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.

• Preface ixChapter 1 Magnetic Field 11.1 Overview of history 11.2 Magnetic fields and magnetic forces 61.2.1 First experiments 71.2.2 Topography: invariances and symmetries 81.3 Magnetic fields created by currents 111.3.1 Magnetic field created by a volume current distribution 111.3.2 Magnetic field created by a surface current distribution or by a filiform current element 121.4 Biot–Savart experiment 131.5 From field B to vector potential A 141.6 Symmetry and invariance properties of the magnetic field related to the symmetry and invariances of the current distribution 151.6.1 Distribution of currents having a plane of symmetry 171.6.2 Current distribution and anti-symmetry plane 171.6.3 Invariance 181.7 Calculation of the magnetic field (principle of) 201.7.1 Examples of field calculations 201.8 Circulation properties of B Ampère’s theorem 361.8.1 Integral form of Ampère’s theorem 371.8.2 Local form of Ampère’s theorem 431.9 Magnetic field flux conservation – vector potential 431.9.1 Local relationship 431.9.2 Integral relationship – magnetic flux 441.9.3 Potential vector of the magnetic field 451.10 Transit relationships 521.10.1 Circulation property of B Discontinuity of the tangential component of B 531.10.2 Flow property of B Continuity of the normal component of B 55Chapter 2 Magnetic Forces and their Work 572.1 Introduction: Academy of Sciences 572.2 Action of a magnetic field on a circuit through which a current flows 592.2.1 Ampère/Laplace force 592.3 Current in a conductor subjected to an electromagnetic field 632.3.1 Examples: action of a rectilinear wire, through which a current flows on another rectilinear wire 632.4 Local Ohm’s law 642.5 Hall effect 652.5.1 Hall effect applications (Figure 2.9) 672.6 Ampère/Laplace magnetic forces on a conductor (Figures 2.10 and 2.11) 682.6.1 Ampère definition 722.7 Work of electromagnetic forces 722.7.1 Cut-off flow theorem 732.7.2 Case of a closed circuit through which a constant current I flows: Maxwell’s theorem 732.8 Application to the study of torsor of magnetic forces exerted by an invariable field on a rigid circuit 802.9 Potential energy 812.9.1 Case of a transverse displacement 822.9.2 Case of a rotation 832.10 Example: flux of a turn in a magnetic field 842.10.1 Turn in a transverse displacement 842.10.2 Turn in rotation 852.11 Potential energy of interaction with a magnetic field: magnetic dipole 862.11.1 Magnetic force and moment acting on the loop 882.12 Electrostatic/magnetostatic analogy 89Chapter 3 Magnetic Media 913.1 Introduction: orbital and spin magnetic moments 913.2 Experimental studies 933.3 Microscopic origins of magnetism: basic concepts 953.3.1 Diamagnetism 953.3.2 Paramagnetism 963.3.3 Ferromagnetism 983.4 Macroscopic appearance; magnetization intensity 1003.4.1 Diamagnetic and paramagnetic materials 1013.5 Determining the magnetic field created by a magnetized medium 1013.5.1 Vector potential of a closed circuit, at a point in the vacuum 1033.6 Macroscopic aspects; magnetization currents 1043.6.1 Total magnetic field in the presence of magnetic media 1083.6.2 General equations of magnetostatics in the presence of magnetized media 1093.7 Generalized Ampère’s theorem: magnetic excitation 1103.7.1 Transit relationships 1113.8 Perfect magnetic media or HLI media – homogeneous, linear, isotropic (Figure 3.21) 1133.8.1 Definition 1133.9 Magnetic field equations for perfect materials and vacuum 1163.9.1 Hysteresis loop 1183.9.2 Applications 121Chapter 4 Induction 1234.1 Introduction: variable regimes 1234.2 Properties of electrical induction and magnetic field 1274.3 Phenomenon of electromagnetic induction 1274.3.1 Faraday–Lenz law 1304.3.2 Terminology and classification of induction phenomena 1324.3.3 Static or Neumann induction and motional or Lorentz induction 1364.3.4 Motional or Lorentz induction 1404.4 Different inductions 1494.4.1 Auto-induction electromotive force 1494.4.2 Mutual inductance – coupling coefficient 1504.5 Applications 1544.6 Electromechanical conversion; moving bar in a uniform B-field 1544.6.1 We place ourselves in the laboratory repository 1544.6.2 We place ourselves in the frame of reference to the bar 1564.7 Vector potential and quantum mechanics 1574.8 Appendix: another example of an induction problem 1704.8.1 Coil with tube-shaped conductive core 170Chapter 5 Propagation: Special Relativity 1795.1 Introduction 1795.1.1. Potential of a moving charge: general solution by Liénard and Wiercherts 1805.1.2 Spherical waves 1825.2 Light and electromagnetic waves 1845.2.1 Spherical wave from a point source 1865.2.2 Paradox of advanced actions 1895.3 Relativity 1955.3.1 Galileo’s relativity 1955.3.2 Special relativity 1965.3.3 Charges in motion: from “Coulomb” to “Ampère” 2175.3.4 Note on Lorentz equations 225Conclusion 227Appendices 229Appendix 1 Ampère/Laplace Magnetic Actions Undergone by a Current Loop Placed in an External Magnetic Field 231Appendix 2 Magnetostatic Potential Energy of a Current System (Perfect Media) 241Appendix 3 Operator Expressions in Cartesian Coordinates 249Appendix 4 Some Key Players in Electromagnetism and Special Relativity 261References 277Index 279

