could get :
Try this!!!
-Solution-
And propagation velocity relation:
to get

42. Example 3 Suppose in free space that: E(z,t) = 5.0 e-2zt ax V/m.
Is the wave lossless?
Find H(z,t).

43. Solution to Example 3
Since the wave has an attenuation term (e-2zt) it is clearly not lossless.
To find H,
Therefore,

44. Solution to Example 3 (Cont’d..)
This integral is solved by parts
where we let
We arrive at:

45. 1.6 EM Wave Fundamental and equations
In free space, the constitutive parameters are σ = 0, µr = 1, εr = 1, so the Ampere’s Law and Faraday’s Law equations become :
If there is some point in space a source of time varying E field, a H field is induced in the surrounding region. As this H field also changing with time, it in turn induces an E field. Energy is pass back and forth between E and H fields as they radiate away from the source at the speed of light.

46. EM Wave Fundamental and equations (Cont’d..)
The EM waves radiates spherically, but at a remote distance away from the source they resemble uniform plane wave. In a uniform plane wave, the E and H fields are orthogonal, or transverse to the direction of propagation ( to propagate in TEM mode ).

47. EM Wave Fundamental and equations (Cont’d..)
We will briefly review some of the fundamental features of waves before employing them in the study of electromagnetic. Consider time harmonic waves, represented by sine waves, rather than transient waves (pulses or step functions), generally as:
The E field is a function of position (z) and time (t). It is always pointing in +x or –x direction  x-polarized wave.

48. EM Wave Fundamental and equations (Cont’d..)
Where,
Initial amplitude at z = 0
and exponential terms  attenuation
Angular frequency
Phase constant
Phase shift
And important relations:

49. EM Wave Fundamental and equations (Cont’d..)
E(0,t) = Exax = E0cos(t)ax.
E(0,t) =Exax = E0cos(t + )ax.
E(z, 0) = E0cos(–z)ax.
E(z, 0) = E0e–zcos (–z)ax.

50. EM Wave Fundamental and equations (Cont’d..)
Use Maxwell’s equations to derive formulas governing EM wave propagation.
Consider that the medium is free of any charge, where:
And linear, isotropic, homogeneous, and time invariant (simple media), whereby the Maxwell’s equation can be rewritten as :

51. EM Wave Fundamental and equations (Cont’d..)
Take curl of both sides of Faraday’s Law,
Consider position derivative acting on a time derivative in a homogeneous material,
Exchange the Faraday’s Law for the curl of H
Invoking a vector identity,

52. EM Wave Fundamental and equations (Cont’d..)
So now we have :
But, medium is charge free, so divergence of E is zero,
Helmholtz Wave Equation for E.
This can be broken up into three vector equations ( Ex, Ey and Ez)

53. Task 1
By starting with Maxwell’s equations for simple and charge free media, derive the Helmholtz Wave Equation for magnetic field, H
Solve this! Please submit the solution after class!
-Solution-

54. EM Wave Fundamental and equations (Cont’d..)
But, with interest on Helmholtz equations for time harmonic fields, the previous Helmholtz wave equation becomes :
because
Generally written in the form:
Where the propagation constant, γ, with real part is attenuation and an imaginary part is phase constant,

55. EM Wave Fundamental and equations (Cont’d..)
The value of  and β in terms of the material’s constitutive parameters:

56. EM Wave Fundamental and equations (Cont’d..)
The Helmholtz wave equation for time harmonic magnetic fields, H :
With loss of generality, we assume that x polarized traveling in the +az direction, where Es has only an x component with function of z, since for plane wave that the fields do not vary in transverse direction, where in this case is xy plane. Then :

57. EM Wave Fundamental and equations (Cont’d..)
By substituting this equation into the Helmholtz wave equation for E field, it becomes:
Hence,
With first and second term is zero :

58. EM Wave Fundamental and equations (Cont’d..)
This is a scalar wave equation, a linear homogeneous differential equation. A possible solution for this equation is :
If we let, then,
So, this equation, becomes:
Which has two solutions,

59. EM Wave Fundamental and equations (Cont’d..)
The general solution is linear superposition of these two:
For the represents the E field amplitude of +z traveling wave at z=0, by reinserting the vector, multiply by time factor apply Euler’s identity and use the propagation constant to get :

60. EM Wave Fundamental and equations (Cont’d..)
For the represents the E field amplitude of negative z traveling wave at z=0, and similarly by previous approach to get :
The general instantaneous solution is superposition of these two solutions above.
or generally as :

61. EM Wave Fundamental and equations (Cont’d..)
The magnetic field can be found by applying Faraday’s Law :
Evaluating the curl of E to find:
We can solve for H ,
If we assume that the wave propagates along +az and Hs has only an y component, as what we had assume for Es

62. EM Wave Fundamental and equations (Cont’d..)
We would have been led to the expression,
By making comparison, we can find a relationship between and , or and
intrinsic impedance (in ohms)

