**UFRJ**

# Eletromagnetismo - livro

(Parte **1** de 3)

Υ Bo Thidé

Bo Thidé ELECTROMAGNETIC FIELD THEORY

Also available

Tobia Carozzi, Anders Eriksson, Bengt Lundborg, Bo Thidé and Mattias Waldenvik

Bo Thidé

Swedish Institute of Space Physics and

Department of Astronomy and Space Physics Uppsala University, Sweden

This book was typeset in LATEX2ε on an HP9000/700 series workstation and printed on an HP LaserJet 5000GN printer.

Copyright ©1997, 1998, 1999 and 2000 by Bo Thidé Uppsala, Sweden All rights reserved.

Electromagnetic Field Theory ISBN X-X-XXX-X

Contents

Preface xi

1.1 Electrostatics | 1 |

1.1.1 Coulomb’s law | 1 |

1.1.2 The electrostatic field | 2 |

1.2 Magnetostatics | 4 |

1.2.1 Ampère’s law | 4 |

1.2.2 The magnetostatic field | 6 |

1.3 Electrodynamics | 8 |

1.3.1 Equation of continuity | 9 |

1.3.2 Maxwell’s displacement current | 9 |

1.3.3 Electromotive force | 10 |

1.3.4 Faraday’s law of induction | 1 |

1.3.5 Maxwell’s microscopic equations | 14 |

1.3.6 Maxwell’s macroscopic equations | 14 |

1.4 Electromagnetic Duality | 15 |

1 Classical Electrodynamics 1

fixed mixing angle | 17 |

Example 1.3 The complex field six-vector | 18 |

Bibliography | 20 |

Example 1.1 Duality of the electromagnetodynamic equations 16 Example 1.2 Maxwell from Dirac-Maxwell equations for a Example 1.4 Duality expressed in the complex field six-vector 19

2.1 The wave equation | 24 |

2.1.1 The wave equation for E | 24 |

2.1.2 The wave equation for B | 24 |

2.1.3 The time-independent wave equation for E | 25 |

2.2 Plane waves | 26 |

2.2.1 Telegrapher’s equation | 27 |

2 Electromagnetic Waves 23 i

2.2.2 Waves in conductive media | 29 |

2.3 Observables and averages | 30 |

Bibliography | 31 |

i CONTENTS

3.1 The electrostatic scalar potential | 3 |

3.2 The magnetostatic vector potential | 34 |

3.3 The electromagnetic scalar and vector potentials | 34 |

3.3.1 Electromagnetic gauges | 36 |

3 Electromagnetic Potentials 3

Gauge transformations | 36 |

Lorentz equations for the electromagnetic potentials . 36

netic potentials | 38 |

The retarded potentials | 41 |

Bibliography | 41 |

3.3.2 Solution of the Lorentz equations for the electromag-

4.1 The magnetic field | 45 |

4.2 The electric field | 47 |

Bibliography | 49 |

4 The Electromagnetic Fields 43

5.1 The special theory of relativity | 51 |

5.1.1 The Lorentz transformation | 52 |

5.1.2 Lorentz space | 53 |

Metric tensor | 54 |

Scalar product and norm | 5 |

Invariant line element and proper time | 56 |

Four-vector fields | 57 |

The Lorentz transformation matrix | 57 |

The Lorentz group | 58 |

5.1.3 Minkowski space | 58 |

5.2 Covariant classical mechanics | 61 |

5.3 Covariant classical electrodynamics | 62 |

5.3.1 The four-potential | 62 |

5.3.2 The Liénard-Wiechert potentials | 63 |

5.3.3 The electromagnetic field tensor | 65 |

Bibliography | 67 |

5 Relativistic Electrodynamics 51 Radius four-vector in contravariant and covariant form 54 Downloaded fromhttp://w.plasma.u.se/CED/Book Draft version released 13th November 2000 at 2:01.

