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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: ISTE Invoiced
- Sprache: Englisch
Introduction to Classical Electrodynamics 3 presents the applications of electromagnetic theory in modern physics.
This book covers waveguides, resonant cavities and transmission lines. It also analyzes electromagnetic radiation from localized and extended oscillating sources, as well as radiation from (non-)relativistic point charges. The book examines particles in circular motion, Thomson scattering, Cherenkov radiation, and Kirchhoff scattering, and presents in detail the problems of diffraction by circular apertures and conducting spheres. Finally, it deals with the principles of special relativity, the covariant formulation of electromagnetism, as well as the concepts of electromagnetic mass, radiation reaction force and the Abraham–Lorentz model.
This book is intended for physics and mathematics students, as well as engineering students interested in the challenges of electromagnetic theory. The discussion is supplemented with numerous applications derived from the theoretical concepts presented.
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Veröffentlichungsjahr: 2025
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Dedicated to the memory of my parentsDedicated to the memory of Daniel ArnaudonDedicated to Prof. Amitabha Chakrabarti
Series Editor
Frédérique de Fornel
Introduction to Classical Electrodynamics 3
Boucif Abdesselam
First published 2026 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUKwww.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USAwww.wiley.com
© ISTE Ltd 2026The rights of Boucif Abdesselam to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group.
Library of Congress Control Number: 2025945758
British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-83669-002-3
The manufacturer’s authorized representative according to the EU General Product Safety Regulation is Wiley-VCH GmbH, Boschstr. 12, 69469 Weinheim, Germany, e-mail: [email protected].
Preface
There is no place in this new kind of physics both for the field and matter, for the field is the only reality.
Albert Einstein
An electric charge is an abstract concept, comparable to the concept of mass, introduced to explain certain behaviors in physics. However, unlike mass, an electric charge can take two experimental forms that are arbitrarily called positive and negative. Two charges of the same nature, both positive or both negative, for instance, repel each other, while two opposite charges attract each other. This phenomenon is called electrostatic interaction.
The history of electric charges goes back to the ancient Greeks who observed that rubbing certain objects against fur produced an imbalance in the total electric charge. These “electrified” objects would also attract other light objects, like hair. What is more, if the objects are rubbed against each other for long enough, it is even possible to create a spark.
Electric charges can be experimentally measured. Particles observed in nature have charges that are integer multiples of an elementary charge, which is a fundamental physical constant. An elementary charge is the electric charge on a proton or, equivalently, the opposite of the electric charge on an electron. It is expressed in Coulombs (C), or in A × s in the international system of units. The value of an electric charge, considered to be invisible, and its discrete nature were first discovered in 1909 by Robert Andrews Millikan.
The elementary charge, denoted by e, has an approximate value e = 1.602176634 × 10−19C. The exact value of the elementary charge is currently defined by: , where h is Planck’s constant, with h = 6.626070040 × 10−34Js; α is the fine structure constant, with α = 7.2973525664 × 10−3; μ0 is the vacuum magnetic permeability, with μ0 = 4π × 10−7Hm−1; and c is the speed of light in a vacuum, with c = 299, 792, 458 m/s. The existence of quarks was postulated in the beginning of 1960 and they are believed to possess a fractional electric charge or , but they are confined within hadrons, whose charge and may be zero, and may be equal to the elementary charge, or may be an integer multiple of the elementary charge. Quarks have not yet been separately detected, but they are assumed to have existed in a free state in the very first instants of the Universe (the quark epoch). Charge is an invariant in the theory of special relativity. A particle of charge q, whatever its velocity, will always retain the same charge q.
