Lecture Notes on Electromagnetic Field Theory

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Renato Orta
Lecture Notes
on
Electromagnetic Field Theory
PRELIMINARY VERSION
November 2012
DEPARTMENT OF ELECTRONICS
AND
TELECOMMUNICATIONS
POLITECNICO DI TORINO
Contents
Contents 1
1 Fundamental concepts 3
1.1 Maxwell's Equations...................................
3
1.2 Phasors..........................................
5
1.3 Constitutive relations..................................
9
1.4 Boundary conditions...................................
11
2 Waves in homogeneous media 14
2.1 Plane waves........................................
14
2.2 Cylindrical waves.....................................
20
2.3 Spherical waves......................................
21
2.4 Waves in non homogeneous media...........................
22
2.5 Propagation in good conductors.............................
23
3 Radiation in free space 28
3.1 Green's functions.....................................
28
3.2 Elementary dipoles....................................
34
3.3 Radiation of generic sources...............................
41
4 Antennas 49
4.1 Antenna parameters...................................
49
4.1.1 Input impedance.................................
50
4.1.2 Radiation pattern,Directivity and Gain....................
51
4.1.3 E®ective area,e®ective height..........................
54
4.2 Friis transmission formula................................
57
4.3 Examples of simple antennas..............................
59
4.3.1 Wire antennas..................................
60
4.3.2 Aperture antennas................................
67
1
CONTENTS
5 Waveguides 76
5.1 Waveguide modes.....................................
77
5.2 Equivalent transmission lines..............................
80
5.3 Rectangular waveguide..................................
86
5.3.1 Design of a single mode rectangular waveguide................
92
5.3.2 Tunneling e®ects.................................
94
5.3.3 Irises and waveguide discontinuities.......................
100
A Mathematical Basics 1
A.1 Coordinate systems and algebra of vector ¯elds....................
1
A.2 Calculus of vector ¯elds.................................
10
A.3 Multidimensional Dirac delta functions.........................
17
B Solved Exercises 20
B.1 Polarization and Phasors.................................
20
B.2 Plane Waves.......................................
23
B.3 Antennas.........................................
28
B.4 Waveguides........................................
36
Bibliography 45
2
Chapter 1
Fundamental concepts
1.1 Maxwell's Equations
All electromagnetic phenomena of interest in this course can be modeled by means of Maxwell's
equations
8
>
>
<
>
>
:
r£E(r;t) = ¡
@
@t
B(r;t) ¡M(r;t)
r£H(r;t) =
@
@t
D(r;t) +J(r;t)
(1.1)
Let us review the meaning of the symbols and the relevant measurement units.
E(r;t) electric ¯eld V/m
H(r;t) magnetic ¯eld A/m
D(r;t) electric induction C/m
2
B(r;t) magnetic induction Wb/m
2
J(r;t) electric current density (source) A/m
2
M(r;t) magnetic current density (source) [V/m
2
]
Customarily,only electric currents are introduced;it is in particular stated that magnetic
charges and currents do not exist.However,it will be seen in later chapters,that the introduction
of ¯ctitious magnetic currents has some advantages:
²
The radiation of some antennas,such as loops or horns,is easily obtained
²
Maxwell's equations are more symmetric
3
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 4
²
(surface) magnetic currents are necessary for the formulation of the equivalence theorem,a
fundamental tool for the rigorous modelling of antennas
In circuit theory,one has two types of ideal generators,i.e.current and voltage ones:likewise,in
electromagnetism one introduces two types of sources.
Concerning the symmetry of Maxwell's equations,we cite the principle of duality:performing
the exchanges
E $H
B $¡D
J $¡M
Maxwell's equations transform into each others.
Experiments show that the electric charge is a conserved quantity.This implies that electric
current density and electric charge (volume density are related by a continuity equation
r¢ J(r;t) +
@½(r;t)
@t
= 0 (1.2)
By analogy,we assume that also magnetic charges are conserved,so that a similar continuity
equation must be satis¯ed:
r¢ M(r;t) +

m
(r;t)
@t
= 0 (1.3)
It can be proved that eqs.(1.1),(1.2) (1.3) imply the well known divergence equations
r¢ B(r;t) = ½
m
(r;t) r¢ D(r;t) = ½(r;t) (1.4)
Some authors prefer to assume eqs.(1.1),1.4) as fundamental equations and obtain the conservation
of charge (1.2) (1.3) as a consequence.
Maxwell's equations can be written in di®erential form as above,so that they are assumed to
hold in every point of a domain,or in integral form,so that they refer to a ¯nite volume.The
integral form can be obtained by integrating eq.(1.1) over an open surface §
o
with boundary ¡
and applying Stokes theorem:
I
¡
E ¢ ^¿ ds = ¡
d
dt
Z
§
o
B¢ ^º d§
o
¡
Z
§
o
M¢ ^º d§
o
I
¡
H¢ ^¿ ds =
d
dt
Z
§
o
D¢ ^º d§
o
+
Z
§
o
J ¢ ^º d§
o
(1.5)
In words,the ¯rst equation says that the line integral of the electric ¯eld,i.e.the sumof all voltage
drops along a closed loop,equals the time rate of change of the magnetic induction °ux plus the
total magnetic current.The second equation says that the line integral of the magnetic ¯eld along
a closed loop equals the time rate of change of the electric induction °ux plus the total electric
current.
Then we integrate eq.(1.4) over a volume V with surface § and apply the divergence theorem:
I
§
B¢ ^º d§ =
Z
V
½
m
dV
I
§
D¢ ^º d§ =
Z
V
½ dV (1.6)
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 5
Figure 1.1.Open surface §
o
for the application of Stokes theorem.Notice that the orien-
tations of ^º and ^¿ are related by the right-hand-rule:if the thumb points in the direction
of ^º,the ¯ngers point in that of ^¿.
Figure 1.2.Closed surface § for the application of the divergence theorem.
which is the usual formulation of Gauss theorem.
The same procedure on eq.(1.2) produces:
I
§
J ¢ ^º d§+
d
dt
Z
V
½ dV = 0 (1.7)
This says that the total current out of a volume equals the time rate of decrease of the internal
charge.
1.2 Phasors
Field variables can have any time dependence but a particularly important one is the so called
time-harmonic regime.Consider a general time-harmonic electric ¯eld in a particular point:
E(t) = E
x0
cos(!
0
t +'
x
)^x +E
y0
cos(!
0
t +'
y
)^y +E
z0
cos(!
0
t +'
z
)^z
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 6
The three cartesian components are sinusoidal functions of time with di®erent amplitudes and
phases,but the same frequency.This equation can be transformed in the following way.
E(t) = R
n
E
x0
e
j(!
0
t+'
x
)
o
^x +R
n
E
y0
e
j(!
0
t+'
y
)
o
^y +R
n
E
z0
e
j(!
0
t+'
z
)
o
^z
= R
©¡
E
x0
e
j'
x
^x +E
y0
e
j'
y
^y +E
z0
e
j'
z
^z
¢
e
j!
0
t
ª
= R
©
E e
j!
0
t
ª
(1.8)
The complex vector E is called the phasor of the time-harmonic vector E(t) and is measured in
the same units.It can be decomposed into real and imaginary parts:
E = E
0
+jE
00
with
E
0
= E
x0
cos'
x
^x +E
y0
cos'
y
^y +E
z0
cos'
z
^z
and
E
00
= E
x0
sin'
x
^x +E
y0
sin'
y
^y +E
z0
sin'
z
^z
Consider again eq.(1.8):
E(t) = R
©
E e
j!
0
t
ª
= R
©
(E
0
+jE
00
) e
j!
0
t
ª
= Rf(E
0
+jE
00
) (cos!
0
t +j sin!
0
t)g
= E
0
cos!
0
t ¡E
00
sin!
0
t
We have obtained a representation of the time-harmonic electric ¯eld as the sum of two vectors,
with arbitrary directions,in time quadrature one with respect to the other:in other words both
these vectors are sinusoidal functions of time,with the same frequency but with a relative delay of
a quarter of a period.This representation is useful to study the polarization of the time-harmonic
vector,i.e.the form of the curve traced by the vector E(t) as a function of time.It can be shown
t
 
