Theory of Electromagnetic Fields

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THE CERN ACCELERATOR SCHOOL
Theory of Electromagnetic Fields
Part I:Maxwell's Equations
Andy Wolski
The Cockcroft Institute,and the University of Liverpool,UK
CAS Specialised Course on RF for Accelerators
Ebeltoft,Denmark,June 2010
Theory of Electromagnetic Fields
In these lectures,we shall discuss the theory of electromagnetic
elds,with an emphasis on aspects relevant to RF systems in
accelerators:
1.Maxwell's equations
 Maxwell's equations and their physical signicance
 Electromagnetic potentials
 Electromagnetic waves and their generation
 Electromagnetic energy
2.Standing Waves
 Boundary conditions on electromagnetic elds
 Modes in rectangular and cylindrical cavities
 Energy stored in a cavity
3.Travelling Waves
 Rectangular waveguides
 Transmission lines
Theory of EM Fields 1 Part I:Maxwell's Equations
Theory of Electromagnetic Fields
I shall assume some familiarity with the following topics:
 vector calculus in Cartesian and polar coordinate systems;
 Stokes'and Gauss'theorems;
 Maxwell's equations and their physical signicance;
 types of cavities and waveguides commonly used in
accelerators.
The fundamental physics and mathematics is presented in
many textbooks;for example:
I.S.Grant and W.R.Phillips,\Electromagnetism,"
2nd Edition (1990),Wiley.
Theory of EM Fields 2 Part I:Maxwell's Equations
Summary of relations in vector calculus
In cartesian coordinates:
gradf  rf 

@f
@x
;
@f
@y
;
@f
@z
!
(1)
div
~
A  r
~
A 
@A
x
@x
+
@A
y
@y
+
@A
z
@z
(2)
curl
~
A  r
~
A 