There are two branches of study:
Electrostatics
is the branch of physics that studies the phenomena created by static electric charges for the observer. In physics, an electrically charged particle creates a vector field that is called an
electric field
. In the presence of a charged particle, the local properties of a space are therefore modified. This is therefore an accurate translation of the concept of a field. If there is another charge in this same field, it will be subject to the action of the electric force (one of the four forces of nature) exerted at a distance by the particle: the electric field is in a way the mediator of this distant action. If there is a movement of electric charges, generally electrons, within a conductive material under the effect of a potential difference at the ends of the material, an electric current is produced. The concept of electric current was first introduced by Benjamin Franklin. He was the first to imagine electricity as a type of invisible fluid present in all matter. He postulated that friction of an insulating surface causes the fluid to move and that a flow of this fluid constitutes an electric current. A material is made up of a large number of electric charges, but these charges balance out among themselves, that is, there is the same number of electrons (negative) as protons (positive). At normal temperatures, the material is electrically neutral. When a static electricity effect is produced, this means that there has been a displacement of charges from material A to material B. This is the
electrification phenomenon
. Excess or missing charges, that is,
uncompensated charges
, are responsible for electrical effects on a body (the example of the rubbed stick). There are two kinds of materials:
A material is said to be a
perfect conductor
if, when electrified, the uncompensated charges move freely through the material.
If the uncompensated charges do not move freely and remain fixed in the place where they were deposited, we say that it is a
perfect insulator (or dielectric)
.
In reality, a real material is obviously lies between these two limit states:
Magnetism
represents all physical phenomena in which objects exert attractive or repulsive forces on other materials. The electric currents and magnetic moments of the fundamental elementary particles are at the origin of the magnetic field generated by these forces. The magnetic field is therefore a vector quantity (characterized by an intensity and a direction) defined for each point in space. It is determined by the position and orientation of magnets, electromagnets and the displacement of electric charges. The presence of this field is seen in the existence of a force acting upon the moving electric charges (called the
Lorentz force
), and various effects that affect certain materials (paramagnetism, diamagnetism, or ferromagnetism, depending on the case). Magnetic susceptibility is the quantity that determines how much a magnetic field interacts with a material.
Electricity and magnetism are intimately related. In 1820, Ørsted discovered the relationship between electricity and magnetism: a wire carrying an electric current is capable of deflecting the magnetized needle of a compass. Subsequently, Ampère, based on Ørsted’s work, discovered and formulated a number of laws governing the relationship between magnetism and electrodynamics. In 1831, Faraday discovered that if an electric current produces a magnetic field, the reverse is also true. An electric current can be produced by setting a magnetic field in motion, according to Lenz’s law.
This marked the birth of electromagnetism. It is therefore the field of physics that studies the interactions between electrically charged particles, whether at rest or in motion, using the concept of the electromagnetic field. Electromagnetism can be defined as the branch that studies the electromagnetic field and its interactions with charged particles. The electric field and the magnetic field are the two components of the electromagnetic field described by electromagnetism. The equations that describe the evolution of the electromagnetic field are called Maxwell equations. Electric and magnetic fields waves can propagate freely in space and in most materials. These are called electromagnetic waves and correspond to all manifestations of light across wavelengths (radio waves, microwaves, infrared, visible domain, ultraviolet rays, X-rays and gamma rays). Maxwell wrote: The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws. From this point of view, optics can be seen as an application of electromagnetism. The electromagnetic interaction is also one of the four fundamental interactions. Together with quantum mechanics, it enables us to understand the existence, cohesion and stability of atoms and molecules.
From the point of view of fundamental physics, the theoretical development of classical electromagnetism led to the theory of special relativity at the beginning of the 20th century. The need to reconcile electromagnetic theory and quantum mechanics led to the construction of quantum electrodynamics, in which electromagnetic interaction is interpreted as an exchange of particles called photons. In particle physics, electromagnetic interaction and weak interaction are unified within the electroweak theory.