E"
E'
Figure 1.3.Elliptically polarized ¯eld
that,in general,this curve is an ellipse inscribed in a parallelogram that has E
0
and E
00
as half
medians,as shown in Fig.1.3.We see easily from the previous equation that
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 7
²
for t = 0,E = E
0
²
for t = T=4,E = ¡E
00
²
for t = T=2,E = ¡E
0
²
for t = 3T=4,E = E
00
where T = 2¼=!is the period.Hence the sense of rotation is from E
0
to ¡E
00
.In this case the
¯eld is said to be elliptically polarized.
There are particular cases.When E
0
and E
00
are parallel or one of the two is zero,the paral-
lelogram degenerates into a line and the polarization is linear.The two cases can be condensed in
the single condition (cross product,i.e.vector product):
E
0
£E
00
= 0
The other particular case is that in which
jE
0
j = jE
00
j and E
0
¢ E
00
= 0
The parallelogram becomes a square and the ellipse a circle:the ¯eld is circularly polarized.
The plane de¯ned by the two vectors E
0
and E
00
is called polarization plane.Suppose we
introduce a cartesian reference in this plane with the z axis orthogonal to it.The phasor E has
only x and y components,
E = E
x
^x +E
y
^y
where E
x
and E
y
are complex numbers.This means that the original time-harmonic ¯eld is
represented as the sum of two sinusoidally varying orthogonal vectors,with arbitrary magnitudes
and phases.On this basis,the type of polarization is ascertained with the following rules:
²
if the phase di®erence between the two components ± = arg E
y
¡ arg E
x
is 0 or ¼ the
polarization is linear,along a line that forms an angle à = arctan(jE
y
j=jE
x
j) (if ± = 0) or
à = ¡arctan(jE
y
j=jE
x
j) (if ± = ¼)
²
if ± = §¼=2 and jE
y
j = jE
x
j the polarization is circular,clockwise if ± = ¼=2,counterclock-
wise if ± = ¡¼=2
²
in the other cases,the polarization is elliptical
To illustrate these concepts consider the following example.
Example
The phasor of a magnetic ¯eld is H = (1 +j)^x +(1 ¡3j)^y.Determine the type of polarization,
write the expression of the time varying ¯eld H(t) and draw the polarization curve de¯ned by this
vector.
Solution Find real and imaginary part of the phasor
H
0
= ^x +^y H
00
= ^x ¡3^y
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 8
Compute
H
0
£H
00
= (^x +^y) £(^x ¡3^y) = (¡3 ¡1)^z 6= 0
H
0
¢ H
00
= (^x +^y) ¢ (^x ¡3^y) = 1 ¡3 = ¡2 6= 0
The polarization is neither linear nor circular,hence it is elliptical counterclockwise (H(t) goes
from H
0
to ¡H
00
).
The time varying ¯eld is
H(t) = (^x +^y) cos!
0
t ¡(^x ¡3^y) sin!
0
t
The plot is shown in Fig.1.4
-4
-3
-2
-1
0
1
2
3
4
-4
-3
-2
-1
0
1
2
3
4
y
x
Figure 1.4.Polarization curve de¯ned by H(t) above
¥
The time-harmonic regime is important because of the property
d
dt
E(r;t) =
d
dt
R
©
E(r) e
j!
0
t
ª
= R
©
j!
0
E(r) e
j!
0
t
ª
(1.9)
so that time derivatives become algebraic operations.If we substitute the representation (1.8) into
(1.1) we obtain,after canceling common factors exp(j!
0
t):
r£E(r) = ¡j!
0
B(r) ¡M(r)
r£H(r) = j!
0
D(r) +J(r)
(1.10)
If the time dependence is of general type,eq.(1.8) is generalized by the spectral representation
(inverse Fourier transform)
E(r;t) =
1

Z
1
¡1
E(r;!) e
j!t
d!(1.11)
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 9
In words,a generic time varying electric ¯eld is represented as a sum of an in¯nite number of
harmonic components of all frequencies and amplitude E(r;!)d!=(2¼),where
E(r;!) =
Z
1
¡1
E(r;t) e
¡j!t
dt (1.12)
(direct Fourier transform).E(r;!) is a spectral density of electric ¯eld,hence it is measured in
V/(m Hz).Due to the fact that E(r;t) is real,E(r;¡!) = E
¤
(r;!),so that the previous equation
can also be written
E(r;t) = 2R
½
1

Z
1
0
E(r;!) e
j!t
d!
¾
in terms of positive frequencies only.
The importance of the spectral representation is related to the property
d
dt
E(r;t) =
d
dt
1

Z
1
¡1
E(r;!) e
j!t
d!=
1

Z
1
¡1
j!E(r;!) e
j!t
d!
If we take the Fourier transform of (1.1),we get
r£E(r;!) = ¡j!B(r;!) ¡M(r;!)
r£H(r;!) = j!D(r;!) +J(r;!)
(1.13)
While!
0
is a speci¯c frequency value,(1.13) must be satis¯ed for all frequencies.We refer to
these system as Maxwell's equations in the frequency domain.The variables will be interpreted as
phasors or Fourier transforms,depending on the application.
1.3 Constitutive relations
Clearly Maxwell's equations as written in the previous section form an underdetermined system:
indeed there are only two equations but four unknowns,the two ¯elds and the two inductions.It
is necessary to introduce the constitutive relations,i.e.the equations linking the inductions to the
¯elds.From a general point of view,matter becomes polarized when it is introduced into a region
in which there is an electromagnetic ¯eld,that is,the electric charges at molecular and atomic
level are set in motion by the ¯eld and produce an additional ¯eld that is summed to the original
one.The inductions describe the electromagnetic behaviour of matter.
The simplest case is that of free space in which
B(r;t) = ¹
0
H(r;t)
D(r;t) = ²
0
E(r;t)
(1.14)
where ²
0
,dielectric permittivity,and ¹
0
magnetic permeability,have the values
¹
0
= 4¼ ¢ 10
¡7
H=m
²
0
=
1
¹
0
c
2
¼
1
36¼
¢ 10
¡9
F=m
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 10
and the speed of light in free space c has the value
c = 2:99792458 ¢ 10
8
m=s:
In the case of linear,isotropic,dielectrics,the constitutive relations (1.14) are substituted by
B(r;!) = ¹(!) H(r;!)
D(r;!) ="(!) E(r;!)
(1.15)
where
¹(!) = ¹
0
¹
r
(!)
²(!) = ²
0
²
r
(!)
and ¹
r
(!),²
r
(!) (pure numbers) are the relative permittivity and permeabilities.All non fer-
romagnetic materials have values of ¹
r
very close to 1.Notice that since molecular and atomic
charges have some inertia,they cannot respond instantaneously to the applied ¯eld,so that the
response depends on the time rate of change of the excitation.The description of such a mechanism
is best performed in the frequency domain,where"(!) and ¹(!) play the role of transfer functions.
The fact that they depend on frequency is called dispersion:hence free space is non dispersive.In
general"(!) and ¹(!) are complex:
"="
0
¡j"
00
¹ = ¹
0
¡j¹
00
It can be shown that the volume density of dissipated power in a medium is related to their
imaginary part
P
diss
=
1
2
"
00
jEj
2
+
1
2
¹
00
jHj
2
Notice that"
00

00
are positive in a passive medium because of the phasor time convention
exp(j!
0
t).Some authors use the convention exp(¡j!
0
t):in this case passive media have neg-
ative"
00

00
.Clearly with the time convention we adopt,negative"
00

00
characterize active
media,such as laser media.
When the dielectric contains free charges,the presence of an electric ¯eld E(r;!) gives rise to
a conduction current density J
c
(r;!):
J
c
(r;!) = °(!) E(r;!)
where °(!) is the conductivity of the dielectric,measured in S/m.The previous equation is the
microscopic form of Ohm's law of circuit theory.
The conduction current enters into the second Maxwell's equation (1.13),which becomes
r£H= j!"E+°E+J
where the term J is the (independent) source.It is customary to incorporate the conduction
current into the polarization current by means of an equivalent permittivity.Indeed we can write
j!"E+°E = j!
³
"¡j
°
!
´
E = j!"
eq
E
with"
eq
="
0
¡j("
00
+°=!).In practice the subscript eq is always dropped:the imaginary part of
"takes into account all loss mechanisms,both those due to molecular and atomic transitions and
those due to Joule e®ect.
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 11
In the case of low loss dielectrics one often introduces the loss tangent
tan± =
"
00
"
0
so that we can write
"="
0
(1 ¡j tan±)
Values of tan±'10
¡3
:10
¡4
characterize low loss dielectrics.
It is interesting to note that for fundamental physical reasons,there is relationship between
the real and the imaginary part of the dielectric permittivity and of the magnetic permeability.
Indeed,in the case of the permittivity,just as a consequence of causality,"
0
(!) ¡"
0
and"
00
(!) are
Hilbert transforms of each other,that is
"
0
(!) ¡"
0
=
1
¼
P
Z
1
¡1
"
00
(®)
® ¡!

"
00
(!) =
1
¼
P
Z
1
¡1
"
0
(®) ¡"
0
® ¡!

These equations are called Kramers-KrÄonig relations.The symbol P denotes the Cauchy principal
value of the integral,that is,for f(0) 6= 0
P
Z
1
¡1
f(x)
x
dx = lim
a!0
½
Z
¡a
¡1
f(x)
x
dx +
Z
1
a
f(x)
x
dx
¾
The constitutive equations (1.15) imply that the inductions are parallel to the applied ¯elds,which
is true in the case of isotropic media but not in the case of crystals.These media are said to be
anisotropic and are characterized by a regular periodical arrangement of their atoms.In this case
the permittivity constitutive equations must be written in matrix form:
0
@
D
x
D
y
D
z
1
A
=
0
@
"
xx
"
xy
"
xz
"
yx
"
yy
"
yz
"
zx
"
zy
"
zz
1
A
0
@
E
x
E
y
E
z
1
A
In the case of an isotropic dielectric,the matrix is a multiple of the identity:"="I.
1.4 Boundary conditions
Maxwell's equations (1.13) are partial di®erential equations (PDE),valid in every point of a given
domain,which can be ¯nite or in¯nite.If it is ¯nite,we must supply information about the nature
of the material that forms the boundary.In mathematical terms,we must specify the boundary
conditions,i.e.the values of the state variables on the boundary.
Often the boundary is a perfect electric conductor (PEC),that is a material with in¯nite
conductivity.Note that copper is such a good conductor that up to microwave frequencies it can
be modeled as a PEC.If the conductivity is in¯nite,the electric ¯eld must vanish everywhere in
the volume of a PEC,otherwise the conduction current would be in¯nite.The ¯rst Maxwell's
equation shows that also the magnetic ¯eld is identically zero,provided the frequency is not zero.
Since we are essentially interested in time-varying ¯elds,we conclude that in a PEC both ¯elds
vanish identically.At the surface,since the conduction current cannot have a normal component,
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 12
only the tangential component of the electric ¯eld must be zero.If ^º is the unit normal at the
boundary,this condition can be written
^º £E = 0 on the boundary (1.16)
Indeed,^º £E is tangential to the boundary,as shown in Fig.1.5.
￿￿￿￿￿￿￿ ￿￿￿￿￿


tg
E
￿￿￿


Figure 1.5.Boundary condition at the surface of a perfect electric conductor.Relationship between
the tangential component of the electric ¯eld E
tg
and ^º £E
Sometimes the permittivity or the permeability change abruptly crossing a surface in the do-
main.By applying some integral theorems to Maxwell's equations,it can be shown that the
following continuity conditions hold
^º £(H(r
§+
) ¡H(r
§¡
)) = 0 ^º £(E(r
§+
) ¡E(r
§¡
)) = 0 (1.17)
^º ¢ (B(r
§+
) ¡B(r
§¡
)) = 0 ^º ¢ (D(r
§+
) ¡D(r
§¡
)) = 0 (1.18)
where ^º is the normal to the surface and r
§§
are in¯nitely close points,lying on opposite sides of
the surface,as shown in Fig.1.6.It can also be proved that if a surface current J
s
or M
s
exist,
then the ¯elds are discontinuous