^x ^y ^z
@
@x
@
@y
@
@z
A
x
A
y
A
z








(3)
r
2
f 
@
2
f
@x
2
+
@
2
f
@y
2
+
@
2
f
@z
2
(4)
Note that ^x,^y and ^z are unit vectors parallel to the x,y and z
axes,respectively.
Theory of EM Fields 3 Part I:Maxwell's Equations
Summary of relations in vector calculus
Gauss'theorem:
Z
V
r
~
AdV =
I
S
~
A d
~
S;(5)
for any smooth vector eld
~
A,where the closed surface S
bounds the volume V.
Stokes'theorem:
Z
S
r
~
A d
~
S =
I
C
~
A d
~
`;(6)
for any smooth vector eld
~
A,where the closed loop C bounds
the surface S.
A useful identity:
rr
~
A  r(r
~
A) r
2
~
A:(7)
Theory of EM Fields 4 Part I:Maxwell's Equations
Maxwell's equations
r
~
D= r
~
B=0
r
~
E =
@
~
B
@t
r
~
H=
~
J +
@
~
D
@t
James Clerk Maxwell
1831 { 1879
Note that  is the electric charge density;and
~
J is the current
density.
The constitutive relations are:
~
D ="
~
E;
~
B =
~
H;(8)
where"is the permittivity,and  is the permeability of the
material in which the elds exist.
Theory of EM Fields 5 Part I:Maxwell's Equations
Physical interpretation of r
~
B =0
Gauss'theorem tells us that for any smooth vector eld
~
B:
Z
V
r
~
BdV =
I
S
~
B  d
~
S;(9)
where the closed surface S bounds the region V.
Applied to Maxwell's equation r
~
B =0,Gauss'theorem tells
us that the total ux entering a bounded region equals the
total ux leaving the same region.
Theory of EM Fields 6 Part I:Maxwell's Equations
Physical interpretation of r
~
D =
Applying Gauss'theorem to Maxwell's equation r
~
D =,we
nd that:
Z
V
r
~
DdV =
I
S
~
D d
~
S =Q;(10)
where Q =
R
V
dV is the total charge within the region V,
bounded by the closed surface S.
The total ux of electric displacement crossing a closed surface
equals the total electric charge enclosed by that surface.
In particular,at a distance r from the
centre of any spherically symmetric
charge distribution,the electric
displacement is:
~
D =
Q
4r
2
^r;(11)
where Q is the total charge within
radius r,and ^r is a unit vector in the
radial direction.
Theory of EM Fields 7 Part I:Maxwell's Equations
Physical interpretation of r
~
H =
~
J +
@
~
D
@t
Stokes'theorem tells us that for any smooth vector eld
~
H:
Z
S
r
~
H  d
~
S =
I
C
~
H  d
~
`;(12)
where the closed loop C bounds the surface S.
Applied to Maxwell's equation
r
~
H =
~
J +
@
~
D
@t
,Stokes'
theorem tells us that the
magnetic eld
~
H integrated
around a closed loop equals the
total current passing through
that loop.For the static case
(constant currents and elds):
I
C
~
H  d
~
`=
Z
S
~
J  d
~
S =I:(13)
Theory of EM Fields 8 Part I:Maxwell's Equations
The displacement current and charge conservation
The term
@
~
D
@t
in Maxwell's equation r
~
H =
~
J +
@
~
D
@t
is known
as the displacement current density,and has an important
physical consequence.
Since,for any smooth vector eld
~
H:
r r
~
H  0;(14)
it follows that:
r
~
J +r
@
~
D
@t
=r
~
J +
@
@t
=0:(15)
This is the continuity equation,that expresses the local
conservation of electric charge.The signicance is perhaps
clearer if we use Gauss'theorem to express the equation in
integral form:
I
S
~
J  d
~
S =
dQ
dt
;(16)
where Q is the total charge enclosed by the surface S.
Theory of EM Fields 9 Part I:Maxwell's Equations
Physical interpretation of r
~
E =
@
~
B
@t
Applied to Maxwell's equation r
~
E =
@
~
B
@t
,Stokes'theorem
tells us that a time-dependent magnetic eld generates an
electric eld.
In particular,the total electric eld around a closed loop equals
the rate of change of the total magnetic ux through that loop:
Z
S
r
~
E  d
~
S =
I
C
~
E  d
~
`=
@
@t
Z
S
~
B  d
~
S:(17)
This is Faraday's law of electromagnetic
induction:
E =
@
@t
;(18)
where E is the electromotive force (the
integral of the electric eld) around a
closed loop,and  is the total magnetic
ux through that loop.
Theory of EM Fields 10 Part I:Maxwell's Equations
Solving Maxwell's equations
Maxwell's equations are of fundamental importance in
electromagnetism,because they tell us the elds that exist in
the presence of various charges and materials.
In accelerator physics (and many other branches of applied
physics),there are two basic problems:
 Find the electric and magnetic elds in a system of charges
and materials of specied size,shape and electromagnetic
characteristics.
 Find a system of charges and materials to generate electric
and magnetic elds with specied properties.
Theory of EM Fields 11 Part I:Maxwell's Equations
Example:elds induced by a bunch in an accelerator
Theory of EM Fields 12 Part I:Maxwell's Equations
Linearity and superposition
Neither problem is particularly easy to solve in general;but
fortunately,there are ways to decompose complex problems
into simpler ones...
Maxwell's equations are linear:
r