Note to readers
Electromagnetic field theory is often poorly or insufficiently taught in university physics programs. The heavy dependence on mathematics, especially vector calculus, integral calculus, complex functions and special functions, etc., add to the difficulty in teaching this subject and may obscure certain phenomena such that students get mired in mathematical difficulties and lose sight of practical applications and vice versa. This modest contribution aims to lay out the essential ideas of the theory in the most logical order. Special importance is given to the mathematical developments of this theory. The experimental side of the theory is only lightly approached. We will not insist on units of measurement. These three volumes will contain many examples (the most classic). I have tried to give as much detail as possible in the calculations. Across the three volumes, the discussion is essentially divided into three main topics: (1) charges and/or dipoles, as the source of the electrostatic field; (2) currents and/or magnetization as sources of the magnetic field; (3) electrodynamics, where the electric field and magnetic field are of equal importance. This book discusses many applications of electromagnetic theory. For instance, there are the solutions of Laplace and Poisson equations in various coordinate systems, dielectric medias, magnetized medias, waveguides, transmission lines, electromagnetic radiation, radiation by point-like charges, Kirchhoff diffraction, special relativity, etc. For the reader’s convenience, we have listed all of the mathematical tools used in the book into six appendices, which will be available to download online: vector analysis, tensors, special/hypergeometric functions, etc. References are presented at the end of each volume. Formulas are given here in the international system of units. We have tried to adopt the notations most commonly used in the literature. Graphs and 3D representations have been produced using Maple.
This three-volume set is meant for physics students, engineering students, as well as mathematics students who are interested in problems in electromagnetic theory. It can also be used by anyone interested in physics in general, or who simply wants to learn more out of intellectual curiosity. There are many books on the same subject and therefore it seems reasonable to repeat some classical developments that we find excellent, rather than striving for originality at any cost.
Volume 1 will cover the following topics:
electrostatic field in vacuum (Gauss’ theorem, etc.);
conductor at equilibrium;
electrokinetics;
magnetic field in vacuum (Laplace force, Biot–Savart law, the Ampère theorem, etc.);
electric and magnetic moments;
boundary problems for static potentials in vacuum.
Volume 2 will study the following points:
static potential with a source – Poisson equation (image method, Green function, etc.);
electromagnetic induction phenomena;
electromagnetic waves in vacuum;
electromagnetic energies;
polarization, polarized media (dielectric);
magnetization, magnetized media;
electromagnetic fields in matter.
Volume 3 will analyze the following concepts:
waveguides;
transmission lines;
electromagnetic radiation;
radiation from point-like charges;
Kirchhoff diffraction;
theory of special relativity;
covariant formulation.
The online appendices will be available at the following URL: www.iste.co.uk/abdesselam/electrodynamics1.zip.
The appendices will cover:
some vector analysis results (Appendix 1);
tensors (Appendix 2);
the Dirac function (Appendix 3);
orthogonal functions (Appendix 4);
beta and gamma functions (Appendix 5);
special functions (Appendix 6).
For online appendices, please see: www.iste.co.uk/abdesselam/electrodynamics1.zip.
Acknowledgments
This work would never have seen the light of day without the encouragement of many people. I am particularly grateful to my students, colleagues and many others. I thank Fréderique de Fornel for her patience and advice. Finally, I dedicate this book to my wife, Nadia, and my children, Abir, Ahmed Nour, Mohammed Sami and Samah. I would like to express my gratitude to them.
Boucif Abdesselam
October 2025
Chapter 1Waveguides
1.1. The different types of waveguides
Guided propagation involves channeling an electromagnetic wave within a space bounded by metallic or dielectric interfaces from a transmitter to a receiver. The main advantage of guided propagation is the transmission of the wave shielded from external disturbances (atmospheric and others); and, sometimes, with low attenuation. Wave guiding is achieved using devices called waveguides, whose structure varies depending on the frequency range of the waves:
hollow waveguides with a rectangular cross-section for millimeter and centimeter waves;
optical fibers for microwaves;
coaxial cables for high frequencies.
Waveguides are generally composed of a vacuum or dielectric medium, bounded by metallic or dielectric walls. In such a medium, the waves must satisfy the Maxwell equations and the boundary conditions on the metallic or dielectric walls.
The most widely used waveguides are the following:
Metallic waveguides
: these are metal tubes with a circular or rectangular cross-section.
