Electrodynamics: The Field-Free Approach by Kjell Prytz

Electrodynamics: The Field-Free Approach pdf

Download Electrodynamics: The Field-Free Approach – This book (Electrodynamics: The Field-Free Approach) is intended as an undergraduate textbook in electrodynamics at basic or advanced level.

The objective is to attain a general understanding of the electrodynamic theory and its basic experiments and phenomena in order to form a foundation for further studies in the engineering sciences as well as in modern quantum physics.

Electrodynamics: The Field-Free Approach PDF

The outline of the book is obtained from the following principles:

•         Base the theory on the concept of force and mutual interaction

•         Connect the theory to experiments and observations accessible to the student

•         Treat the electric, magnetic and inductive phenomena cohesively with respect to force, energy, dipoles and material

•         Present electrodynamics using the same principles as in the preceding mechanics course

•         Aim at explaining that theory of relativity is based on the magnetic effect

•         Introduce field theory after the basic phenomena have been explored in terms of force

Although electrodynamics is described in this book from its 1st principles, prior knowledge of about one semester of university studies in mathematics and physics is required, including vector algebra, integral and differential calculus as well as a course in mechanics, treating Newton’s laws and the energy principle.

The target groups are physics and engineering students, as well as professionals in the field, such as high school teachers and employees in the telecom industry. Chemistry and computer science students may also benefit from the book.