+

r


r

1 1
 
2 2
 
Figure 1.6.Boundary conditions at the surface of separation between di®erent dielectric media
^º £(H(r
§+
) ¡H(r
§¡
)) = J
s
^º £(E(r
§+
) ¡E(r
§¡
)) = M
s
Since it can be proved that when a PEC is present in a magnetic ¯eld,the induced current °ows
only on the surface of it and the magnetic ¯eld is identically zero in the PEC,we can write
^º £H(r
§+
) = J
s
(1.19)
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 13
Notice that this is an equation that does not force a condition on Hbut allows the computation of
J
s
.Hence the boundary condition to be enforced at the surface of a PEC is only (1.16).Notice also
that the units of a surface electric current are A/m and those of a surface magnetic current V/m.
This is necessary for the validity of the previous equations,but it is also obvious for geometrical
reasons,as Fig.1.7 shows.
s
J

￿￿￿


Figure 1.7.Surface electric current induced on a perfect electric conductor (PEC).° is a curve
lying on the PEC surface.J
s
is a current density per unit length measured along ° (A/m)
Sometimes an approximate boundary condition of impedance type is used:it is a linear relation
between the tangential electric and magnetic ¯elds,called also a Leontovich boundary condition,
that can be written
^º £E(r
§+
) = Z
s
(^º £H(r
§+
)) £ ^º (1.20)
in terms of a suitable surface impedance.This condition is typically applied when the boundary is
a real metal and one desires a more accurate model than just a PEC.The double vector product
on the right hand side makes the tangential electric and magnetic ¯eld orthogonal on the surface.
If the surface is not smooth but has a sub-wavelength wire structure,the surface impedance is not
a scalar but a tensor (matrix).
If the domain is in¯nite and sources are all at ¯nite distance from the origin,the only necessary
boundary condition is that the ¯eld is only outgoing at large distance from the origin.
Sometimes the geometry of dielectric and metal bodies possesses sharp edges or sharp vertices
(e.g.plates,cubes,cones),as shown in Fig.1.8.In this case some ¯eld components can become
in¯nite at the geometric singularity:however the ¯elds must always be locally square integrable.
In physical terms this condition means that the electromagnetic energy contained in a ¯nite neigh-
borhood of the singularity must always be ¯nite.Apart from these cases of true singularities of
the geometry,¯elds are always regular,i.e.di®erentiable.This is to be noted in particular when
the geometry singularity is only apparent because it is related to the particular coordinate system.
For example if we use cylindrical coordinates in a homogeneous medium the ¯elds must be regular
in the origin even if the coordinates have a singularity there.
￿￿￿￿￿￿￿￿￿
￿￿￿￿￿
￿￿￿￿
Figure 1.8.
Chapter 2
Waves in homogeneous media
At this point we have ¯nished the preliminaries.We have decided to use the electric and magnetic
¯eld as state variables and we have the relevant equations:
8
>
>
<
>
>
:
r£E = ¡j!¹H¡M
r£H = j!"E+J
+ boundary conditions
(2.1)
where for brevity we have dropped the dependence of all variables on r,!,but it is understood.
The line\boundary conditions"contains the form of the domain where we want to compute the
¯elds created by the given sources J;Mand information on the nature of the material that forms
the boundary.In general permittivity and permeability are functions of r and provide information
on the shape of the bodies in the domain and on their nature.In this way the problemis well posed
and it can be shown it has a unique solution,provided there is at least a region in the domain
where energy is dissipated.Needless to say the problem (2.1) can be very complicated and can be
solved only by approximate numerical techniques.For this reason it is useful to proceed by small
steps,by analyzing ¯rst a very idealized problem that is so simple to allow an analytical solution.
2.1 Plane waves
Let us start by assuming that the domain of interest is in¯nite and the medium is homogeneous
and lossless,so that";¹ are real constants.The only boundary condition to enforce is that the
¯eld is regular everywhere,in particular at in¯nity.
Moreover we assume that sources are identically zero.This is,at ¯rst sight,a strange assump-
tion since it would seem to imply that also the ¯elds must be identically zero!However,if there
are no sources,Maxwell's equations become a homogeneous system of di®erential equations:it is
well known that homogeneous di®erential equations have nontrivial solutions.Let us review some
examples:
²
d
dx
f(x) = 0;x 2 R)f(x) = const
14
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 15
²
Harmonic oscillator
µ
d
2
dt
2
+!
2
0

f(t) = 0;t 2 R)f(t) = Acos!
0
t +Bsin!
0
t
²
Transmission line
8
>
<
>
:
¡
d
dz
V = j!LI
¡
d
dz
I = j!C V
;z 2 R)
8
<
:
V (z) = V
+
0
e
¡jkz
+V
¡
0
e
+jkz
I(z) = Y
1
V
+
0
e
¡jkz
¡Y
1
V
¡
0
e
+jkz
These are actually generalizations of the simple case of a linear system of algebraic equations
Ax = 0
which has nontrivial solutions if the matrix A is non invertible.
So the problem we want to solve is
(
r£E = ¡j!¹H
r£H = j!"E
(2.2)
Equations (2.2) are written in a coordinate-free language.However,in order to solve them it
is necessary to select a coordinate system.Several choices are at our disposal,the more common
being cartesian,cylindrical and spherical coordinates.The corresponding solutions of (2.2) will
be plane,cylindrical and spherical waves,respectively.The simplest case is the ¯rst and we start
with that.
Recalling the expression of the r operator in cartesian coordinates
r = ^x
@
@x
+^y
@
@y
+^z
@
@z
and that the medium is homogeneous,it is clear that (2.2) is a linear system of constant coe±cient
equations.On the basis of the experience with ordinary di®erential equations,we can expect that
the solution is of exponential type,hence we assume tentatively
E(r) = E
0
exp(¡jk
x
x) exp(¡jk
y
y) exp(¡jk
z
z) (2.3)
and likewise for the magnetic ¯eld,where E
0
and k
x
;k
y
;k
z
are constants to be found.The constants
k
x
;k
y
;k
z
have dimensions rad/m and are wavenumbers along the three coordinate axes.It is
convenient to work with a vector formalism,even if the coordinate system is ¯xed.By recalling
that r = x^x+y^y +z^z and de¯ning the wavevector k = k
x
^x+k
y
^y +k
z
^z,the assumed form of the
solution is
E(r) = E
0
exp(¡jk ¢ r) H(r) = H
0
exp(¡jk ¢ r) (2.4)
Before substituting it into (2.2) it is useful to compute
rexp(¡jk ¢ r) =
µ
^x
@
@x
+^y
@
@y
+^z
@
@z

exp(¡jk
x
x) exp(¡jk
y
y) exp(¡jk
z
z)
= (¡jk
x
^x ¡jk
y
^y ¡jk
z
^z) exp(¡jk
x
x) exp(¡jk
y
y) exp(¡jk
z
z)
= ¡jkexp(¡jk ¢ r)
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 16
Moreover we recall the identity of vector calculus
r£(A(r)Á(r)) = Á(r)r£A(r) +rÁ(r) £A(r)
so that the substitution of (2.4) into (2.2) yields
¡jk £E
0
exp(¡jk ¢ r) = ¡j!¹H
0
exp(¡jk ¢ r)
¡jk £H
0
exp(¡jk ¢ r) = j!"E
0
exp(¡jk ¢ r)
Canceling common factors
¡jk £E
0
= ¡j!¹H
0
¡jk £H
0
= j!"E
0
Note that the fact that these equations do not contain the variable r any longer con¯rms the
correctness of the assumption (2.4).
Recalling the properties of vector products we learn that E
0
;H
0
;k form a righthanded triple
of mutually orthogonal vectors.Next,to proceed,we eliminate H
0
between the two equations.To
this end,we solve the ¯rst equation with respect to H
0
:
H
0
=
k £E
0