Dielectric waveguides and optical fibers
: a dielectric waveguide is a device that is made up of the juxtaposition of several dielectric materials with different refractive indices. The material with the highest refractive index is generally placed in the center of the guide. An electromagnetic wave that enters the guide essentially stays within this high-index material.
Coaxial cables
: a coaxial cable takes the form of an
inner conductor
, surrounded by an
insulation
and then a
braided and wrapped conductive shield
. All of this is wrapped within a
protective cover
. Coaxial cables will be studied in
Chapter 2
.
1.2. Description of a guided wave
1.2.1. Propagation equation and field expressions
We will see if it is possible to propagate a monochromatic wave with angular frequency ω, within the guide and along the Oz axis. The medium inside the guide, which will support the propagation of the electromagnetic wave, must have the lowest possible absorbance – it generally consists of air or a dielectric material. To simplify, let us assume that this propagation medium is an LIH (linear, isotropic, and homogeneous) dielectric, with electric permittivity ε and magnetic permeability μ (in the case of a vacuum, we have ε = ε0 and μ = μ0). The behavior of an electromagnetic wave within the waveguide is studied by solving the Maxwell equations (in the propagation medium), taking into account the boundary conditions imposed on the wave (on the fields) on the internal faces delimiting the propagation medium.
The propagation medium is a perfect LIH dielectric material (i.e. without free charges and no free current). At any point in this medium, the Maxwell equations are written as:
where and . In the case of a conducting medium, the free current term (which appears in ) can be integrated into the displacement field, which becomes a generalized displacement field , related to the electric field through the complex dielectric permittivity . To obtain the propagation equation for the electromagnetic field, we apply the double curl to the electric and magnetic fields: , because . The last equation in [1.1] also yields . Thus, from this, we can derive the equation for the propagation of the electric field. The same procedure leads to the propagation equation for the magnetic excitation field in the medium. The propagation equations for the fields are written as:
To study propagation in a waveguide, we first need to choose an analytical expression for the electromagnetic field that satisfies both the propagation equations [4.2] and the boundary conditions imposed on the medium. In the case of a cylindrical guide, cylindrical coordinates should be chosen, while in the case of a rectangular guide, a Cartesian system is preferable. In both cases, the choice of plane-wave propagation is not possible (with the exception of certain special situations) because a plane wave cannot satisfy the multiple boundary conditions imposed (the plane-wave amplitude is assumed to be constant). To do this, the fields must be decomposed into longitudinal fields (parallel to the direction of the guide, chosen here as the Oz axis) and in transverse fields (perpendicular to Oz). In the expressions for and , the dependence in z (assumed to be sinusoidal) is separated from the dependence in .
We therefore consider an electromagnetic wave propagating along the Oz axis as a plane wave:
but with variable amplitude. Here, kz is the wavenumber or the propagation constant in the Oz direction. The problem is therefore reduced to solving the Maxwell equations for the amplitudes and , which depend on the transverse coordinates: the problem thus becomes a 2D problem. Let us start by decomposing the fields into transverse and longitudinal components:
where and are perpendicular to Oz. We also define the transverse gradient . We want to show here that the transverse field (and, consequently, ) is fully determined if we know the Ez components of the electric field and Hz of the magnetic field.
This result is also true for the transverse field . First, let us observe that:
The first term is oriented along , while the other two are transverse. Next, we write the Maxwell equations for the fields [1.3], separating the transverse and longitudinal components. For the Maxwell–Faraday equation, we obtain:
which, after separating the components, gives:
Taking the cross product of equation [1.8] with yields:
Similarly, by taking the dot product of equation [1.7] with , we find:
Let us repeat the same procedure for the Maxwell–Ampère equation:
which leads to:
and consequently:
The other two Maxwell equations can be reduced to:
Equations [1.9–1.10] and [1.14–1.17] make it possible to express and in terms of Ez and Hz. To see this, take the cross product of with equation [1.9] and then substitute equation [1.14] into the result:
which gives:
where . Repeating the same procedure for [1.14] and taking into account [1.9], we get:
which allows us to derive :
Equations [1.19] and [1.21] therefore allow us to determine all transverse components and , as soon as the two longitudinal components Ez and Hz are known. We can thus limit ourselves to solving the propagation equation only for Ez and Hz, and then deduce and using equations [1.19] and [1.21]. In particular, when , the electromagnetic field is zero if the components Ez and Hz are simultaneously zero.