Download Electrodynamics: The Field-Free Approach

Table of Content

Contents
1 Basic Principles ………………………………. 1
1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Electrodynamic Force …………………………… 5
2.1 Electric Charges at Rest–Electric Force . . . . . . . . . . . . . . . . . 5
2.2 Uniform Motion–Magnetic Force . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Electric Current . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.2 Measurement of Parallel Motion . . . . . . . . . . . . . . . . 8
2.2.3 Measurement of Perpendicular Motion. . . . . . . . . . . . 9
2.2.4 Magnetic Force for General Uniform Motion . . . . . . . 11
2.2.5 Evaluation of the Magnetic Force Formulas . . . . . . . . 15
2.3 Accelerated Motion–Inductive Force . . . . . . . . . . . . . . . . . . . 17
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Electrodynamic Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1 Electric Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 The Voltage Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3 Magnetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3.1 Magnetic Force from Magnetic Energy . . . . . . . . . . . 39
3.4 General Inductance—Interaction Between
Two Current Elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.5 Faraday-Henry’s Induction Law . . . . . . . . . . . . . . . . . . . . . . 41
3.6 Electrodynamic Force—Updated. . . . . . . . . . . . . . . . . . . . . . 42
3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
xi
4 Macroscopic Systems of Unbound Charges . . . . . . . . . . . . . . . . . 47
4.1 Electric Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1.1 Electrically Charged Wire and Point Charge. . . . . . . . 48
4.1.2 Force Between Two Electrically Charged Wires . . . . . 49
4.1.3 Force Between a Point Charge and an Electrically
Charged Plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.1.4 Electric Force Between a Point Charge
and a Charged Sphere . . . . . . . . . . . . . . . . . . . . . . . 52
4.1.5 Capacitor and Capacitance . . . . . . . . . . . . . . . . . . . . 54
4.1.6 Electric Energy Stored in a Capacitor . . . . . . . . . . . . 56
4.1.7 Electric Force Between Two Charged Plates. . . . . . . . 57
4.2 Magnetic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2.1 Magnetic Force Between a Point Charge
and a Long Straight Current . . . . . . . . . . . . . . . . . . . 59
4.2.2 Magnetic Force Between Point Charge
and Large Current-Carrying Plate . . . . . . . . . . . . . . . 60
4.2.3 Magnetic Force Between a Straight Conductor
and a Large Plate . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2.4 Magnetic Force and Energy Between Two
Parallel Current-Carrying Plates . . . . . . . . . . . . . . . . 62
4.2.5 Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.6 Induction in a Moving Rod Interacting
with Current-Carrying Plate . . . . . . . . . . . . . . . . . . . 71
4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5 Conductors and Resistive Effects . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.1 The Metal as a Conductor . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2 Relaxation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3 Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.4 Heat Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.5 The Principle of Charge Conservation . . . . . . . . . . . . . . . . . . 84
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6 Electric Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.1 Measurement of Capacitance Using an RC Circuit . . . . . . . . . 93
6.2 Measurement of Inductance Using an RL Circuit . . . . . . . . . . 94
6.3 The Oscillation Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.3.1 The RLC Circuit with Constant Input Voltage . . . . . . 95
6.3.2 Forced Oscillation Circuit . . . . . . . . . . . . . . . . . . . . 97
6.3.3 Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
xii Contents
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7 Electric and Magnetic Dipoles. . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.1 Electric Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.1.1 Interaction Between Dipole and Point Charge. . . . . . . 107
7.1.2 Dipole-Dipole Interaction. . . . . . . . . . . . . . . . . . . . . 108
7.1.3 Interaction Between a Charged Plate and a Dipole . . . 109
7.1.4 Generalized Electric Dipole Moment . . . . . . . . . . . . . 110
7.2 Magnetic Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.2.1 Interaction Between a Magnetic Dipole
and a Large Current-Carrying Plate . . . . . . . . . . . . . . 113
7.2.2 Induced Voltage in a Rotating Loop Interacting
with a Current-Carrying Plate . . . . . . . . . . . . . . . . . . 115
7.2.3 Generalized Magnetic Dipole Moment—Interaction
Between Rotating Cylinders . . . . . . . . . . . . . . . . . . . 116
7.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
8 Material Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.1 Electric Response Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.1.1 Electric Force Between a Charged Capacitor
and a Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.1.2 The Dielectric Constant—Not a Constant. . . . . . . . . . 134
8.1.3 Bound Charges. . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.1.4 Three Examples of Polarisation . . . . . . . . . . . . . . . . 138
8.2 Magnetic Response Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.2.1 Magnetization Currents . . . . . . . . . . . . . . . . . . . . . . 144
8.2.2 Magnetization from a Magnetic Influence . . . . . . . . . 147
8.3 General Multipole Interactions . . . . . . . . . . . . . . . . . . . . . . . 157
8.4 Measurement of Electric and Magnetic Material
Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
8.4.1 Measurements on Solids . . . . . . . . . . . . . . . . . . . . . 158
8.4.2 Measurements on Liquids . . . . . . . . . . . . . . . . . . . . 158
8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
9 Motional Consequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
9.1 Modelling the Electrodynamic Interaction . . . . . . . . . . . . . . . 174
9.2 Magnetism as a Motional Consequence . . . . . . . . . . . . . . . . . 174
9.3 Induction as a Motional Consequence . . . . . . . . . . . . . . . . . . 176
Contents xiii
9.4 Special Theory of Relativity. . . . . . . . . . . . . . . . . . . . . . . . . 177
9.4.1 Relative Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
9.4.2 Time Dilation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
9.4.3 Relativistic Momentum . . . . . . . . . . . . . . . . . . . . . . 180
9.4.4 Relativistic Energy . . . . . . . . . . . . . . . . . . . . . . . . . 181
9.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
9.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
10 Field Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
10.1 The Concept of a Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
10.2 The Electric and the Magnetic Fields. . . . . . . . . . . . . . . . . . . 196
10.3 Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
10.4 Material Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
10.5 Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
10.5.1 General Vector Field . . . . . . . . . . . . . . . . . . . . . . . . 200
10.5.2 Divergence and Curl for Static Electric
and Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . 203
10.5.3 Boundary Conditions for Static Electric
and Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . 206
10.6 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
10.6.1 Accelerating Charges—The Time Variation
of the Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . 207
10.6.2 The Continuity Equation—Time Variation
of Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
10.7 Potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
10.8 Power Transportation—The Poynting Vector . . . . . . . . . . . . . 210
10.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
10.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
11 Antenna Theory—The Loop and the Dipole. . . . . . . . . . . . . . . . . 219
11.1 The Loop Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
11.2 The Dipole Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
11.2.1 The Oscillating Electric Dipole. . . . . . . . . . . . . . . . . 223
11.2.2 The Inductive Force . . . . . . . . . . . . . . . . . . . . . . . . 227
11.2.3 Total Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
11.3 Antenna Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
11.4 Power Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
11.4.1 The Dipole Antenna . . . . . . . . . . . . . . . . . . . . . . . . 230
11.4.2 The Loop Antenna . . . . . . . . . . . . . . . . . . . . . . . . . 231
11.5 The Wave Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
xiv Contents
11.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
11.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
Appendix A: Electric Multipoles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
Appendix B: Magnetic Multipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
Appendix C: Magnetic Energy in the Presence of a Material. . . . . . . . 253
Appendix D: Solutions to Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 255
Appendix E: General Magnetic Force Formula . . . . . . . . . . . . . . . . . . 353
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