(2.5)
and substitute into the second one
k £(k £E
0
) +!
2
¹"E
0
= 0
The double vector product can be expanded
k(k ¢ E
0
) ¡(k ¢ k) E
0
+!
2
¹"E
0
= 0
The ¯rst term is zero because of the orthogonality of k and E
0
,hence
¡
k ¢ k ¡!
2
¹"
¢
E
0
= 0 (2.6)
We are interested in nontrivial solutions of this equation,so that the following condition must hold
k ¢ k =!
2
¹"(2.7)
A relationship between frequency and wavenumbers is called in general a dispersion relation.We
can read it from right to left or vice versa:in the ¯rst instance it tells us what the frequency
must be so that the ¯eld distribution (2.4) with a speci¯c wavevector k is a solution of Maxwell's
equations.From this point of view,the value of!can be considered to be a resonance frequency of
the structure.Notice that the requirement that the solution be regular everywhere (in particular
at in¯nity) forces the wavector to be real.Apart from this condition there is no constrain on the
possible values of the wavenumbers,hence the resonance frequencies of the system are in¯nite in
number and even distributed continuously.We start seeing a property that characterizes all ¯eld
problems.Whereas lumped element circuits have a ¯nite number of resonances,distributed systems
always have an in¯nite number of them.Moreover,if the structure has ¯nite size its resonances
are denumerably in¯nite:this means that they can be labeled with integers!
1
;:::;!
n
;:::.If the
structure,as in this case,has in¯nite size,then the resonances should be labeled with a continuous
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 17
variable.To simplify the notation,we omit this labeling variable and remember that!can take
every real value.
The dispersion equation can also be read from left to right:in this case the frequency is
considered ¯xed and we look for the wavevectors that satisfy eq.(2.7).It is convenient to introduce
a unit vector ^s,called direction of propagation,directed along k,so that k = k^s;then the dispersion
relation becomes
k =!
p
"¹ (2.8)
Clearly the direction ^s can be whatever,only the wavenumber k is speci¯ed.In other words for
any given frequency there are an in¯nite number of waves with arbitrary directions of propagation.
Considering again eq.(2.6),we see that if the dispersion relation is satis¯ed,the vector E
0
can be arbitrarily chosen,provided it is orthogonal to ^s.For a given ^s,there are two linearly
independent waves,but they are degenerate,because they have the same value of the wavenumber
k.The corresponding magnetic ¯eld is obtained by the impedance relation (2.5):
H
0
=
!
p
"¹^s £E
0

=
r
"
¹
^s £E
0
= Y ^s £E
0
(2.9)
where the wave admittance has been introduced.Its inverse is the wave impedance Z = 1=Y.In
free space it has the value
Z
0
=
r
¹
0
"
0
¼ 120¼ ¼ 377 ­
In conclusion a solution of the problem (2.2) is
E
^s
(r) = E
0
exp(¡jk^s ¢ r)
H
^s
(r) = H
0
exp(¡jk^s ¢ r)
(2.10)
where ^s is the direction of propagation,k =!
p
"¹,H
0
= Y ^s £ E
0
and E
0
;H
0
;^s are mutually
orthogonal:electromagnetic waves are transverse.Since the problem is linear,the general solution
of (2.2) can be written as a linear combination of waves of the type (2.10) with all possible directions
^s.
Wavefronts are de¯ned to be the surfaces on which the phase ©(r) of the wave is constant.In
this case
©(r) = ¡k^s ¢ r = constant
is the equation of a family of planes perpendicular to ^s:hence the ¯elds (2.10) are called plane
waves because the wavefronts are planes.
Let us now study the polarization of plane waves.This requires going back to time domain via
eq.(1.8)
E(r;t) = RfE
0
exp(¡jk^s ¢ r) exp(j!
0
t)g
= E
0
0
cos (!
0
t ¡k^s ¢ r) ¡E
00
0
sin(!
0
t ¡k^s ¢ r)
(2.11)
We see clearly from this equation that the type of polarization in every point of space is speci¯ed
by E
0
,which is the electric ¯eld in the origin.What changes from point to point is the time
evolution,due to the propagation delay.The constant phase surfaces of the time varying ¯eld are
!
0
t ¡k^s ¢ r = const,from which we ¯nd
^s ¢ r =
!
0
k
t ¡const
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 18
￿
E
H
￿￿￿￿￿￿￿￿￿￿
r

s
￿￿￿￿￿￿￿

u
Figure 2.1.Wavefronts of a plane wave with direction of propagation ^s.They move at the phase
velocity along ^s.The vector ^u denotes an arbitrary direction
This means that the wavefronts are not ¯xed but are moving.Indeed,consider a speci¯c wavefront,
say the zero-phase one,i.e.the one for which const=0,as shown in Fig.2.1;the vector r that denotes
its points is such that its projection on ^s increases linearly with time.In other words,the plane
moves as a whole with speed
v
ph
=
!
0
k
=
!
0
!
0
p

=
1
p

=
c
p
"
r
¹
r
(2.12)
This velocity is called phase velocity of the wave,because it has been de¯ned by means of the
constant phase surfaces.Now consider an arbitrary straight line with direction ^u,an let P be its
intersection with the zero-phase wavefront.The velocity of P is
v
ph
(^u) =
1
^u ¢ ^s
c
p
"
r
¹
r
Clearly,for all directions ^u 6= ^s this velocity is larger than c=
p
"
r
¹
r
.Notice,however,that even
when this velocity is larger than the speed of light in empty space,the theory of relativity is not
violated.Indeed no matter or energy is moving in the direction of ^u,but only a mathematical
point,e.g.a maximum or a node of the oscillation.This concept is at the basis of the fact that
the phase velocity in a waveguide is always greater that the speed of light in vacuum.
Fig.2.2 shows the ¯elds of a linearly polarized plane wave propagating in the ^s = ^z direction:
remember that the electric and magnetic ¯eld are orthogonal.For clarity,the ¯eld vectors have
been drawn only for a number of points on the z axis,even if they are de¯ned in every point of
space.
Wave propagation is always associated to energy °ow.The Poynting vector has the meaning
of power density (per unit surface) associated to the wave.Let us compute the Poynting vector S
in the case of a plane wave:
S =
1
2
(E£H
¤
) =
1
2
(E
0
exp(¡jk^s ¢ r) £H
0
¤
exp(jk^s ¢ r))
=
1
2
(E
0
£H
0
¤
) =
1
2
(E
0
£(^s £E
0
¤
)Y
¤
)
=
1
2
jE
0
j
2
Z
^s
(2.13)
where we used the impedance relation (2.9) and the property
E
0
£(^s £E
0
¤
) = jE
0
j
2
^s ¡(E
0
¢ ^s) E
¤
0
= jE
0
j
2
^s (2.14)
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 19
￿
￿
￿
,
z t
E( )
,
z t
H( )
Figure 2.2.Snapshot of a linearly polarized plane wave propagating in the ^s =^z direction
because of the orthogonality between E
0
and ^s.Note that the Poynting vector is real,hence
no reactive power is associated to plane waves in a lossless medium.We see that the active
power density (magnitude of S) associated to a plane wave is constant:this implies that the total
power,obtained by integration over the whole wavefront,is in¯nite.Hence,a single plane wave
is not a physically realizable ¯eld.This property,however,does not destroy the usefulness of the
concept.Indeed,since Maxwell's equations are linear,the superposition principle holds and linear
combinations of plane waves are also solutions.It turns out that a continuous sum (integral) of
plane waves not necessarily has in¯nite power:indeed all physically realizable ¯elds can always be
represented as integrals of plane waves.
We recall that to each plane wave not only a power °ow is associated,but also a °ow of
linear momentum and of angular momentum.In particular the linear momentum °ow,which has
direction ^s,is responsible for the radiation pressure,that explains,for instance,the shape of comet
tails and has been considered as a possible\engine"for interplanetary travels.
The properties of plane waves do not change much if the dielectric is lossy.In this case the
permittivity is complex and the dispersion relation (2.8) becomes
k =!
p
("
0
¡j"
00
)¹ = ¯ ¡j®;® ¸ 0
where ¯ is the true phase constant,measured in rad/mand ® is the attenuation constant,measured
in Nepers/m.The electric ¯eld,for instance,obeys the propagation law
E(r) = E
0
exp(¡j¯^s ¢ r) exp(¡®^s ¢ r) (2.15)
Clearly the magnitude of the ¯eld is no longer constant in space and the wavefronts are also surfaces
of constant ¯eld magnitude.Obviously,it must be remarked that the value of the phase velocity
cannot be computed by (2.12),but it is given by
v
ph
=
!
¯
and the wavelength is
¸ =