~
B
1
+
~
B
2

=r
~
B
1
+r
~
B
2
;(19)
and:
r

~
H
1
+
~
H
2

=r
~
H
1
+r
~
H
2
:(20)
This means that if two elds
~
B
1
and
~
B
2
satisfy Maxwell's
equations,so does their sum
~
B
1
+
~
B
2
.
As a result,we can apply the principle of superposition to
construct complicated electric and magnetic elds just by
adding together sets of simpler elds.
Theory of EM Fields 13 Part I:Maxwell's Equations
Example:plane electromagnetic waves in free space
Perhaps the simplest system is one in which there are no
charges or materials at all:a perfect,unbounded vacuum.
Then,the constitutive relations are:
~
D ="
0
~
E;and
~
B =
0
~
H;(21)
and Maxwell's equations take the form:
r
~
E =0 r
~
B=0
r
~
E =
@
~
B
@t
r
~
B=
1
c
2
@
~
E
@t
where 1=c
2
=
0
"
0
.
There is a trivial solution,in which all the elds are zero.But
there are also interesting non-trivial solutions,where the elds
are not zero.To nd such a solution,we rst\separate"the
electric and magnetic elds.
Theory of EM Fields 14 Part I:Maxwell's Equations
Example:plane electromagnetic waves in free space
If we take the curl of the equation for r
~
E we obtain:
rr
~
E  r(r
~
E) r
2
~
E =
@
@t
r
~
B:(22)
Then,using r
~
E =0,and r
~
B =
1
c
2
@
~
E
@t
,we nd:
r
2
~
E 
1
c
2
@
2
~
E
@t
2
=0:(23)
This is the equation for a plane wave,which is solved by:
~
E =
~
E
0
e
i(
~
k~r!t)
;(24)
where
~
E
0
is a constant vector,and the phase velocity c of the
wave is given by the dispersion relation:
c =
!
jkj
=
1
p

0
"
0
:(25)
Theory of EM Fields 15 Part I:Maxwell's Equations
Example:plane electromagnetic waves in free space
Similarly,if we take the curl of the equation for r
~
B we
obtain:
r
2
~
B 
1
c
2
@
2
~
B
@t
2
=0:(26)
This is again the equation for a plane wave,which is solved by:
~
B =
~
B
0
e
i(
~
k~r!t)
;(27)
where
~
B
0
is a constant vector,and the phase velocity c of the
wave is again given by the dispersion relation (25):
c =
!
jkj
=
1
p

0
"
0
:
Theory of EM Fields 16 Part I:Maxwell's Equations
Example:plane electromagnetic waves in free space
Although it appears that we obtained independent equations
for the electric and magnetic elds,we did so by taking
derivatives.Therefore,the original Maxwell's equations impose
constraints on the solutions.
For example,substituting the solutions (24) and (27) into
Maxwell's equation:
r
~
E =
@
~
B
@t
;(28)
we nd:
~
k 
~
E
0
=!
~
B
0
:(29)
This imposes a constraint on both the directions and the
relative magnitudes of the electric and magnetic elds.
Theory of EM Fields 17 Part I:Maxwell's Equations
Example:plane electromagnetic waves in free space
Similarly we nd:
r
~
B =
1
c
2
@
~
E
@t
)
~
k 
~
B
0
=
!
c
2
~
E
0
(30)
r
~
E =0 )
~
k 
~
E
0
=0 (31)
r
~
B =0 )
~
k 
~
B
0
=0 (32)
These equations impose constraints on the relative amplitudes
and directions of the electric and magnetic elds in the waves:

~
E
0
,
~
B
0
,and
~
k are mutually perpendicular;
 The eld amplitudes are related by
E
0
B
0
=c.
Theory of EM Fields 18 Part I:Maxwell's Equations
Example:plane electromagnetic waves in free space
Theory of EM Fields 19 Part I:Maxwell's Equations
Example:plane electromagnetic waves in free space
Note that the wave vector
~
k can be chosen freely.We can refer
to a wave specied by a particular value of
~
k as a\mode"of
the electromagnetic elds in free space.
The frequency of each mode is determined by the dispersion
relation (25):
c =
!
jkj
=
1
p