6.1 Charged Particles in an Electromagnetic Field | 69 |

6.1.1 Covariant equations of motion | 69 |

Lagrange formalism | 69 |

Hamiltonian formalism | 72 |

6.2 Covariant Field Theory | 76 |

The electromagnetic field | 80 |

tensor | 81 |

Other fields | 84 |

Bibliography | 85 |

6 Interactions of Fields and Particles 69 6.2.1 Lagrange-Hamilton formalism for fields and interactions 7 Example 6.1 Field energy difference expressed in the field

7.1 Electric polarisation and the electric displacement vector | 87 |

7.1.1 Electric multipole moments | 87 |

7.2 Magnetisation and the magnetising field | 90 |

7.3 Energy and momentum | 91 |

7.3.1 The energy theorem in Maxwell’s theory | 92 |

7.3.2 The momentum theorem in Maxwell’s theory | 93 |

Bibliography | 95 |

7 Interactions of Fields and Matter 87

8.1 The radiation fields | 97 |

8.2 Radiated energy | 9 |

8.2.1 Monochromatic signals | 100 |

8.2.2 Finite bandwidth signals | 100 |

8.3 Radiation from extended sources | 102 |

8.3.1 Linear antenna | 102 |

8.4 Multipole radiation | 104 |

8.4.1 The Hertz potential | 104 |

8.4.2 Electric dipole radiation | 108 |

8.4.3 Magnetic dipole radiation | 109 |

8.4.4 Electric quadrupole radiation | 110 |

8.5 Radiation from a localised charge in arbitrary motion | 1 |

8.5.1 The Liénard-Wiechert potentials | 112 |

8.5.2 Radiation from an accelerated point charge | 114 |

8 Electromagnetic Radiation 97

force | 123 |

Example 8.1 The fields from a uniformly moving charge . 121 Example 8.2 The convection potential and the convection Draft version released 13th November 2000 at 2:01. Downloaded fromhttp://w.plasma.u.se/CED/Book

Radiation for small velocities | 125 |

8.5.3 Bremsstrahlung | 127 |

iv CONTENTS

celeration times | 130 |

8.5.4 Cyclotron and synchrotron radiation | 132 |

Cyclotron radiation | 134 |

Synchrotron radiation | 134 |

Radiation in the general case | 137 |

Virtual photons | 137 |

8.5.5 Radiation from charges moving in matter | 139 |

Vavilov- Cerenkov radiation | 142 |

Bibliography | 147 |

Example 8.3 Bremsstrahlung for low speeds and short ac-

F.1 The Electromagnetic Field | 149 |

F.1.1 Maxwell’s equations | 149 |

Constitutive relations | 149 |

F.1.2 Fields and potentials | 149 |

Vector and scalar potentials | 149 |

Lorentz’ gauge condition in vacuum | 150 |

F.1.3 Force and energy | 150 |

Poynting’s vector | 150 |

Maxwell’s stress tensor | 150 |

F.2 Electromagnetic Radiation | 150 |

F Formulae 149

F.2.2 The far fields from an extended source distribution | 150 |

F.2.3 The far fields from an electric dipole | 150 |

F.2.4 The far fields from a magnetic dipole | 151 |

F.2.5 The far fields from an electric quadrupole | 151 |

F.2.6 The fields from a point charge in arbitrary motion | 151 |

F.2.7 The fields from a point charge in uniform motion | 151 |

F.3 Special Relativity | 152 |

F.3.1 Metric tensor | 152 |

F.3.2 Covariant and contravariant four-vectors | 152 |

F.3.3 Lorentz transformation of a four-vector | 152 |

F.3.4 Invariant line element | 152 |

F.3.5 Four-velocity | 152 |

F.3.6 Four-momentum | 153 |

F.3.7 Four-current density | 153 |

F.3.8 Four-potential | 153 |

F.2.1 Relationship between the field vectors in a plane wave 150 Downloaded fromhttp://w.plasma.u.se/CED/Book Draft version released 13th November 2000 at 2:01.