¯
=
v
ph
f
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 20
Indeed,both the phase velocity and the wavelength are de¯ned on the basis of the phase of the
wave and ¯ is exactly the phase rate-of-change,measured,as said above,in rad/m.
Let us compute the power °ow.
S =
1
2
(E£H
¤
) =
1
2
(E
0
exp(¡jk^s ¢ r) £H
0
¤
exp(jk
¤
^s ¢ r))
=
1
2
(E
0
£(^s £E
0
¤
)Y
¤
exp(¡2®^s ¢ r)) =
1
2
Y
¤
jE
0
j
2
exp(¡2®^s ¢ r)^s
(2.16)
where we have used again the property (2.14).In this case the Poynting vector is complex.The
active power density of the wave decreases during propagation because part of it is transferred to
the dielectric in the form of heat.
All the plane waves considered up to now are called uniform because their propagation direc-
tion ^s is real.If we go through all the steps of the derivation,we realize that even if ^s is complex
(whatever this means!) the expression (2.10) is a valid solution of Maxwell's equations,although
only in a halfspace.Such a generalization leading to non uniform plane waves is required when
solving a scattering problem where a plane wave is incident on the interface separating two di-
electrics.It is to be remarked that also the plane wave (2.15) is a valid solution only in a halfspace.
Indeed,if ^s ¢ r!¡1,the ¯eld diverges,which is not physically acceptable.
2.2 Cylindrical waves
To solve eqs.(2.2) it is also possible to use a cylindrical coordinate system instead of a cartesian
one.The mathematics is considerably more complicated in this case.The reason is that the unit
vectors of the cylindrical coordinate system are not constant but change from point to point.As
a consequence,the expression of the di®erential operators is no longer with constant coe±cients
and the solutions are no longer of exponential type,but are expressed in terms of Bessel functions.
These special functions of mathematical physics were actually introduced,along with many others,
in order to solve the wave equation.
If a cylindrical or spherical coordinate systemis used,Maxwell's equations (2.2) are not attacked
directly but are ¯rst transformed into a single second order equation.We write them again for
convenience
(
r£E = ¡j!¹H
r£H = j!"E
Since it is a system of equations,we can eliminate one of the two unknowns.We solve the ¯rst
equation with respect to H and substitute in the second
8
>
>
<
>
>
:
H =
r£E
¡j!¹
r£(r£E) =!
2
"¹E
As expected,the second equation contains only the electric ¯eld:the price to pay for it is that it
is second order in the space derivatives;it is called the curl-curl equation.However its form can
be simpli¯ed recalling the identity
r£(r£E) = r(r¢ E) ¡r
2
E = ¡r
2
E
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 21
where we have used the fact that (2.2) do not have sources,hence also the charge density ½(r) is
zero and the electric ¯eld has zero divergence,r¢ E = ½ = 0.We obtain in this way the vector
Helmholtz equation
r
2
E+!
2
"¹E = 0 (2.17)
Even if it is written in coordinate-free language,its meaning is easily understood in cartesian
coordinates only,where
E(r) = E
x
(r)^x +E
y
(r)^y +E
z
(r)
Since the unit vectors are not function of r,each cartesian component of the electric ¯eld satis¯es
Helmholtz equation,which then becomes scalar:
r
2
à +!
2
"¹Ã = 0
where Ã(r
) denotes any component of E
.It is interesting to note that even if we are using cartesian
components to represent the electric ¯eld,we are not forced to use necessarily cartesian coordinates
to specify the observation point,i.e.the components of r.By using the classical method of the
separation of variables in cylindrical coordinates,we can ¯nd
Ã
(
½;';z
) =
Ã
0
H
(2)
m
(
k
½
½
)e
¡jk
z
z
e
¡jm'
(2.18)
where m= 0;§1;§2;:::,k
½
2 [0;1) and k
z
identify the various outgoing cylindrical waves.These
three parameters play the role of k
x
,k
y
,k
z
in the case of plane waves.The function H
(2)
m
(k
½
½) is
a Hankel function of second kind and order m.Its asymptotic expansion is
H
(2)
m
(k
½
½) »
s
2
¼k
½
½
exp
h
¡j(k
½
½ ¡m
¼
2
¡
¼
4
)
i
The dispersion relation is
k
z
=
q
!
2
"¹ ¡k
2
½
Notice that all three components have this form,but the values of Ã
0
for each must be interrelated
so that the resulting vectors E and H satisfy Maxwell's equations.
We are not going to describe in detail the properties of these waves.To explain the name,it
is enough to say that the wavefronts are cylinders having ^z as axis,at least in the case m= 0 and
k
½
=!
p
"¹.
2.3 Spherical waves
The case of spherical waves is similar,froma certain point view,to that of cylindrical waves.Again
the mathematics is fairly complicated and new special functions are introduced.In this case the
scalar Helmholtz equation is solved in spherical coordinates and the result is
Ã(r;#;') = Ã
0
h
(2)
l
(kr)P
m
l
(cos#)e
¡jm'
where l = 0;1;2;:::and ¡l · m · l identify the various outgoing spherical waves.The functions
h
(2)
l
(k½) are spherical Hankel functions of second kind and order l,whereas P
m
l
(cos#) are associated
Legendre polynomials of degree l and order m.Again,the various solutions for the three cartesian
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 22
1
1


2
2


3
3


4
4


t
r
i
Figure 2.3.Scattering from a strati¯ed dielectric:i,incident wave;r re°ected wave;t,
transmitted wave.For clarity,the couple of plane waves existing in each of the internal
layers has not been indicated
components must be related so that the resulting E and H satisfy Maxwell's equations.The
asymptotic expansion of the spherical Hankel functions is
h
(2)
l
(kr) »
1
kr
exp
h
¡j(kr ¡m
¼
2
¡
¼
4
)
i
Hence the wavefronts are spheres with center in the origin and this justi¯es the name.
2.4 Waves in non homogeneous media
The case that has been considered,namely a homogeneous medium¯lling the whole space is highly
idealized.In a realistic situation,"(r),¹(r) are not constant and obviously the plane waves (2.10)
are not solution of Maxwell's equations (2.2).In order to consider a simple case,let us assume
that the medium is piecewise homogeneous and that the interfaces between the di®erent materials
are planar:the structure is called a strati¯ed dielectric.In the left half space an incident plane
wave is assumed.In each layer,plane waves are solutions of (2.2),but the continuity conditions
(1.17) must be obeyed.It can be proved that in each one of the internal layers two plane waves
are present,one forward (incident on the following interface) the other backward (re°ected from
the following interface);in the right half space only one,because the medium extends to in¯nity
and there is no other interface.All these plane waves have the same transverse (to z) component
of the wavevector and their amplitudes can be easily determined so that the continuity conditions
are satis¯ed.The single wave in the fourth medium is the transmitted ¯eld,the second one in the
¯rst medium is the re°ected ¯eld,as sketched in Fig.2.3.
If the interfaces are not planar,the problem becomes much more di±cult.Consider,for ex-
ample,the case of Fig.2.4,where a plane wave is incident on a cylinder with parameters"
2

2
,
embedded in a homogeneous medium with parameters"
1

1
.It can be shown that the continuity
conditions require that an in¯nite number of plane waves are excited,each one with the right am-
plitude.Collectively.these are called scattered waves.Hence the di±culty of the problem stems
from the necessity to solve a linear system with an in¯nite number of unknowns.
If the medium is not even piecewise homogeneous but arbitrarily inhomogeneous,no analytical
solution is at our disposal.It is,however,to be mentioned that when the variations of"(r),¹(r)
are small on the wavelength scale,a well known approximate method can be used,i.e.Geometrical
Optics.Whereas the plane waves discussed up to now can be de¯ned global plane waves since each
one is de¯ned over the whole space,the elementary geometrical optics ¯eld is a local plane wave.
For instance,a spherical wave in free space can be approximated by a collection of local plane
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 23
i
s
s
s
s
1
1


2
2


Figure 2.4.Scattering froma non planar interface:i,incident wave;s,scattered waves.For clarity,
the plane waves existing inside the cylinder have not been indicated
waves because its wavefront (a sphere) can be approximated locally by the relevant tangent plane.
The k vectors of these local plane waves de¯ne a vector ¯eld,whose ¯eld lines are the geometrical
optics rays.It turns out that rays are also the ¯eld lines of the Poynting vector ¯eld:hence a plot
of the rays provides information about the power °ow.
Geometrical optics is a very powerful technique,but sometimes yields de¯nitely wrong results.
This happens when rays cross in a point or along a line,because in this case it predicts a ¯eld of
in¯nite intensity.These singularities are called caustics and the focus of a converging lens is an
example:in such a point the electromagnetic ¯eld can be very large but is certainly ¯nite.Hence
geometrical optics can be safely used only away from caustics.
2.5 Propagation in good conductors
Apart from the case of optical ¯bers,guided wave propagation is possible in structures containing
metal conductors.Examples are coaxial cables,parallel wire transmission lines,microstrip lines,
waveguides with any cross section.Since the metals used in the applications (such as copper) are
characterized by a very large conductivity,in a ¯rst approximation they can be considered to be
perfect conductors (PEC),an assumption that greatly simpli¯es the study.However,in order to
build more accurate numerical models of real devices,it is necessary to take into account the ¯nite
conductivity of real metals.In this section we consider the propagation of plane waves in good
conductors,in order to draw some conclusions pertaining to transmission systems.
Metals are characterized by so a large conductivity that the displacement currents can be safely
neglected with respect to the conduction currents,so that some simpli¯cations in the general
formulas of Section 2.1 are possible.Starting with the wavenumber,
k
m
=!
p
("¡j°=!)¹ ¼!
p
(¡j°=!)¹
if
°
!"
À1 (good conductor)
Recalling that
p
¡j = §
1 ¡j
p
2
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 24
and that Imk · 0 for a passive medium,we ¯nd
k
m
=
1 ¡j
p
2
p
!°¹ =
1 ¡j
±
(2.19)
where we have introduced the so called skin depth
± =
r
2
!¹°
(2.20)
which,of course,should not be confused with the loss angle,introduced in Section 1.3,indicated
with the same symbol.This relation can also be written
p
f± =
r
1
¼¹°
= const
where f is the frequency and the constant depends only on the material.For instance,in the case
of copper,° = 5:8 ¢ 10
7
S/m and ¹ = ¹
0
= 4¼ ¢ 10
¡7
H/m,hence
p
f± = 0:0661
p
Hzm (2.21)
The reason for the name will be explained below.
The wave impedance is computed with the same approximation:
Z
m
=
r
¹
"¡j°=!
¼
r
¹
¡j°=!
=
s
j¹!
°
=
1 +j
p
2
r

°
that is
Z
m
= R
s
(1 +j) (2.22)
where we have introduced the surface resistance
R
s
=
r