0
"
0
:
A single mode represents a plane wave of a single frequency,
with innite extent in space and time.More realistic waves can
be obtained by summing together (superposing) dierent
modes.
Theory of EM Fields 20 Part I:Maxwell's Equations
Electromagnetic potentials
Sometimes,problems can be simplied by working with the
electromagnetic potentials,rather than the elds.
The potentials  and
~
A are dened as functions of space and
time,whose derivatives give the elds:
~
B = r
~
A;(33)
~
E = r 
@
~
A
@t
:(34)
Note that because the elds are obtained by taking derivatives
of the potentials,there is more than one set of potential
functions that will produce the same elds.This feature is
known as gauge invariance.
Theory of EM Fields 21 Part I:Maxwell's Equations
Electromagnetic potentials:the Lorenz gauge
To dene the potentials uniquely,we need to specify not just
the elds,but also an additional condition { known as a gauge
condition { on the potentials.
For time-dependent elds (and potentials),the conventional
choice of gauge is the Lorenz gauge:
r
~
A+
1
c
2
@
@t
=0;(35)
where c is the speed of light.
The Lorenz gauge is convenient because it allows us to write
wave equations for the potentials in the presence of sources,in
a convenient form.
Theory of EM Fields 22 Part I:Maxwell's Equations
Wave equations for the potentials in the Lorenz gauge
If we take Maxwell's equation r
~
D =,and substitute for the
electric eld in terms of the potentials (34),we nd:
r
~
E =r
2
 
@
@t
r
~
A =

"
0
:(36)
Then,using the Lorenz gauge (35),we nd:
r
2
 
1
c
2
@
2

@t
2
=

"
0
:(37)
The Lorenz gauge allows us to write a wave equation for the
scalar potential ,with a source term given by the charge
density ;and without the appearance of either the vector
potential
~
A or the current density
~
J.
Theory of EM Fields 23 Part I:Maxwell's Equations
Wave equations for the potentials in the Lorenz gauge
We can nd a similar wave equation for the vector potential,
~
A.
Substituting
~
B =r
~
A into Maxwell's equations,we obtain:
r
~
B =rr
~
A  r(r
~
A) r
2
~
A =
0
~
J +
1
c
2
@
~
E
@t
:(38)
Then,using the Lorenz gauge (35),and substituting for the
electric eld
~
E =r 
@
~
A
@t
,we nd:
r
2
~
A
1
c
2
@
2
~
A
@t
2
=
0
~
J:(39)
Using the Lorenz gauge allows us to write a wave equation for
the vector potential,with a source term given by the current
density
~
J,and without the appearance of either the charge
density ,or the scalar potential .
Theory of EM Fields 24 Part I:Maxwell's Equations
Wave equations for the potentials in the Lorenz gauge
The wave equations for the potentials are useful,because they
allow us to calculate the elds around time-dependent charge
and current distributions.
The general solutions to the wave
equations can be written:
(~r;t) =
1
4"
0
Z
(~r
0
;t
0
)
j~r ~r
0
j
dV
0
;(40)
~
A(~r;t) =

0
4
Z
~
J(~r
0
;t
0
)
j~r ~r
0
j
dV
0
;(41)
where:
t
0
=t 
j~r ~r
0
j
c
:(42)
Theory of EM Fields 25 Part I:Maxwell's Equations
Generating electromagnetic waves:the Hertzian dipole
It is easy to show that,in the static case,the expression for the
scalar potential gives the result expected from Coulomb's law.
A more interesting exercise is to calculate the elds around an
innitesimal oscillating dipole (a Hertzian dipole).We can
model the current associated with a Hertzian dipole oriented
parallel to the z axis as:
~
I =I
0
e
i!t
^z:(43)
We can think of the current as being associated with a charge
oscillating between two points either side of the origin,along
the z axis.
Theory of EM Fields 26 Part I:Maxwell's Equations
Generating electromagnetic waves:the Hertzian dipole
Since the current is located only at
the origin,it is straightforward to
perform the integral (41) to nd
the vector potential:
~
A(~r;t) =