The components Ez and Hz satisfy the Helmholtz equation:
Substituting expressions [1.3] into [1.22], we find:
where:
We must solve this equation, taking into account the boundary conditions imposed in the Oxy plane due to the presence of the waveguide walls (conductive walls or interfaces between two dielectric media).
1.2.2. Classification of propagation modes
Unlike waves propagating in vacuum, electromagnetic waves are not always transverse, that is, the electric and magnetic fields are not necessarily perpendicular to the direction of propagation (the Oz axis). Different propagation modes can be distinguished:
TM (transverse magnetic) mode
: the field is perpendicular to the direction of propagation, but
E
z
≠ 0. We must therefore solve
equation [1.23]
for
E
z
. This mode is also called an
E
type wave (since
E
z
is non-zero).
TE (transverse electric) wave
: the electric field is perpendicular to the direction of propagation, but
H
z
≠ 0. We must therefore solve
equation [1.23]
for
H
z
. This mode is also called an
H
type wave (since
H
z
is non-zero).
TEM (transverse electric and magnetic) mode
: the fields and are perpendicular to the direction of propagation, as if the wave were propagating in a vacuum (i.e.
E
z
=
H
z
= 0).
According to [1.19] and [1.21], for a TEM wave to exist, it is required that and . For and , we must require that γ2 = 0. The dispersion relation for TEM waves is therefore:
(the propagation constant is fixed for TEM waves). We also see from [1.7] and [1.16] that the transverse electric field satisfies the following relations:
These are the two-dimensional electrostatic equations in the absence of charges. We know that this implies that derives from a scalar potential, that is, there exists a function such that . Since , this potential satisfies the 2D Laplace equation:
Relations similar to [1.26] are satisfied by the transverse magnetic excitation field . In fact, equations [1.12] and [1.17] yield (for TEM waves):
which also indicates that the magnetic excitation field derives from a potential , satisfying the 2D Laplace equation:
For TEM waves, the fields and are related (see [1.13–1.14]) in this way:
with . In other words, the field lines are perpendicular to the field lines. A consequence of equations [1.30] is that a TEM wave cannot propagate inside a hollow waveguide surrounded by a conductor (as will be demonstrated explicitly later for rectangular waveguides) because the electrostatic field is always zero inside a conductor. This result is quite general and can be extended to any hollow waveguide with conductive walls and a cross-section of any shape. To see this, let us consider two closed contours C1 and C2 of the Oxy plane, surrounding the guide at positions z1 and z2, respectively. Let us choose the following surfaces: (1) S1 the surface bounded by C1 in the Oxy plane; (2) S2 the surface bounded by C2 in the Oxy plane; (3) Senv is the surface that envelopes the guide, bounded on one side by C1 and on the other by C2. The integrals , and are zero, because is in the Oxy plane for the first two and zero on the surface for the third. This means that:
This last integral can be transformed into a volume integral using the divergence theorem:
which gives and then in τ. Thus, a waveguide made up of a hollow conductor with no other conductor on its inside cannot conduct a TEM electromagnetic wave. These modes can exist in the presence of two or more conductors.