=
1
°±
for which we can write
R
s
p
f
=
r
¼¹
°
= const
i
E
r
E
t
E
x
z
￿￿￿￿￿
￿￿￿￿￿￿￿￿￿￿
t
H
r
H
i
H
Figure 2.5.Good conductor in a plane wave ¯eld.In the free space region both an incident and
a re°ected wave exist,in the metal only the transmitted one.The wavevector of the transmitted
wave is drawn dashed,to indicate that it is complex.
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 25
Again,in the case of copper,
R
s
p
f
= 2:6090 ¢ 10
¡7
­=
p
Hz
Consider now a (highly idealized) conductor in the form of a half space,which faces free space,
with a linearly plane wave incident normally on it,as shown in Fig.2.5.The tangential electric
and magnetic ¯elds are continuous at the interface,then the ratio of their magnitudes is the same
in z = 0
¡
and in z = 0
+
.But in z = 0
+
this ratio is Z
m
by de¯nition,so we can easily understand
that the expressions of the electric ¯elds are
E
i
= E
0
e
¡jk
0
z
^x
E
r
= ¡E
0
e
jk
0
z
^x
E
t
= (1 +¡)E
0
e
¡jk
m
z
^x
where the re°ection coe±cient is
¡ =
Z
m
¡Z
0
Z
m
+Z
0
Since jZ
m
j ¿Z
0
,¡ is close to ¡1.Indeed,
1 +¡ =
2Z
m
Z
m
+Z
0
¼
2Z
m
Z
0
= 2(1 +j)
s
¼"
0
f
°
from which
¡ ¼ ¡1 +2(1 +j)
s
¼"
0
f
°
In the case of copper,
¡ ¼ ¡1 +2(1 +j) ¢ 6:9252 ¢ 10
¡10
p
f
(frequency in Hz).We can also say that the metal enforces an impedance type boundary condition
(see (1.20)) with Z
m
as surface impedance.
The magnetic ¯eld is
H
i
= Y
0
E
0
e
¡jk
0
z
^y
H
r
= ¡Y
0
¡E
0
e
jk
0
z
^y
H
t
= Y
0
(1 ¡¡)E
0
e
¡jk
m
z
^x ¼ 2Y
0
E
0
e
¡jk
m
z
^y
(2.23)
Note that the total magnetic ¯eld at the interface is approximately twice the incident one because
¡ is very close to ¡1.
The electric ¯eld in the metal produces a conduction current in the ^x direction
J
c
= °E
t
= 2(1 +j)
p
¼"
0
f°E
0
e
¡jk
m
z
^x
In the case of copper,this becomes
J
c
= 2(1 +j) ¢ 0:402
p
fE
0
e
¡jz=±
e
¡z=±
^x
(frequency in Hz) where we have used (2.19).The magnitude of this current density is maximum
at the interface and then decays exponentially in the metal.At a depth z = ±,it has reduced by a
factor 1=e = 0:368.Eq.(2.21) allows a simple computation of ± for various frequencies,reported in
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 26
Table 2.1.Skin depth for copper at various frequencies
Frequency
Skin depth
50 Hz
9.3 mm
1 kHz
2.1 mm
1 MHz
66.1 ¹m
1 GHz
2.1 ¹m
Table 2.1.We see clearly that as the frequency increases,the current density remains appreciable
only in a very thin layer close to the metal surface,which can be considered as its\skin".Even if
this analysis strictly refers to a metal half space,we can use it to draw qualitative conclusions in
the case of ¯nite thickness conductors or even round conductors,provided the thickness is much
larger than the skin depth.At the power frequency of 50Hz,the skin depth is so large that the
current has a uniform distribution in ordinary wires.At the frequency of 1MHz,instead,most of
the conductor copper is not used.At microwave frequencies,a few microns of copper deposited on
an insulator perform as an excellent conductor.
The consequence of the skin depth change with frequency is that the resistance of a conductor is
an increasing function of frequency:indeed,the\e®ective"cross-section of the conductor decreases
as the frequency increases.This phenomenon is generally called skin e®ect.
Let us compute the impedance of the structure of Fig.2.5,viewed as a current carrying con-
ductor.Since the ¯elds and the current density does not depend on y,we consider a strip of unit
length in this direction.We compute ¯rst the total current I,°owing in the ^x direction,per unit
length along y:
I =
Z
1
0
J
c
(x;z) ¢ ^xdz =
Z
1
0
J
c0
e
¡j(1¡j)z=±
dz =
J
c0
±
1 +j
(2.24)
Notice that the dimensions of I are correctly A/m,since J
c0
is a surface current density with value
J
c0
= 2(1 +j)
p
¼"
0
f°E
0
(2.25)
Next,consider a unit length in the ^x direction of this conductor and compute the potential di®er-
ence along this length by integrating the electric ¯eld E
x
along the x axis (y = 0,z = 0).Note
that E
x
does not depend on x,hence E
x
itself coincides numerically with this potential di®erence.
Finally,the impedance per unit width in the y direction and unit length in the x direction is given
by
Z
pul
=
E
x
(0;0)
I
=
J
c0

J
c0
±=(1 +j)
=
1 +j
°±
= Z
m
where we used (2.22).In conclusion we have this remarkable result:the impedance seen by a
current °owing through a square of unit sides coincides with the wave impedance in the metal.
Notice that,apart from the similarity in the symbols,
Z
m
=
E
x
H
y
hence it is a completely di®erent concept.Moreover,since the conductor we are considering has
unit width in the y direction,unit length in the x direction (and in¯nite thickness in the z direction)
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 27
the previous analysis shows why often the value of the surface resistance R
s
is expressed in ­=¤
(read Ohm per square).
As a ¯nal remark,we note that the material becomes a perfect conductor when °!1.In
these conditions,the skin depth vanishes and the value of the current density at the interface tends
to in¯nity,according to (2.25).Nevertheless,we see from (2.24) that the total current is ¯nite and
its value,independent of ° is
I =
J
c0
±
1 +j
= 2
r
"
0
¹
0
E
0
where (2.20) and (2.25) have been used.This means that in a perfect conductor the current density
can be written
J
c
(x;z) = 2
r
"
0
¹
0
E
0
±(z) = J
¾
±(z)
On the other hand,from(2.23) we see that in the limit °!1the magnitude of the total magnetic
¯eld at the z = 0
¡
interface coincides with J
¾
.Taking the directions of the vectors into account,
we conclude that if a perfect conductor is immersed in an electromagnetic ¯eld,on its surface a
current density J
¾
(A/m) appears,such that
J
¾
= ^º £H
where ^º is the normal to the PEC surface,pointing toward free space.In practice,this is the proof
of Eq.(1.19).
Another example that we consider now is that of sea water:because of the salt contained in it,
the conductivity is ° = 5 S/m,whereas the relative permittivity,up to the microwave region,does
not change very much and will be taken to be"
r
= 80.We compute the complex wavenumber and
the attenuation constant by the general equation
k
m
=!
p
("¡j°=!)¹
The results are the following:
²
At f = 100Hz,°=(2¼f"
0
"
r
) = 1:1234 ¢ 10
7
,so sea water behaves as a good conductor;
k = (4:4429 ¡j4:4429) ¢ 10
¡2
m
¡1
® = 0:3859dB/m
²
At f = 10kHz,°=(2¼f"
0
"
r
) = 1:1234 ¢ 10
5
,so sea water behaves as a good conductor;
k = (0:4429 ¡j0:4429)m
¡1
® = 3:8590dB/m
²
At f = 1GHz,°=(2¼f"
0
"
r
) = 1:1234,so the displacement currents cannot be neglected;
k = (209:7536 ¡j94:1066)m
¡1
® = 817:3998dB/m
²
At f = 10GHz,°=(2¼f"
0
"
r
) = 0:1123,so the displacement currents cannot be neglected;
k = (1877:5266 ¡j105:1341)m
¡1
® = 913:1833dB/m
Obviously,at microwave frequency,the attenuation of sea water precludes the possibility of commu-
nicating with submarines during subsurface navigation.This becomes possible at low frequencies,
where,however,the available bandwidth is very narrow.
Chapter 3
Radiation in free space
The fundamental problem in electromagnetics is computing the ¯elds created by a speci¯ed set of
sources in a given region of space.This means that the functions"(r),¹(r) are assigned,as well
as the form of the region boundary and the material of which it consists.Then the sources are
speci¯ed in terms of electric and magnetic current densities J(r),M(r).
In order to understand the basic mechanism of radiation,it is convenient to consider ¯rst a
highly idealized problem,wherein the sources radiate in an in¯nite homogeneous medium.Later
we will see how to apply the results of this chapter to the real antenna problem.
3.1 Green's functions
The radiation problem is mathematically formulated as
(
r£E = ¡j!¹
0
H¡M
r£H = j!"
0
E+J
(3.1)
in an in¯nite homogeneous domain that we assume to be free space.These equations are linear with
constant coe±cients and the independent variable is r.We can interpret them as the equations of
a Linear Space Invariant system (LSI),where the source currents play the role of input and the
radiated ¯elds that of output,see Fig.3.1.The box represents a system with two inputs and two
outputs.
E(r)
H(r)
J(r)
M(r)
EJ
G
HJ
G
HM
G
EM
G
Figure 3.1.Linear system view of the radiation phenomenon
28
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 29
LSI systems are clearly a multidimensional generalization of Linear Time Invariant (LTI) sys-
tems.Let us review brie°y the properties of the latter.LTI systems,as shown in Fig.3.2 are
completely characterized in time domain by their impulse response h(t),that is the output that is
obtained when the input is a Dirac delta function ±(t).An arbitrary (continuous) input x(t) can
be represented as a linear combination of pulses thanks to the sifting property of the delta function
x(t) =
Z
1
¡1
x(t
0
)±(t ¡t
0
)dt
0
Because of linearity,the response y(t) to x(t) can be found by convolution
y(t) = h(t) ¤ x(t) =
Z
1
¡1
h(t ¡t
0
)x(t
0
)dt
0
Alternatively,an LTI systemcan be characterized by its transfer function:when the input is x(t) =
exp(j!t),the output is proportional to it and the constant of proportionality is,by de¯nition,H(!),
so that y(t) = H(!) exp(j!t).It can be proved that the impulse response and the transfer function
of a system are related by a Fourier transform
H(!) =
Z
1
¡1
h(t) exp(¡j!t)dt
( )
x t
( )
h t
( )
y t
( )
X

