Electromagnetism: Links to Special Relativity

This book is dedicated to the study of the theory of electromagnetism. It is not intended to cover all aspects of the topic, but instead will give a certain perspective, that of its relationship with special relativity. Indeed, special relativity is intrinsic to electromagnetism; thus, this paradigm eliminates some false paradoxes. Electromagnetism also discusses the limit of classical mechanics, and covers problems that arise when phenomena related to the propagation of electromagnetic waves are encountered. These are problems that even the greatest scientists of the last two hundred years have not been able to entirely overcome. This book is directed towards the undergraduate level, and will also support the readers as they move on to advanced technical training, such as an engineering or master's degree.

Table of Contents:
Preface ix Chapter 1 Magnetic Field 1 1.1 Overview of history 1 1.2 Magnetic fields and magnetic forces 6 1.2.1 First experiments 7 1.2.2 Topography: invariances and symmetries 8 1.3 Magnetic fields created by currents 11 1.3.1 Magnetic field created by a volume current distribution 11 1.3.2 Magnetic field created by a surface current distribution or by a filiform current element 12 1.4 Biot–Savart experiment 13 1.5 From field B to vector potential A 14 1.6 Symmetry and invariance properties of the magnetic field related to the symmetry and invariances of the current distribution 15 1.6.1 Distribution of currents having a plane of symmetry 17 1.6.2 Current distribution and anti-symmetry plane 17 1.6.3 Invariance 18 1.7 Calculation of the magnetic field (principle of) 20 1.7.1 Examples of field calculations 20 1.8 Circulation properties of B Ampère’s theorem 36 1.8.1 Integral form of Ampère’s theorem 37 1.8.2 Local form of Ampère’s theorem 43 1.9 Magnetic field flux conservation – vector potential 43 1.9.1 Local relationship 43 1.9.2 Integral relationship – magnetic flux 44 1.9.3 Potential vector of the magnetic field 45 1.10 Transit relationships 52 1.10.1 Circulation property of B Discontinuity of the tangential component of B 53 1.10.2 Flow property of B Continuity of the normal component of B 55 Chapter 2 Magnetic Forces and their Work 57 2.1 Introduction: Academy of Sciences 57 2.2 Action of a magnetic field on a circuit through which a current flows 59 2.2.1 Ampère/Laplace force 59 2.3 Current in a conductor subjected to an electromagnetic field 63 2.3.1 Examples: action of a rectilinear wire, through which a current flows on another rectilinear wire 63 2.4 Local Ohm’s law 64 2.5 Hall effect 65 2.5.1 Hall effect applications (Figure 2.9) 67 2.6 Ampère/Laplace magnetic forces on a conductor (Figures 2.10 and 2.11) 68 2.6.1 Ampère definition 72 2.7 Work of electromagnetic forces 72 2.7.1 Cut-off flow theorem 73 2.7.2 Case of a closed circuit through which a constant current I flows: Maxwell’s theorem 73 2.8 Application to the study of torsor of magnetic forces exerted by an invariable field on a rigid circuit 80 2.9 Potential energy 81 2.9.1 Case of a transverse displacement 82 2.9.2 Case of a rotation 83 2.10 Example: flux of a turn in a magnetic field 84 2.10.1 Turn in a transverse displacement 84 2.10.2 Turn in rotation 85 2.11 Potential energy of interaction with a magnetic field: magnetic dipole 86 2.11.1 Magnetic force and moment acting on the loop 88 2.12 Electrostatic/magnetostatic analogy 89 Chapter 3 Magnetic Media 91 3.1 Introduction: orbital and spin magnetic moments 91 3.2 Experimental studies 93 3.3 Microscopic origins of magnetism: basic concepts 95 3.3.1 Diamagnetism 95 3.3.2 Paramagnetism 96 3.3.3 Ferromagnetism 98 3.4 Macroscopic appearance; magnetization intensity 100 3.4.1 Diamagnetic and paramagnetic materials 101 3.5 Determining the magnetic field created by a magnetized medium 101 3.5.1 Vector potential of a closed circuit, at a point in the vacuum 103 3.6 Macroscopic aspects; magnetization currents 104 3.6.1 Total magnetic field in the presence of magnetic media 108 3.6.2 General equations of magnetostatics in the presence of magnetized media 109 3.7 Generalized Ampère’s theorem: magnetic excitation 110 3.7.1 Transit relationships 111 3.8 Perfect magnetic media or HLI media – homogeneous, linear, isotropic (Figure 3.21) 113 3.8.1 Definition 113 3.9 Magnetic field equations for perfect materials and vacuum 116 3.9.1 Hysteresis loop 118 3.9.2 Applications 121 Chapter 4 Induction 123 4.1 Introduction: variable regimes 123 4.2 Properties of electrical induction and magnetic field 127 4.3 Phenomenon of electromagnetic induction 127 4.3.1 Faraday–Lenz law 130 4.3.2 Terminology and classification of induction phenomena 132 4.3.3 Static or Neumann induction and motional or Lorentz induction 136 4.3.4 Motional or Lorentz induction 140 4.4 Different inductions 149 4.4.1 Auto-induction electromotive force 149 4.4.2 Mutual inductance – coupling coefficient 150 4.5 Applications 154 4.6 Electromechanical conversion; moving bar in a uniform B-field 154 4.6.1 We place ourselves in the laboratory repository 154 4.6.2 We place ourselves in the frame of reference to the bar 156 4.7 Vector potential and quantum mechanics 157 4.8 Appendix: another example of an induction problem 170 4.8.1 Coil with tube-shaped conductive core 170 Chapter 5 Propagation: Special Relativity 179 5.1 Introduction 179 5.1.1. Potential of a moving charge: general solution by Liénard and Wiercherts 180 5.1.2 Spherical waves 182 5.2 Light and electromagnetic waves 184 5.2.1 Spherical wave from a point source 186 5.2.2 Paradox of advanced actions 189 5.3 Relativity 195 5.3.1 Galileo’s relativity 195 5.3.2 Special relativity 196 5.3.3 Charges in motion: from “Coulomb” to “Ampère” 217 5.3.4 Note on Lorentz equations 225 Conclusion 227 Appendices 229 Appendix 1 Ampère/Laplace Magnetic Actions Undergone by a Current Loop Placed in an External Magnetic Field 231 Appendix 2 Magnetostatic Potential Energy of a Current System (Perfect Media) 241 Appendix 3 Operator Expressions in Cartesian Coordinates 249 Appendix 4 Some Key Players in Electromagnetism and Special Relativity 261 References 277 Index 279

About the Author :
Christian Gontrand is a professor at INL/INSA Lyon, France, focusing on 3D circuits. He was formerly a head professor in the Smart Power Integration team at AMPERE lab and had the technical charge of the CIMIRLY from 1988 to 1996.