1.3. TE and TM modes in a hollow conducting waveguide
Consider a hollow waveguide surrounded by a conducting wall. In this case, TEM modes do not exist and only TE and TM waves are possible. To minimize energy losses inside these guides during propagation, it is assumed that the interior is vacuum (or air). Therefore, we take ε = ε0 and μ = μ0 in what follows. Let be the normal to the guide’s conductive wall (directed toward the interior). Across this wall, the tangential component of must be continuous. The same applies to the normal component of . Since the wall is a perfect conductor, the fields must vanish inside the conductor. Hence, by continuity, and must be zero on the inner side of the wall. Taking into account the decomposition [1.4], we deduce that:
and represent the transverse and longitudinal components, respectively, of and they must both vanish independently. If we now calculate the scalar product of equation [1.14] with , we obtain:
According to [1.35], the first term is zero on the wall. The mixed product of the second term is written as ; it is zero according to [1.33]. Therefore, the right-hand side of equation [1.36], which is the derivative of Hz in the direction, must also be zero:
In the same way, let us take the vector product of equation [1.9] and :
The two terms on the left in [1.38] are obviously zero, since , which gives:
The problem therefore boils down to solving the Helmholtz equation [1.23], taking into account the boundary conditions [1.34] and [1.37]. As the boundary conditions are different for Ez and Hz, the solutions and eigenvalues must be different between the TM and TE modes.
1.3.1. Rectangular waveguide
Consider a waveguide with a rectangular cross-section (with sides a and b). The inner region of the waveguide is defined by 0 ≤ x ≤ a and 0 ≤ y ≤ b.
The length of the waveguide along Oz is assumed to be infinite, as shown in Figure 1.1.
Figure 1.1.Rectangular waveguide of height a and width b
1.3.1.1. TM waves (Hz = 0, Ez ≠ 0)
Let us first consider the TM modes. The Helmholtz equation is written as follows:
where . The boundary conditions are:
The solution to [1.40] is obtained by separation of variables. We assume that the unknown function can be written in the form , where (respectively, ) depends only on x (respectively, y). Substituting into the Helmholtz equation and dividing by XY, we obtain . The first term depends only on x, the second only on y and the third is constant. This equality can only hold if the first two terms are also constants, that is, , and , where the separation constants must satisfy the relation:
In particular, when ω = 0, we obtain stationary solutions. If at least one of the wavenumbers is purely imaginary, the solutions are decreasing (exponential). If all three wavenumbers are real, there is pure propagation (sinusoidal solutions). The values of these wavenumbers are determined by the boundary conditions imposed on the walls in the Ox and Oy directions. The general solutions are given by:
and therefore:
where A1, A2, B1 and B2 are constants. Since the rectangular waveguide is composed of perfectly conducting walls, the longitudinal component of the electric field on these walls is zero (as already mentioned in [1.41]). In order to satisfy the boundary conditions at x = 0 and y = 0, it is essential that A2 = B2 = 0. Consequently, it follows that:
where E0 is the amplitude of the field. In addition, to satisfy the other two boundary conditions in x = a and y = b, the wavenumbers kx and ky must be quantified as follows:
Note that if m or n is zero, the electric field Ez is zero. Therefore, the desired solution is:
This represents what is called the waveguide eigenmode, denoted by TMmn. Thus, the most general solution of the Helmholtz equation [1.22] is obtained as a linear combination of all possible eigenmodes. Since:
we observe the following:
k
z
can only take discrete values.
k
z
is real only if:
In other words, for a given value of ω, the (m, n) eigenmode can only propagate in the waveguide if condition [14.49] is satisfied. Otherwise, kz becomes purely imaginary, that is, kz = iδ with δ > 0; this introduces the factor e−δz in the expression of Ez, preventing any propagation (the wave is therefore attenuated). Therefore, there is a lower limit for ω, called the cutoff frequency and denoted ωm,n, below which no propagation is possible.
For the TMmn mode, we obtain:
Modes that are excited at a frequency lower than their cutoff frequency therefore do not propagate, but are still present over a certain distance near the excitation point of the waveguide.
Using the notation [1.50], equation [1.48] can also be written as:
This is the dispersion relation of the waveguide; this formula resembles that of a plasma, as illustrated in Figure 1.2.
Figure 1.2.(a) Dispersion relations for some modes of the rectangular waveguide. (b) Phase velocity and group velocity for the (m, n) mode.