Preface

The intent of this book is to serve as an undergraduate textbook in electrodynamics at a basic or advanced level. The objective is to attain a general understanding of electrodynamic theory and its basic experiments and phenomena in order to form a foundation for further studies in the engineering sciences as well as in modern quantum physics. The outline of the book is based on the following principles: • Introduce each phenomenon with relevant and complete experiments • Focus on experiments and observations accessible to the student • Base the theory on the concept of force and mutual interaction • Present electrodynamics using the same principles as in the preceding mechanics course • Treat electric, magnetic and inductive phenomena cohesively with respect to force, energy, dipoles and material • Aim at explaining that theory of relativity is based on the magnetic effect • Introduce field theory after the basic phenomena have been explored in terms of force.

Overview : Electrodynamics: The Field-Free Approach

The book starts by considering the different types of forces that occur between electric charges. These may be directly related to their motion, i.e. charges at rest, in uniform motion and in acceleration. The forces, known as electric, magnetic and inductive, are treated cohesively and formulated through observations and measurements. The subsequent chapters are more or less direct applications of the force formulas. Chapter 3 introduces the energy concept as a direct consequence of force through the principle of work. The inductive force is then utilized to derive magnetic energy. Neumann’s formula for inductance is fully derived and used to express magnetic energy. Chapter 4 covers macroscopic systems whose characteristics are obtained through a summation of mutual interactions between infinitesimal elements of charge. Calculation techniques for capacitance and inductance are introduced and shown to be useful concepts in case the system is homogeneous. Chapters 5 and 6 deal with the conductor and electric circuits which constitute the experimental environment from which electrodynamics was developed and technical applications originated. The microscopic description of electric conduction, the origin of resistance and its relation to heat are treated first. Then the resonance circuit which includes the other two circuit components, capacitance and inductance, is introduced. Chapter 7 introduces electric and magnetic dipoles, which are significant concepts since nature generally may be described in terms of such objects. The expressions for electric and magnetic dipole–dipole interaction energies are then central to providing both force and torque. Chapter 8 investigates how different electrically and magnetically neutral materials respond to electric and magnetic influences. It is then assumed that the material is composed of dipoles. The material parameters are introduced and techniques for measuring them are described. A mathematically rigorous treatment of the dipole, or generally multipole, interactions is presented in the accompanying Appendices A and B. In Chap. 9, it is shown conceptually how the magnetic and inductive dynamics arise as motional consequences of the electric force assuming that interactions take time; they are mediated at the speed of light. Alternatively, one may utilize the knowledge of electric and magnetic forces to derive the speed of light.Electrodynamics: The Field-Free Approach