0
4
(I
0
`)
e
i(kr!t)
r
^z;(44)
where:
k =
!
c
:(45)
Note that`is the length of the
dipole:strictly speaking,we take
the limit`!0,but with the
amplitude I
0
`remaining constant.
Theory of EM Fields 27 Part I:Maxwell's Equations
Generating electromagnetic waves:the Hertzian dipole
Having obtained the vector potential,we can nd the magnetic
eld from
~
B =r
~
A.For the curl in spherical polar
coordinates,see Appendix A.The result is:
B
r
= 0;(46)
B

= 0;(47)
B

=

0
4
(I
0
`)k sin

1
kr
i

e
i(kr!t)
r
:(48)
The electric eld can be obtained from r
~
B =
1
c
2
@
~
E
@t
.
The result is:
E
r
=
1
4"
0
2
c
(I
0
`)

1 +
i
kr

e
i(kr!t)
r
2
;(49)
E

=
1
4"
0
(I
0
`)
k
c
sin

i
k
2
r
2
+
1
kr
i

e
i(kr!t)
r
;(50)
E

= 0:(51)
Theory of EM Fields 28 Part I:Maxwell's Equations
Generating electromagnetic waves:the Hertzian dipole
At distances from the dipole large compared with the
wavelength,kr 1,and we can nd approximate expressions
for the dominant eld components:
E

 i
1
4"
0
(I
0
`)
k
c
sin
e
i(kr!t)
r
;(52)
B

 i

0
4
(I
0
`)k sin
e
i(kr!t)
r
:(53)
This is known as the\far eld regime".Note that the elds
take the form of a wave propagating in the radial direction:the
electric and magnetic elds are perpendicular to each other,
and to the direction of the wave (as we found for the case of
the plane wave).
Theory of EM Fields 29 Part I:Maxwell's Equations
Generating electromagnetic waves:the Hertzian dipole
The relative amplitudes of the electric and magnetic elds are
also as we found for a plane wave.
Note that the eld amplitudes fall o as 1=r;and that there is a
directional dependence on sin,so that the amplitudes are zero
in the direction of the current ( =0

,and  =180

),and are
maximum in the plane perpendicular to the current ( =90

).
Theory of EM Fields 30 Part I:Maxwell's Equations
Fields around a Hertzian dipole
http://www.amanogawa.com
Theory of EM Fields 31 Part I:Maxwell's Equations
Fields around a Hertzian dipole
http://www.falstad.com
Theory of EM Fields 32 Part I:Maxwell's Equations
The Scream (Edvard Munch,1893)
Theory of EM Fields 33 Part I:Maxwell's Equations
Waves in conducting media
There are some important dierences between the behaviour of
electromagnetic waves in free space,and the behaviour of
electromagnetic waves in conductors.
One signicant dierence is that the electric eld in the wave
drives a ow of electric current in the conductor:this leads to
ohmic energy losses,and results in attentuation of the wave.
A key concept is the skin depth:this is the distance over which
the amplitude of the wave falls by a factor 1=e.
We shall derive an expression for the skin depth in terms of the
properties of the conductor.
Theory of EM Fields 34 Part I:Maxwell's Equations
Waves in conducting media
To start with,we dene an ohmic conductor as a material in
which the current density is proportional to the electric eld:
~
J =
~
E:(54)
The constant  is the conductivity of the material.
In practice, depends on many factors,including (in the case
of an oscillating electric eld) on the frequency of oscillation of
the eld.However,we shall regard  as a constant.
Theory of EM Fields 35 Part I:Maxwell's Equations
Waves in conducting media
In an ohmic conductor with absolute permittivity",absolute
permeability ,and conductivity ,Maxwell's equations take
the form:
r
~
E =0 r
~
B=0
r
~
E =
@
~
B
@t
r
~
B=
~
E +
1
v
2
@
~
E
@t
where 1=v
2
=".
We can derive wave equations for the electric and magnetic
elds as before;but with the additional term in ,the wave
equation for the electric eld takes the form:
r
2
~
E 
@
~
E
@t