F.3.9 Field tensor | 153 |

F.4 Vector Relations | 153 |

F.4.1 Spherical polar coordinates | 154 |

Base vectors | 154 |

Directed line element | 154 |

Solid angle element | 154 |

Directed area element | 154 |

Volume element | 154 |

F.4.2 Vector formulae | 154 |

General relations | 154 |

Special relations | 156 |

Integral relations | 157 |

Bibliography | 157 |

M.1 Scalars, Vectors and Tensors | 159 |

M.1.1 Vectors | 159 |

Radius vector | 159 |

M.1.2 Fields | 161 |

Scalar fields | 161 |

Vector fields | 161 |

Tensor fields | 162 |

Example M.1 Tensors in 3D space | 164 |

M.1.3 Vector algebra | 167 |

Scalar product | 167 |

Example M.2 Inner products in complex vector space | 167 |

space | 168 |

Example M.4 Metric in general relativity | 168 |

Dyadic product | 169 |

Vector product | 170 |

M.1.4 Vector analysis | 170 |

The del operator | 170 |

Example M.5 The four-del operator in Lorentz space | 171 |

The gradient | 172 |

tances in 3D | 172 |

The divergence | 173 |

M Mathematical Methods 159 Example M.3 Scalar product, norm and metric in Lorentz Example M.6 Gradients of scalar functions of relative dis- Draft version released 13th November 2000 at 2:01. Downloaded fromhttp://w.plasma.u.se/CED/Book

Example M.7 Divergence in 3D | 173 |

The Laplacian | 173 |

Example M.8 The Laplacian and the Dirac delta | 173 |

The curl | 174 |

Example M.9 The curl of a gradient | 174 |

Example M.10 The divergence of a curl | 175 |

M.2 Analytical Mechanics | 176 |

M.2.1 Lagrange’s equations | 176 |

M.2.2 Hamilton’s equations | 176 |

Bibliography | 177 |

vi CONTENTS Downloaded fromhttp://w.plasma.u.se/CED/Book Draft version released 13th November 2000 at 2:01.

1.1 Coulomb interaction | 2 |

1.2 Ampère interaction | 5 |

1.3 Moving loop in a varying B field | 12 |

5.1 Relative motion of two inertial systems | 52 |

5.2 Rotation in a 2D Euclidean space | 59 |

5.3 Minkowski diagram | 59 |

6.1 Linear one-dimensional mass chain | 76 |

8.1 Radiation in the far zone | 98 |

8.2 Radiation from a moving charge in vacuum | 112 |

8.3 An accelerated charge in vacuum | 114 |

8.4 Angular distribution of radiation during bremsstrahlung | 128 |

8.5 Location of radiation during bremsstrahlung | 129 |

8.6 Radiation from a charge in circular motion | 133 |

8.7 Synchrotron radiation lobe width | 135 |

8.8 The perpendicular field of a moving charge | 138 |

8.9 Vavilov- Cerenkov cone | 144 |

M.1 Surface element of a material body | 164 |

M.2 Tetrahedron-like volume element of matter | 165 |

List of Figures vii

To the memory of LEV MIKHAILOVICH ERUKHIMOV dear friend, remarkable physicist and a truly great human.

Preface

This book is the result of a twenty-five year long love affair. In 1972, I took my first advanced course in electrodynamics at the Theoretical Physics department, Uppsala University. Shortly thereafter, I joined the research group there and took on the task of helping my supervisor, professor PER-OLOF FRÖMAN, with the preparation of a new version of his lecture notes on Electricity Theory. These two things opened up my eyes for the beauty and intricacy of electrodynamics, already at the classical level, and I fell in love with it.

Ever since that time, I have off and on had reason to return to electrodynamics, both in my studies, research and teaching, and the current book is the result of my own teaching of a course in advanced electrodynamics at Uppsala University some twenty odd years after I experienced the first encounter with this subject. The book is the outgrowth of the lecture notes that I prepared for the four-credit course Electrodynamics that was introduced in the Uppsala University curriculum in 1992, to become the five-credit course Classical Electrodynamics in 1997. To some extent, parts of these notes were based on lecture notes prepared, in Swedish, by BENGT LUNDBORG who created, developed and taught the earlier, two-credit course Electromagnetic Radiation at our faculty.

Intended primarily as a textbook for physics students at the advanced undergraduate or beginning graduate level, I hope the book may be useful for research workers too. It provides a thorough treatment of the theory of electrodynamics, mainly from a classical field theoretical point of view, and includes such things as electrostatics and magnetostatics and their unification into electrodynamics, the electromagnetic potentials, gauge transformations, covariant formulation of classical electrodynamics, force, momentum and energy of the electromagnetic field, radiation and scattering phenomena, electromagnetic waves and their propagation in vacuum and in media, and covariant Lagrangian/Hamiltonian field theoretical methods for electromagnetic fields, particles and interactions. The aim has been to write a book that can serve both as an advanced text in Classical Electrodynamics and as a preparation for studies in Quantum Electrodynamics and related subjects. In an attempt to encourage participation by other scientists and students in xii PREFACE the authoring of this book, and to ensure its quality and scope to make it useful in higher university education anywhere in the world, it was produced within a World-Wide Web (W) project. This turned out to be a rather successful move. By making an electronic version of the book freely down-loadable on the net, I have not only received comments on it from fellow Internet physicists around the world, but know, from W ‘hit’ statistics that at the time of writing this, the book serves as a frequently used Internet resource. This way it is my hope that it will be particularly useful for students and researchers working under financial or other circumstances that make it difficult to procure a printed copy of the book.