63. EM Wave Fundamental and equations (Cont’d..)
By plugged in the equation of intrinsic impedance into equation of magnetic field, we could get:
Where E and H are out of phase by θn at any instant of time due to the complex intrinsic impedance of the medium. Thus at any time, E leads H or H lags E by θn . The ratio of magnitude of conduction current density to displacement current density in a lossy medium is:
loss tangent

64. EM Wave Fundamental and equations (Cont’d..)
Tan δ is used to determine how lossy a medium is, i.e.
Good (lossless or perfect) dielectric if
Good conductor if

65. EM Wave Fundamental and equations (Cont’d..)
Basically, we can use Fleming’s Left Hand Rule to determine the E, H and propagation direction ;
Propagation direction (first finger)
H direction (second finger)
E direction (thumb)

66. EM Wave Fundamental and equations (Cont’d..)
By knowing the EM wave’s direction of propagation, given as unit vector ap, is the same as the cross product of Es with unit vector, aE and Hs with unit vector aH :
And also, a pair of simple formulas can be derived:
In addition to properties of cross product,

67. EM Wave Fundamental and equations (Cont’d..)
Representation of waves; in (a), the wave travels in the ap = +az direction and has Es = E0+ e–z ax and Hs = (E0+/)e–z ay. In (b), the wave travels in the ap = –az direction and has Es = E0–ez ax along with Hs = –(E0–/)ez ay.

68. Example 4 Suppose in free space :
H(x,t) = 100 cos(2π x 107t – βx + π/4) az mA/m. Find E(x,t).

69. Solution to Example 4 We could find: So then,
Since free space is stated,
and then

70. Example 5 Suppose: E (x,y,t) = 5 cos(π x 106t – 3.0x + 2.0y) az V/m.
Find :
H (x,y,t).
The direction of propagation, ap

71. Solution to Example 5 We could find:
Assume nonmagnetic material and therefore have:
So that,

72. Solution to Example 5 (Cont’d..)
The direction of propagation :
Where,
And with the exponential terms canceling in the top and bottom of the equation for ap, we have:

73. 1.7 EM Wave Propagation In Different Media
We will consider time harmonic field propagating in different types of media :-
Lossless, Charge-Free
Dielectrics
Conductors
Lossless, Charge - Free
Charge free, ρv=0, medium has zero conductivity, σ=0. This is the case where waves traveling in vacuum or free space (free of any charges). Perfect dielectric is also considered as lossless media.

74. EM Wave Propagation In Different Media (Cont’d..)
Evaluate the propagation constant,
So,
Where,
Since , the signal does not attenuate as it travels  lossless medium.
The propagation velocity,

75. EM Wave Propagation In Different Media (Cont’d..)
Evaluate the intrinsic impedance,
For lossless materials, E and H are always in phase. Again,
intrinsic impedance of free space

76. Example 6 In a lossless, nonmagnetic material with :
εr = 16, and H = 100 cos(ωt – 10y) az mA/m.
Determine :
The propagation velocity
The angular frequency
The instantaneous expression for the electric field intensity.

77. Solution to Example 6 The propagation velocity: The angular frequency:
From given H field :

78. Solution to Example 6 (Cont’d..)
So, the time harmonic H field is:
Where,
Finally, the instantaneous expression for E field is:

79. EM Wave Propagation In Different Media (Cont’d..)
Dielectric
Treating a dielectric as lossless is often a good approximation, but all dielectrics are to some degree lossy  finite conductivity, polarization loss etc. With finite conductivity, the E field gives rise to conduction current density results in power dissipation.
Thus, it will give a complex permittivity, complex propagation constant with attenuation constant greater than zero. The intrinsic impedance is also complex, resulting a phase difference between E and H fields.

80. EM Wave Propagation In Different Media (Cont’d..)
Conductor
In any decent conductor, the loss tangent, σ/ωε>>1 or σ>>ωε so that σ ≈ ∞, so that:
where interior brackets becomes:
Therefore,

81. EM Wave Propagation In Different Media (Cont’d..)
The intrinsic impedance approximated by:
By considering leading to:
E leads H by 450
and also..

82. EM Wave Propagation In Different Media (Cont’d..)
A wave from air (free space) penetrates rapidly in a good conductor, with wavelength clearly much shorter.
A large attenuation means the fields cannot penetrate far into the conductor.
In a good conductor, the large attenuation means the penetration depth can be quite small, confining the fields near the surface or skin, of the conductor  skin depth.