1
v
2
@
2
~
E
@t
2
=0:(55)
The rst order derivative with respect to time in equation (55)
describes the attenuation of the wave.
Theory of EM Fields 36 Part I:Maxwell's Equations
Waves in conducting media
We can write a solution to the wave equation (55) in the usual
form:
~
E(~r;t) =
~
E
0
e
i(
~
k~r!t)
:(56)
But now,if we substitute this into equation (55),we nd that
the dispersion relation is:

~
k
2
+i! +
!
2
v
2
=0:(57)
In general,the wave vector
~
k will be complex.We can write:
~
k =~ +i
~
;(58)
where ~ and
~
 are real vectors,that we shall assume are
parallel.
Theory of EM Fields 37 Part I:Maxwell's Equations
Waves in conducting media
In terms of the real vectors  and ,the electric eld (56) can
be written:
~
E(~r;t) =
~
E
0
e

~
~r
e
i(~~r!t)
:(59)
The amplitude of the wave falls by a factor 1=e in a distance
 =1=. is known as the skin depth.
The dispersion relation (57) can be solved to nd the
magnitudes of the vectors ~ and
~
.The algebra is left as an
exercise!The result is:
 =
!
v
0
@
1
2
+
1
2
s
1 +

2
!
2
"
2
1
A
1
2
;(60)
and:
 =
!
2
:(61)
Theory of EM Fields 38 Part I:Maxwell's Equations
Waves in conducting media
Theory of EM Fields 39 Part I:Maxwell's Equations
Waves in conducting media
Equations (60) and (61) are exact expressions for the real and
imaginary parts of the wave vector for an electromagnetic wave
in an ohmic conductor.
In the case that  !"(a good conductor),we can make the
approximations:
 
!
v
r

2!"
;(62)
and (using v =1=
p
"):
 =
1


s
2
!
:(63)
Note that the skin depth is smaller for larger conductivity:the
better the conductivity of a material,the less well an
electromagnetic wave can penetrate the material.This has
important consequences for RF components in accelerators,as
we shall see in the next lecture.
Theory of EM Fields 40 Part I:Maxwell's Equations
Energy in Electromagnetic Fields:Poynting's Theorem
Electromagnetic waves carry energy.
The energy density in an electric eld is given by:
U
E
=
1
2
"
~
E
2
(64)
The energy density in a magnetic eld is given by:
U
H
=
1
2

~
H
2
(65)
The energy ux (energy crossing unit area per unit time) is
given by the Poynting vector:
~
S =
~
E 
~
H (66)
These results follow from Poynting's theorem...
Theory of EM Fields 41 Part I:Maxwell's Equations
Energy in electromagnetic elds:Poynting's theorem
To derive Poynting's theorem,we start with Maxwell's
equations.First,we use:
r
~
E =
@
~
B
@t
(67)
Take the scalar product on both sides with the magnetic
intensity
~
H:
~
H  r
~
E =
~
H 
@
~
B
@t
(68)
Next,we use:
r
~
H =
~
J +
@
~
D
@t
(69)
Take the scalar product on both sides with the electric eld
~
E:
~
E  r
~
H =
~
E 
~
J +
~
E 
@
~
D
@t
(70)
Theory of EM Fields 42 Part I:Maxwell's Equations
Energy in electromagnetic elds:Poynting's theorem
Now we take equation (68) minus equation (70):
~
H  r
~
E 
~
E  r
~
H =
~
E 
~
J 
~
E 
@
~
D
@t

~
H 
@
~
B
@t
(71)
which can be written as:
@
@t

1
2
"
~
E
2
+
1
2

~
H
2

=r

~
E 
~
H


~
E 
~
J (72)
Equation (72) is Poynting's theorem.Using Gauss'theorem,it
may be written in integral form:
@
@t
Z
V
(
U
E
+U
H
)
dV =
I
A
~
S  d
~
A
Z
V
~
E 
~
J dV (73)
where the closed surface A bounds the volume V,
U
E
=
1
2
"
~
E
2
U
H
=
1
2