( )
H

( )
Y

Figure 3.2.Time domain and frequency domain description of an LTI system
As a preparation for (3.1),let us consider the simpler case of an in¯nite,uniform transmission
line excited by a distribution of voltage and current generators,v
s
(z) and i
s
(z),as shown in Fig.3.3.
Since these generators are distributed continuously,v
s
(z) and i
s
(z) are densities per unit length
of generators described,as usual,in terms of their open circuit voltage (V/m) and short circuit
current (A/m),respectively.The di®erential equations of the system are
8
>
<
>
:
¡
dV
dz
= j!LI +v
s
¡
dI
dz
= j!CV +i
s
(3.2)
+
+
+
+
( )
s
v z
( )
s
i z
Figure 3.3.In¯nite uniform transmission line with distributed voltage and current generators.
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 30
( )
V z
I z
( )
( )
s
v z
( )
s
i z
s
Vi
G
s
Ii
G
s
Iv
G
s
Vv
G
Figure 3.4.Linear system view of the transmission line with distributed generators.
Generally,transmission lines are excited at one end by a generator that acts as a transmitter.
The model shown in Fig.3.3 refers to a situation of electromagnetic compatibility,where a line
is excited by an electromagnetic wave that couples to the line along a certain segment of it.It
is easy to recognize that this is the one dimensional analogue of Maxwell's equations (3.1).The
problem is again LSI and can be schematized as in Fig.3.4.Hence,as suggested by this picture,
the solution can be expressed as
V (z) = Z
1
Z
1
¡1
G
V i
s
(z ¡z
0
)i
s
(z
0
)dz
0
+
Z
1
¡1
G
V v
s
(z ¡z
0
)v
s
(z
0
)dz
0
I(z) =
Z
1
¡1
G
Ii
s
(z ¡z
0
)i
s
(z
0
)dz
0
+Y
1
Z
1
¡1
G
Iv
s
(z ¡z
0
)v
s
(z
0
)dz
0
The system here has two inputs and two outputs:each output depends on both inputs,so that
in practice there are four Green's functions,each one a pure number.They can be obtained
by applying the spatial Fourier transform to the system equations (3.2).However,by the very
de¯nition of Green's function
²
G
V i
s
(z) is the voltage wave V (z)=Z
1
created by a unit amplitude current generator located
in z = 0
²
G
V v
s
(z) is the voltage wave V (z) created by a unit amplitude voltage generator located in
z = 0
²
G
Ii
s
(z) is the current wave I(z) created by a unit amplitude current generator located in
z = 0
²
G
Iv
s
(z) is the current wave I(z)Y
1
created by a unit amplitude voltage generator located in
z = 0
so that they can be found by simple circuit theory,just recalling that the input impedance of an
in¯nitely long line is Z
1
.The resulting expressions are
G
V i
s
(z) = G
Iv
s
(z) = ¡
1
2
e
¡jkjzj
G
V v
s
(z) = G
Ii
s
(z) = ¡
1
2
sgn(z)e
¡jkjzj
where sgn(z) is the sign function
sgn(z) =
½
1 if z > 0
¡1 if z < 0
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 31
As another preparatory example before tackling (3.1),let us consider the case of sound waves.
It can be shown that the excess pressure p(r) with respect to the background pressure satis¯es the
scalar Helmholtz equation
µ
r
2
+
!
2
V
2
s

p(r) = ¡S(r) (3.3)
where V
s
is the sound velocity and S(r) is a source term.This equation corresponds to the picture
of Fig.3.5,where S(r) is the input and p(r) the output.In this case the system has only one
input and one output but it is multidimensional,since both depend on the three independent
variables x,y,z.In perfect analogy with LTI systems,LSI systems are completely characterized
in space domain by their\impulse response"G(r),which is traditionally called Green's function.
This is the output of the system when the input is a point source located at the origin of the
coordinate system,which can be represented mathematically by a three-dimensional Dirac delta
function S(r) = ±(r) = ±(x)±(y)±(z).The fundamental property of this multidimensional Dirac ±
function is
Z
±(r)dr =
Z
1
¡1
Z
1
¡1
Z
1
¡1
±(x)±(y)±(z)dxdy dz = 1
When the input is an arbitrary function,the output is found by (three-dimensional) convolution
p(r) =
Z
G(r ¡r
0
)S(r
0
)dr
0
Alternatively,an LSI system can be characterized in the spectral domain.When the input is a
harmonic function of x,y,z,that is S(r) = exp(¡j(k
x
x +k
y
y +k
z
z)) = exp(¡jk ¢ r),the output
is proportional to it and the coe±cient of proportionality is,by de¯nition,the transfer function
H(k).Again,transfer function and Green's function of the same system are related by a Fourier
transform:however,in this case,it is triple,since it operates on the three variables x,y,z.The
couple of inverse and direct 3-D Fourier transform is given by
G(r) =
1
(2¼)
3
Z
H(k) exp(¡jk ¢ r)dk
H(k) =
Z
G(r) exp(jk ¢ r)dr
(3.4)
where dk = dk
x
dk
y
dk
z
.It can be shown that in the case of free space,the transfer function is
H(k) =
1
k
2
¡!
2
=V
2
s
(3.5)
and the corresponding Green's function is
G(r) =
exp(¡jk
0
r)
4¼r
(3.6)
( )
S
r
( )
G
r
( )
p
r
( )
S
k
( )
H
k
( )
p
k
Figure 3.5.Space domain and spatial frequency domain description of the sound radiation phenomenon
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 32
with k
0
=!=V
s
denoting the wavenumber.This expression describes a diverging spherical wave.
Indeed,the constant phase surfaces are obviously r = const,a series of concentric spheres with
center in the origin.Moreover,assuming that the source is harmonic with frequency!
0
,the
expression of the Green's function in time domain is
g(r;t) = R
½
exp(¡jk
0
r)
4¼r
exp(j!
0
t)
¾
=
cos(!
0
t ¡k
0
r)
4¼r
from which it is evident that the phase velocity is
V
ph
=!
0
=k
0
= V
s
> 0
As another well known example of LSI system,let the frequency!go to zero in (3.3),so that
the Helmholtz equation becomes Poisson equation.This,for example,relates the electric potential
V (r) to a charge distribution ½(r),which acts as its source:
r
2
V (r) = ¡
½(r)
"
The transfer function associated to this equation is (from (3.5))
H(k) =
1
"k
2
and the corresponding Green's function (from (3.6))
G(r) =
1
4¼"r
We recognize immediately this expression as the potential generated by a point charge q = 1 C in
a dielectric with permittivity".
We are ready now for Maxwell's equations (3.1),which are still more complicated because in
addition to being multidimensional and multiple input/output,they are vector equations:this
means that the output is a vector that is not necessarily parallel to the input.This implies that
each of the four Green's function is not a scalar but a linear operator (a tensor),which,in a basis,
is represented by a 3 £3 matrix.This means that,di®erently from the case of sound waves,the
Green's function is not directly the ¯eld radiated by a point source.The source is really a point
but is also a vector,which can have all possible orientations.From a certain point of view,we can
say that the Green's tensor yields the ¯eld radiated in a given point by a point source in the origin
with all possible orientations.This concept will be better clari¯ed in Section 3.2.
In coordinate-free language
E(r) = ¡j!¹
0
Z
G
EJ
(r ¡r
0
) ¢ J(r
0
)dr
0
¡
Z
G
EM
(r ¡r
0
) ¢ M(r
0
)dr
0
H(r) =
Z
G
HJ
(r ¡r
0
) ¢ J(r
0
)dr
0
¡j!"
0
Z
G
HM
(r ¡r
0
) ¢ M(r
0
)dr
0
(3.7)
To check the dimensions of the various Green's functions,it is useful to note that

0
= k
0
Z
0
!"
0
= k
0
Y
0
= k
0
=Z
0
Hence we recognize that G
EJ
and G
HM
are measured in m
¡1
,G
EM
and G
HJ
in m
¡2
.The
explicit expressions of the various dyadic Green's functions can be obtained by applying the Fourier
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 33
transform technique to (3.1).The most appropriate coordinate system that can be used to show
the result is the spherical one,because the source is a point.It can be shown that the matrices
representing the Green's functions are:
G
EJ
(r;#;') = G
HM
(r;#;') $
0
@
A 0 0
0 B 0
0 0 B
1
A
exp(¡jk
0
r)
4¼r
G
EM
(r;#;') = G
HJ
(r;#;') $¡jk
0
0
@
0 0 0
0 0 ¡C
0 C 0
1
A
exp(¡jk
0
r)
4¼r
(3.8)
where the wavenumber is k
0
=!
p
"
0
¹
0
.Note that the row and column indices are ^r,
^
#,^',
respectively.In dyadic form
G
EJ
(r;#;') = G
HM
(r;#;') =
h
A^r^r +B
^
#
^
#+B^'^'
i
exp(¡jk
0
r)
4¼r
G
EM
(r;#;') = G
HJ
(r;#;') = ¡jk
0
C
h
^'
^

^
#^'
i
exp(¡jk
0
r)
4¼r
where
A = 2
µ
j
1
k
0
r
+
1
(k
0
r)
2

B = 1 ¡
1
2
A = 1 ¡j
1
k
0
r
¡
1
(k
0
r)
2
C = 1 ¡j
1
k
0
r
Consistently with the fact that the source is a point,the Green's function does not depend on the
angular variables.
The behavior of the three coe±cients at large distance from the source,k
0
r!1 (i.e.r À¸)
is
A = O
µ
1
k
0
r