I am grateful not only to Per-Olof Fröman and Bengt Lundborg for providing the inspirationfor my writing this book, but also to CHRISTER WAHLBERG at Uppsala University for interesting discussions on electrodynamics in general and on this book in particular, and to my former graduate students MATTIAS WALDENVIK and TOBIA CAROZZI as well as ANDERS ERIKSSON, all at the Swedish Institute of Space Physics, Uppsala Division, and who have participated in the teaching and commented on the material covered in the course and in this book. Thanks are also due to my long-term space physics colleague HELMUT KOPKA of the Max-Planck-Institut für Aeronomie, Lindau, Germany, who not only taught me about the practical aspects of the of highpower radio wave transmitters and transmission lines, but also about the more delicate aspects of typesetting a book in TEX and LATEX. I am particularly indebted to Academician professor VITALIY L. GINZBURG for his many fas- cinating and very elucidating lectures, comments and historical footnotes on electromagnetic radiation while cruising on the Volga river during our joint Russian-Swedish summer schools.

Finally, I would like to thank all students and Internet users who have downloaded and commented on the book during its life on the World-Wide Web.

I dedicate this book to my son MATTIAS, my daughter KAROLINA, my high-school physics teacher, STAFFAN RÖSBY, and to my fellow members of the CAPELLA PEDAGOGICA UPSALIENSIS.

Uppsala, Sweden BO THIDÉ November, 2000

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Classical Electrodynamics

Classical electrodynamics deals with electric and magnetic fields and interactions caused by macroscopic distributions of electric charges and currents. This means that the concepts of localised charges and currents assume the validity of certain mathematical limiting processes in which it is considered possible for the charge and current distributions to be localised in infinitesimally small volumes of space. Clearly, this is in contradiction to electromagnetism on a truly microscopic scale, where charges and currents are known to be spatially extended objects. However, the limiting processes used will yield results which are correct on small as well as large macroscopic scales.

In this Chapter we start with the force interactions in classical electrostatics and classical magnetostatics and introduce the static electric and magnetic fields and find two uncoupled systems of equations for them. Then we see how the conservation of electric charge and its relation to electric current leads to the dynamic connection between electricity and magnetism and how the two can be unified in one theory, classical electrodynamics, described by one system of coupled dynamic field equations.

1.1 Electrostatics

The theory that describes physical phenomena related to the interactionbetween stationary electric charges or charge distributions in space is called electrostatics.

1.1.1 Coulomb’s law

It has been found experimentally that in classical electrostatics the interaction between two stationary electrically charged bodies can be described in terms of a mechanical force. Let us consider the simple case described by Figure 1.1.1.

2 CLASSICAL ELECTRODYNAMICS

FIGURE 1.1: Coulomb’s law describes how a static electric charge q, located at a point x relative to the origin O, experiences an electrostatic force from a static electric charge q0 located at x0 .

Let F denote the force acting on a charged particle with charge q located at x, due to the presence of a charge q′ located at x0. According to Coulomb’s law this force is, in vacuum, given by the expression

where we have used results from Example M.6 on page 172. In SI units, which we shall use throughout, the force F is measured in Newton (N), the charges q and q0 in Coulomb (C) [= Ampère-seconds (As)], and the length jx x0j in

in vacuum. In CGS units "0 = 1=(4 ) and the force is measured in dyne, the charge in statcoulomb, and length in centimetres (cm).