The phase and group velocities are given, respectively, by:
As in a plasma, the phase velocity is greater than c, while the group velocity (the physical velocity) is less than c.
The other components of the fields and can be derived from the relations [1.19] and [1.21]:
Taking into account the expression [1.48] of the field Ez, we find:
We clearly see that the boundary conditions:
are satisfied. The tangential components of the electric field and the normal components of the magnetic field are automatically zero on the walls. The solutions found in [1.54] and [1.55] can be interpreted as the result of the fields of four waves resulting from reflection on the waveguide walls. Indeed, let us rewrite the sine functions using De Moivre’s formulas:
where and . This expansion can be interpreted as follows: the first term corresponds to an incident wave propagating in the direction of the wave vector . The second term is the wave resulting from the reflection of this incident wave on the face parallel to the Oyz plane at x = a (reflection coefficient equal to −1 and ). The third term is the wave obtained after reflection of the incident wave on the face located at y = b. Finally, the last term comes from a double successive reflection on the faces at x = a and y = b. This decomposition shows that the angles of incidence are quantized on the walls; only these values make it possible to achieve destructive interference on the boundaries. The propagation of the electromagnetic wave in the waveguide is accompanied by the propagation of currents and charges on the inner walls. The surface charge densities on the walls of the waveguide are obtained from the discontinuity equation of the normal component of :
Similarly, the surface currents are found using the discontinuity equation for the tangential component of the field :
The following are some important remarks:
The surface currents are purely directed along the
Oz
direction.
The charges , , and are, respectively, related to the currents , , and by charge conservation laws. For example, we can verify that:
etc.
If
m
is even (respectively, odd), the charges and currents on the opposite walls at
x
= 0 and
x
=
a
have opposite signs (respectively, the same sign).
Similarly, if
n
is even (respectively, odd), the opposite walls at
y
= 0 and
y
=
b
carry charges of opposite signs (respectively, the same sign) and are traversed by currents flowing in opposite directions (respectively, in the same direction).
The field lines of in the
Oxy
plane are obtained from the following equation:
When the surface charge is positive, the electric field points outward from the wall, whereas it points toward the wall when the surface charge is negative.
For the field lines of in the
Oxy
plane, we must solve the equation:
The field configurations for the TM11, TM12, TM21 and TM22 modes are illustrated in Figure 1.3.
EXAMPLE 1.1.– The total charge per unit length of the waveguide carried by all walls at time t and position z for TMmn mode is calculated as follows:
The total current intensity on all walls at time t and position z is determined using a formula analogous to [1.63]:
It is easy to verify that the quantities and are related by the continuity equation .
We see that if m or n is even, then and are zero.
Figure 1.3.Electric field lines (in blue) and magnetic field lines (dashed) for the TM11, TM12, TM21and TM22= modes in the Oxy plane.
Substituting previously found solutions [1.54] and [1.55] for fields and into the expression of the (time-averaged) Poynting vector gives:
If ω is below the cutoff frequency of a particular mode, kz becomes purely imaginary, and the attenuation factor then appears in the expression of the Poynting vector. For propagating modes where kz is real, there is no dependence on z in [1.65]. In both cases, we see from [1.54] and [1.55] that the products of Ez, respectively, with and are purely imaginary, which means that the real parts of and are zero. Only the component of directed along Oz can have a time average. The other two components (in the directions Ox and Oy) of the Poynting vector have zero time averages. Thus, we obtain:
If kz is purely imaginary, we obtain 〈Πz〉 = 0. On the other hand, if kz is real, the average power flux over time is not equal to zero. The total power flux directed along the Oz direction is found by integrating [1.66] over the cross-sectional area of the waveguide (here, kz is assumed to be real):
since m and n are different from zero. To compute [1.67], the following identities were used:
EXAMPLE 1.2. SCALAR AND VECTOR POTENTIALS OF A TM WAVE.– Using the relation between the magnetic field and its vector potential , for a TM wave (Hz = 0), we obtain:
which means that there exists a scalar function