B!1
C!1
In this region,usually called far ¯eld region,the expressions of the Green's functions simplify and
become
G
EJ
(r;#;') = G
HM
(r;#;') » I
tr
exp(¡jk
0
r)
4¼r
G
EM
(r;#;') = G
HJ
(r;#;') » ¡jk
0
^r £I
tr
exp(¡jk
0
r)
4¼r
(3.9)
where I
tr
is the transverse to r identity dyadic,see Appendix.This operator,when applied to an
arbitrary vector,produces as a result the projection of the vector on the plane perpendicular to r.
The operator ^r £I
tr
adds a further 90
±
counterclockwise turn around r.
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 34
Conversely,in the near ¯eld region,k
0
r!0 (i.e.r ¿¸),the coe±cients become
A »
2
(k
0
r)
2
B » ¡
1
(k
0
r)
2
C » ¡j
1
k
0
r
3.2 Elementary dipoles
As discussed previously,the Green's function is the basic tool for the computation of the ¯eld
radiated by any source by means of eq.(3.7).However,it is convenient to start with the simplest
one,i.e.a point source,and this will help in understanding the properties of the Green's functions.
Consider ¯rst an elementary source of electric type located at the origin of the coordinate system,
modeled by the current distribution
J(r) = M
e
±(r)
The vector M
e
is called the electric dipole moment of the current distribution and is measured
in Am (recall that the dimensions of the three dimensional ± function are m
¡3
).An arbitrary
current distribution can be characterized by its moments.This concept is used also in the theory
of probability:if » is a random variable with density function W
»
(x),moments of all orders can
be de¯ned by
m
n
= Ef»
n
g =
Z
1
¡1
x
n
W
»
(x)dx
where Ef g denotes the expectation value.In the case of the current distribution,the role of
W
»
(x) is played by J(r),but the situation is more complicated because its vector nature implies
that the moments beyond the ¯rst are tensors.The ¯rst moment (dipole moment) is a vector,
de¯ned by
M
e
=
Z
J(r)dr (3.10)
In the case of the point source introduced above,thanks to the properties of the delta function,the
previous equation becomes an identity and we understand the reason for the name of the coe±cient.
From a practical point of view,we can imagine to obtain this source by a limiting process,starting
from a rectilinear current I,whose length l is progressively reduced without changing the aspect
ratio (diameter/length of the wire),while,at the same time,the current is increased,so that the
value of the integral (the dipole moment Il) remains constant.
Introducing the dipole current into (3.7),we ¯nd that the radiated ¯elds are given by
E(r) = ¡j!¹
0
Z
G
EJ
(r ¡r
0
) ¢ M
e
±(r
0
)dr
0
= ¡j!¹
0
G
EJ
(r) ¢ M
e
H(r) =
Z
G
HJ
(r ¡r
0
) ¢ M
e
±(r
0
)dr
0
= G
HJ
(r) ¢ M
e
(3.11)
Since we know the expressions of the matrices representing the Green's functions in the spherical
basis,it is necessary to express the vector M
e
in the same basis.Let us assume that the polar
axis of the coordinate system is in the direction of M
e
,i.e.assume M
e
= M
e
^z.Note that
this step is allowed because the Green's function does not depend on the angular variables,as a
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 35
consequence of the isotropy of free space.It is to be remarked,as a general rule,that the Green's
function depends only on the structure and,hence,shares its symmetries.This choice guarantees
the simplest description of the radiated ¯eld.Since the radiated ¯eld must have the direction of
M
e
as a symmetry axis,orienting the polar axis of the coordinate system in this direction allows
the expressions to be independent on'.
M
e
= (M
e
¢ ^r)^r +(M
e
¢
^
#)
^
#+(M
e
¢ ^') ^'
= M
e
(^z ¢ ^r)^r +M
e
(^z ¢
^
#)
^
#+M
e
(^z ¢ ^') ^'
= M
e
cos#^r ¡M
e
sin#
^
#
(3.12)
where we have exploited (A.12).Now recalling the expression of the Green's function (3.8),we
obtain
E(r) = ¡j!¹
0
exp(¡jk
0
r)
4¼r
0
@
A 0 0
0 B 0
0 0 B
1
A
0
@
cos#
¡sin#
0
1
A
M
e
= ¡j
Z
0
M
e
exp(¡jk
0
r)
2r¸
³
Acos#^r ¡Bsin#
^
#
´
(3.13)
where use has been made of

0
= k
0
Z
0
=

¸
Z
0
and Z
0
is the wave impedance.Concerning the meaning of (3.12),it is to be remarked that the
matrix (3.8) represents the Green's function in the spherical basis consisting of the unit vectors
^r,
^
#,^'de¯ned in the observation point r.Hence,even if the source is located in the origin,its
components are evaluated in the basis associated to the point r.
We can proceed similarly for the magnetic ¯eld:
H(r) = ¡jk
0
exp(¡jk
0
r)
4¼r
0
@
0 0 0
0 0 ¡C
0 C 0
1
A
0
@
cos#
¡sin#
0
1
A
= j
M
e
exp(¡jk
0
r)
2r¸
Csin#^'
(3.14)
In conclusion,the electromagnetic ¯eld radiated by an electric dipole has the following expression
E(r) = ¡j
Z
0
M
e
exp(¡jk
0
r)
2r¸
·
2
µ
j
1
k
0
r
+
1
(k
0
r)
2

cos#^r ¡
µ
1 ¡j
1
k
0
r
¡
1
(k
0
r)
2

sin#
^
#
¸
H(r) = j
M
e
exp(¡jk
0
r)
2r¸
µ
1 ¡j
1
k
0
r

sin#^'
(3.15)
This wave has two components of electric ¯eld and only one of magnetic ¯eld.Imagine a geo-
graphical system of coordinates such that the direction of the dipole moment de¯nes the direction
of the earth axis.The angle#is the colatitude (= 90
±
¡latitude),the angle'is the longitude.
Then the electric ¯eld is contained in the meridian planes and the magnetic ¯eld is tangent to the
parallels.This type of wave is called TM (Transverse Magnetic) since the magnetic ¯eld has no
radial component.We recognize also that the radial component of the electric ¯eld is dominant
close to the source,but negligible with respect to the others at large distance.Here the wave is
essentially TEM,since neither ¯eld has a (signi¯cant) radial component.
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 36
Let us compute the energy budget by means of the Poynting theorem.The Poynting vector is
S = E£H
¤
=
Z
0
M
2
e
4r
2
¸
2
h
BC
¤
sin
2
#^r +AC
¤
sin#cos#
^
#
i
Compute the components
BC
¤
=
·
1 ¡j
1
k
0
r
¡
1
(k
0
r)
2
¸·
1 +j
1
k
0
r
¸
= 1 ¡j
1
k
0
r
¡
1
(k
0
r)
2
+j
1
k
0
r
+
1
(k
0
r)
2
¡j
1
(k
0
r)
3
= 1 ¡j
1
(k
0
r)
3
AC
¤
=
·
j
2
k
0
r
+
2
(k
0
r)
2
¸·
1 +j
1
k
0
r
¸
= j
2
k
0
r
+
2
(k
0
r)
2
¡
2
(k
0
r)
2
+j
2
(k
0
r)
3
= j
·
2
k
0
r
+
2
(k
0
r)
3
¸
Substituting in the previous equation we get
S =
Z
0
M
2
e
4r
2
¸
2
·µ
1 ¡j
1
(k
0
r)
3

sin
2
#^r +j
µ
2
k
0
r
+
2
(k
0
r)
3

sin#cos#
^
#
¸
According to Poynting theorem,the surface density of active power °ow is
dP

=
1
2
RfS ¢ ^ºg (3.16)
where ^º is the normal to the surface element.In order to compute the total radiated active power,
we have to evaluate the °ux of the Poynting vector across a closed surface surrounding the source.
For maximum simplicity we choose a sphere of radius r:
P
rad
=
1
2
R
Z
S ¢ ^º d§ =
1
2
Z

0
Z
¼
0
Z
0
M
2
e
4r
2
¸
2
sin
2
#r
2
sin#dµd'
=
1
2
Z
0
M
2
e

2

Z
¼
0
sin
3
#dµ
=
1
2
Z
0
M
2
e


2
(3.17)
Here we have used the following facts
²
the normal to the spherical surface is ^º =^r
²
the area element in spherical coordinates is d§ = r
2
sin#dµd'
²
the integrand does not depend on',so the'integration yields the 2¼ factor
²
the#integration yields
Z
¼
0
sin
3
#dµ =
4
3
The factor 1/2 has been left explicit to make it clear that M
e
is a peak value,that is the time
domain dipole moment is M
e
(t) = M
e
cos(!
0
t).If,on the contrary M
e
is an e®ective value,the
factor 1/2 has to be dropped.
Renato Orta - Electromagnetic Field Theory (Nov.2012) PRELIMINARY VERSION 37
We notice that the total radiated power does not depend on the radius of the sphere chosen
to compute it.Algebraically this is the result of the cancellation between the r
2
factor in the
denominator of the Poynting vector and the one in the area element d§.To get a more physical
explanation,consider the °uxes through two concentric spheres of di®erent radii:if they were
di®erent,power would be lost or generated in the shell,which is impossible by conservation of
energy in a lossless medium.
In (3.17) we took the real part of the integral.The imaginary part is the reactive power
Q =
1
2
I
Z
S ¢ ^º d§ = ¡
1
2
Z

0
Z
¼
0
Z
0
M
2
e
4r
2
¸
2
1
(k
0
r)
3
sin
2
#r
2
sin#dµd'
= ¡
1
2
Z
0
M
2
e
4r
2
¸
2
1
(k
0
r)
3

Z
¼
0
sin
3
#dµ
= ¡
1
2
Z
0
M
2
e


2
1
(k
0
r)
3
It is reasonable that Q depends on r:the reactive power is the energy that twice per period is
exchanged between generator and load,hence crosses the spherical surface of radius r.Moreover,
it may be remarked that in the#direction,the structure is closed as a (virtual) cavity.This
explains why the#component of the Poynting vector is pure imaginary.Indeed,a real part of S
#
would imply a steady energy °ow in that direction;however,this is impossible,since the angular
domain is ¯nite (0 ·#· ¼=2) and the dielectric is lossless.
It is useful to explicitly indicate the dominant components close to the source and far from it.
In the far ¯eld region,r À¸
E(r) » j
Z
0
M
e
exp(¡jk
0
r)
2r¸
sin#
^
#
H(r) » j
M
e
exp(¡jk
0
r)
2r¸
sin#^'
(3.18)
We see that the ¯elds tend to be linearly polarized,orthogonal and proportional to each other and
also orthogonal to the radial direction.These properties are summarized in the impedance relation