1.1.2 The electrostatic field

Instead of describing the electrostatic interaction in terms of a “force action at a distance,” it turns out that it is often more convenient to introduce the concept of a field and to describe the electrostatic interaction in terms of a static vectorial electric field Estat defined by the limiting process

where F is the electrostatic force, as defined in Equation (1.1), from a net charge q0 on the test particle with a small electric net charge q. Since the

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1.1 ELECTROSTATICS 3 purpose of the limiting process is to assure that the test charge q does not influence the field, the expression for Estat does not depend explicitly on q but only on the charge q′ and the relative radius vector x−x0. This means that we can say that any net electric charge produces an electric field in the space that surrounds it, regardless of the existence of a second charge anywhere in this space.1

Using formulae (1.1) and (1.2), we find that the electrostatic field Estat at the field point x (also known as the observation point), due to a field-producing charge q0 at the source point x0, is given by

In the presence of several field producing discrete charges q0 respectively, the assumption of linearity of vacuum2 allows us to superimpose their individual E fields into a total E field

If the discrete charges are small and numerous enough, we introduce the charge density located at x0 and write the total field as

where, in the last step, we used formula Equation (M.68) on page 172. We emphasise that Equation (1.5) above is valid for an arbitrary distribution of charges, including discrete charges, in which case can be expressed in terms of one or more Dirac delta functions.

1In the preface to the first edition of the first volume of his book A Treatise on Electricity and Magnetism, first published in 1873, James Clerk Maxwell describes this in the following, almost poetic, manner: [6]

“For instance, Faraday, in his mind’s eye, saw lines of force traversing all space where the mathematicians saw centres of force attracting at a distance: Faraday saw a medium where they saw nothing but distance: Faraday sought the seat of the phenomena in real actions going on in the medium, they were satisfied that they had found it in a power of action at a distance impressed on the electric fluids.”

2In fact, vacuum exhibits a quantum mechanical nonlinearity due to vacuum polarisation effects manifesting themselves in the momentary creation and annihilation of electron-positron pairs, but classically this nonlinearity is negligible.

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4 CLASSICAL ELECTRODYNAMICS

Since, according to formula Equation (M.78) on page 175, ∇×[rα(x)] 0 for any 3D 3 scalar field (x), we immediately find that in electrostatics

I.e., Estat is an irrotational field.

Taking the divergence of the general Estat expression for an arbitrary charge distribution, Equation (1.5) on the preceding page, and using the representation of the Dirac delta function, Equation (M.73) on page 174, we find that

which is Gauss’s law in differential form.

1.2 Magnetostatics

While electrostatics deals with static charges, magnetostatics deals with stationary currents, i.e., charges moving with constant speeds, and the interaction between these currents.

1.2.1 Ampère’s law

Experiments on the interaction between two small current loops have shown that they interact via a mechanical force, much the same way that charges interact. Let F denote such a force acting on a small loop C carrying a current J located at x, due to the presence of a small loop C0 carrying a current J0

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1.2 MAGNETOSTATICS 5 dlC J x dl0

FIGURE 1.2: Ampère’s law describes how a small loop C, carrying a static electric current J through its tangential line element dl located at x, experiences a magnetostatic force from a small loop C0, carrying a static electric current J0 through the tangential line element dl0 located at x0. The loops can have arbitrary shapes as long as they are simple and closed.

located at x′. According to Ampère’s law this force is, in vacuum, given by the expression

Here dl and dl0 are tangential line elements of the loops C and C0, respectively,

which is a useful relation.

At first glance, Equation (1.8) above appears to be unsymmetric in terms of the loops and therefore to be a force law which is in contradiction with Newton’s third law. However, by applying the vector triple product “bac-cab”

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6 CLASSICAL ELECTRODYNAMICS formula (F.54) on page 155, we can rewrite (1.8) in the following way

Recognising the fact that the integrand in the first integral is an exact differential so that this integral vanishes, we can rewrite the force expression, Equation (1.8) on the previous page, in the following symmetric way

This clearly exhibits the expected symmetry in terms of loops C and C0 .

1.2.2 The magnetostatic field

In analogy with the electrostatic case, we may attribute the magnetostatic interaction to a vectorial magnetic field Bstat. I turns out that Bstat can be defined through

which expresses the small element dBstat(x) of the static magnetic field set up at the field point x by a small line element dl0 of stationary current J0 at the source point x0. The SI unit for the magnetic field, sometimes called the magnetic flux density or magnetic induction, is Tesla (T).

(Parte **1** de 3)