~
H
2
(74)
and:
~
S =
~
E 
~
H (75)
Theory of EM Fields 43 Part I:Maxwell's Equations
Energy in electromagnetic elds:Poynting's theorem
Poynting's theorem in integral form is equation (73):
@
@t
Z
V
(U
E
+U
H
) dV =
I
A
~
S  d
~
A
Z
V
~
E 
~
J dV
We note that the last term on the right hand side represents
the rate at which the electric eld does work on electric
charges within the bounded volume V.It is then natural to
interpret the rst term on the right hand side as the ow of
energy in the electromagnetic eld across the boundary of the
volume V,and the left hand side as the rate of change of the
total energy in the electromagnetic eld.
With this interpretation,Poynting's theorem expresses the local
conservation of energy.
Theory of EM Fields 44 Part I:Maxwell's Equations
Energy in an electromagnetic wave
As an example,let us calculate the average energy density and
the energy ux in a plane electromagnetic wave in free space.
From equations (24) and (27),the (real) electric and magnetic
elds are given by:
~
E =
~
E
0
cos(
~
k  ~r !t);(76)
~
H =
~
H
0
cos(
~
k  ~r !t);(77)
where:
E
0
B
0
=c =
1
p

0
"
0
;)
E
0
H
0
=Z
0
=
s

0
"
0
:(78)
Note (in passing) that Z
0
is the impedance of free space.Impedance will
play an important role when we come to consider waves on boundaries.
Theory of EM Fields 45 Part I:Maxwell's Equations
Energy in an electromagnetic wave
The energy densities in the electric and magnetic elds are:
U
E
=
1
2
"
0
~
E
2
;U
H
=
1
2

0
~
H
2
:(79)
But since H
0
=E
0
=Z
0
,we nd:
U
H
=
1
4
"
0
~
E
2
=U
H
:(80)
In other words,the energy density in the magnetic eld is equal
to the energy density in the electric eld (for a plane
electromagnetic wave in free space).
Theory of EM Fields 46 Part I:Maxwell's Equations
Energy ux
The Poynting vector (which gives the energy ow per unit area
per unit time)
~
S is dened by:
~
S =
~
E 
~
H (81)
Since
~
E
0
and
~
H
0
are perpendicular to each other and to
~
k (the
direction in which the wave is travelling),and the amplitudes of
the elds are related by the impedance Z
0
,we nd that:
~
S =
E
2
0
Z
0
^
k cos
2
(
~
k  ~x !t):(82)
We see that (as expected) the energy ow is in the direction of
the wave vector.
The amount of energy carried by the wave depends on the
square of the electric eld amplitude,divided by the impedance.
Theory of EM Fields 47 Part I:Maxwell's Equations
Summary
 Maxwell's equations describe the constraints on physical
electric and magnetic elds.
 In free space,electromagnetic waves can propogate as
transverse plane waves.For such waves,the wave vector
~
k
denes the\mode"of the electromagnetic elds.
 In conductors,electromagnetic waves are attenuated
because the energy is dissipated by currents driven by the
electric eld in the waves.
 Electromagnetic waves can be generated by oscillating
electric charges.
 Poynting's theorem provides expressions for the energy
density and energy ux in an electromagnetic eld.
Theory of EM Fields 48 Part I:Maxwell's Equations
Appendix A:curl in spherical polar coordinates
In spherical polar coordinates,the curl of a vector eld is given
by:
r
~
A 
1
r
2
sin








^r r
^
 r sin
^

@
@r
@
@
@
@
A
r
rA

r sinA









(83)
Theory of EM Fields 49 Part I:Maxwell's Equations
Appendix B:Exercises for the student
1.Derive the wave equation for the electric eld in a
conductor.Show that the real and imaginary parts of the
wave vector have magnitudes given by equations (60) and
(61).
2.Estimate the skin depth for microwaves in copper.
3.Find an expression for the total power radiated by a
Hertzian dipole.
Theory of EM Fields 50 Part I:Maxwell's Equations