In a special case both the magnetic force and the Faraday-Henry’s law of induction are derived. It is also shown how electromagnetic dynamics is related to relativity, using the fact that magnetism is the motional consequence on which the special theory of relativity is based. Since we build the theory upon the concept of force the material is unique to this book. Chapter 9 also introduces Lorentz transformation in the form of a tutorial. Prerequisites of Chap. 9 are only Chaps. 1–3, thus these four chapters may form a concise course in basic electrodynamics and its relation to relativity. In Chap. 10, electromagnetic field theory is introduced and Maxwell’s equations formulated. The fields are indeed already defined by the force formulas, but expressed in Maxwell’s equations in terms of their divergence and curl. This is motivated by showing that the boundary conditions of the fields are then defined. Using the fields, the Poynting vector may be formulated corresponding to the power transported from an electrodynamic system. An important feature of this book is thus that field theory is introduced after the physical phenomena that constitute electrodynamics have been described, interpreted and formulated in terms of fundamental forces. In Chap. 11, antenna theory is introduced using the principle of retarded interactions, i.e. taking into account that interactions take time.Electrodynamics: The Field-Free Approach

The small loop and the small wire antennas are treated assuming current is uniform and varies harmonically with time. Furthermore, the antenna array is discussed. The basic principles of retarded interactions and array effects are thus developed and may then be applied to natural oscillators as are found in nature. In this way, the reflection law, the refraction law and the phenomenon of Brewster reflection are derived and fully explained. The power delivered by an antenna is also analysed using the Poynting vector derived in a previous chapter. Appendix D contains solutions to the exercises appearing in the book.

Prerequisites and Target Audience

Although electrodynamics is described in this book from its first principles, prior knowledge of about one semester of university studies in mathematics and physics is required, including vector algebra, integral and differential calculus as well as a course in mechanics, treating Newton’s laws and the energy principle. The target groups are teachers, engineering and physics students as well as professionals in the field, e.g. high-school teachers and employees in the telecom industry. Also chemistry and computer science students may benefit from the book.

Study Tips: Electrodynamics: The Field-Free Approach

Learning physics inevitably implies active involvement, especially in problemsolving and experimental studies. We recommend that the discussed experiments also be implemented in practice, not least to avoid tendencies to abstraction. Some of the exercises, marked with an asterisk, are included in the theory of the book and need to be solved before the chapter that follows them. The exercises marked with a ‘C’ are more challenging and normally not suitable for independent problem solving. A solution manual is included in Appendix D.

Acknowledgment: Electrodynamics: The Field-Free Approach

I want to express my gratitude to all students who contributed so much to the courses I have given over the years, including everything from basic and advanced electromagnetic courses to theory of relativity and application courses in Antenna and Microwave engineering at different levels in different study programs. The reflections, comments and questions I have received from them have been of crucial importance for my own development and the genesis of this book. I would also like to thank my colleagues for many intense and fruitful discussions on Physics in general and its role in society. Many interesting conversations with lecturers Göran Nordström on the subject’s didactics and Peter Johansson on the connection between relativity and electrodynamics have been indispensable for me. The latter coined the term ‘motional consequence’ which is diligently used in this book. Many thanks to Dr. Jenny Ivarsson for a thorough scientific review of a first version of this book and to engineering student Nicklas Bjärnhall Prytz for indispensable advice on pedagogical issues as well as for making this book readable in English. This book was originally published in Swedish and is available from its publisher Studentlitteratur. I appreciate all their support during the work, in particular with the drawing of the figures. Finally, I would like to thank the Springer staff for all their support during the translation and revision of the book. 2014 Kjell Prytz

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About the Author

Kjell Prytz is a Senior Lecturer and Associate Professor of Physics, Högskolan Gävle (Gävle University College) since 1996. He has a background as a particle physicist and has worked at CERN, DESY and Celsius. His research focused on the smallest parts, called quarks, and their interactions.
Dr Prytz has been teaching on all possible levels, from the base year to the master level, in practically all fields of physics. In addition to pure physics courses, he has also been responsible for courses in electronics such as microwave and antenna theory.

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