83. Example 7 In a nonmagnetic material,
E(z,t) = 10 e-200z cos(2π x 109t - 200z) ax mV/m.
Find H(z,t)

84. Solution to Example 7
Since  = β, the media is a good metal, with µr = 1 we have:
We could also find the intrinsic impedance,

85. Solution to Example 7 (Cont’d)
So, to calculate H,
The instantaneous expression for the magnetic field intensity.

86. 1.8 EM Wave Reflection & Transmission at Normal Incidence
What happens when a EM wave is incident on a different medium? E.g. Light wave incident with mirror, most of it gets reflected but a portion gets transmitted (rapidly attenuating in the silver backing of the mirror.
Consider a plane wave that are normally incident which means the planar boundary separating the two media is perpendicular to the wave’s propagation direction.
Generally, consider a time harmonic x-polarized electric field incident from medium 1 (µr1, εr1, σr1) to medium 2 (µr2, εr2, σr2)

87. EM Wave Reflection & Transmission at Normal Incidence (Cont’d..)
With incident field:

88. EM Wave Reflection & Transmission at Normal Incidence (Cont’d..)
We have the following sets of equations:
Incident Fields
Reflected Fields
The E field intensities at z=0
Transmitted Fields

89. EM Wave Reflection & Transmission at Normal Incidence (Cont’d..)
The boundary conditions:
Applying these boundary conditions at z=0 to get:
Try this!!
Reflection Coefficient
Transmission Coefficient

90. EM Wave Reflection & Transmission at Normal Incidence (Cont’d..)
By comparison,
Consider a special case when medium 1 is a perfect dielectric (lossless,σ1=0) and medium 2 is a perfect conductor (σ2= ∞).
For this case, showing that the wave is totally reflected  fields in perfect conductor must vanish, so there can be no transmitted wave, E2 = 0.
The totally reflected wave combines with the incident wave to form a standing wave  it stands and does not travel, it consists of two traveling waves Ei and Er of equal amplitudes but in opposite directions.

91. EM Wave Reflection & Transmission at Normal Incidence (Cont’d..)
Standing wave pattern for an incident wave in a lossless medium reflecting off a second medium at z=0 where  = 0.5.

92. Example 8
A uniform planar waves is normally incident from media 1 (z < 0, σ = 0, µr = 1.0, εr = 4.0) to media 2 (z > 0, σ = 0, µr = 8.0, εr = 2.0).
Calculate the reflection and transmission coefficients seen by this wave.

93. Solution to Example 8 The reflection coefficient ; This leads to:
and the transmission coefficient,

94. Example 9
Suppose media 1 (z < 0) is air and media 2 (z > 0) has εr = 16. The transmitted magnetic field intensity is known to be:
Ht = 12 cos (ωt - β2z) ay mA/m.
Determine the instantaneous value of the incident electric field.

95. Solution to Example 9 We know that,
From transmitted H field, we could find the transmitted E field,

96. Solution to Example 9 (Cont’d..)
Since we know the relation between transmitted E field and incident E field,
Thus,

97. 1.9 EM Wave Reflection & Transmission at Oblique Incidence
A uniform plane waves traveling in the ai direction is obliquely incident from medium 1 onto medium 2.
Plane of incidence  plane containing both a normal to the boundary and the incident’s wave propagation.
In figure, the propagation direction is ai and the normal is az, so the plane incidence is the x z plane. The angle of incidence, reflection and transmission is the angle that makes the field a normal to the boundary.

98. EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..)
When EM Wave in plane wave form obliquely incident on the boundary, it can be decomposed into:
Perpendicular Polarization, or transverse electric (TE) polarization  The E Field is perpendicular or transverse to the plane of incidence.
Parallel Polarization, or transverse magnetic (TM) polarization  The E Field is parallel to the plane of incidence, but the H Field is transverse.
We need to decompose into its TE and TM components separately, and once the reflected an the transmitted fields for each polarization determined, it can be recombined for final answer.

99. EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..)
TE Polarization

100. EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..)
For TE polarization, the fields are summarized as follows:
Incident Fields
Reflected Fields
Transmitted Fields

101. EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..)
By applying Snell’s Law,
Try to solve this!!!
We could get:

102. EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..)
TM Polarization

103. EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..)
By similar geometric arguments, TM fields are:
Incident Fields
Reflected Fields
Transmitted Fields

104. EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..)
By applying Snell’s Law and employing the boundary conditions, we could get the following expressions relating the field amplitudes:

105. EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..)
For TM polarizations, there exists an incidence angle at which all of the wave is transmitted into the second medium  Brewster Angle, θi = θBA , where:
When a randomly polarized wave such as light is incident on a material at the Brewster angle, the TM polarized portion is totally transmitted but a TE component is partially reflected.

106. Example 10
A 100 MHz TE polarized wave with amplitude 1.0 V/m is obliquely incident from air (z < 0) onto a slab of lossless, nonmagnetic material with εr = 25 (z > 0). The angle of incidence is 40. Calculate:
the angle of transmission,
the reflection and transmission coefficients,
the incident, reflected and transmitted for E fields.

107. Solution to Example 10
(a)
(b)

108. Solution to Example 10 (Cont’d)
For incident field:
Thus,
For reflected field:

109. Solution to Example 10 (Cont’d)
Leading to:
Thus,
Finally for transmitted field:

110. Solution to Example 10 (Cont’d)
To get:
Therefore,

111. Electromagnetic Waves
End