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.

• Preface ixChapter 1 Magnetic Field 11.1 Overview of history 11.2 Magnetic fields and magnetic forces 61.2.1 First experiments 71.2.2 Topography: invariances and symmetries 81.3 Magnetic fields created by currents 111.3.1 Magnetic field created by a volume current distribution 111.3.2 Magnetic field created by a surface current distribution or by a filiform current element 121.4 Biot–Savart experiment 131.5 From field B to vector potential A 141.6 Symmetry and invariance properties of the magnetic field related to the symmetry and invariances of the current distribution 151.6.1 Distribution of currents having a plane of symmetry 171.6.2 Current distribution and anti-symmetry plane 171.6.3 Invariance 181.7 Calculation of the magnetic field (principle of) 201.7.1 Examples of field calculations 201.8 Circulation properties of B Ampère’s theorem 361.8.1 Integral form of Ampère’s theorem 371.8.2 Local form of Ampère’s theorem 431.9 Magnetic field flux conservation – vector potential 431.9.1 Local relationship 431.9.2 Integral relationship – magnetic flux 441.9.3 Potential vector of the magnetic field 451.10 Transit relationships 521.10.1 Circulation property of B Discontinuity of the tangential component of B 531.10.2 Flow property of B Continuity of the normal component of B 55Chapter 2 Magnetic Forces and their Work 572.1 Introduction: Academy of Sciences 572.2 Action of a magnetic field on a circuit through which a current flows 592.2.1 Ampère/Laplace force 592.3 Current in a conductor subjected to an electromagnetic field 632.3.1 Examples: action of a rectilinear wire, through which a current flows on another rectilinear wire 632.4 Local Ohm’s law 642.5 Hall effect 652.5.1 Hall effect applications (Figure 2.9) 672.6 Ampère/Laplace magnetic forces on a conductor (Figures 2.10 and 2.11) 682.6.1 Ampère definition 722.7 Work of electromagnetic forces 722.7.1 Cut-off flow theorem 732.7.2 Case of a closed circuit through which a constant current I flows: Maxwell’s theorem 732.8 Application to the study of torsor of magnetic forces exerted by an invariable field on a rigid circuit 802.9 Potential energy 812.9.1 Case of a transverse displacement 822.9.2 Case of a rotation 832.10 Example: flux of a turn in a magnetic field 842.10.1 Turn in a transverse displacement 842.10.2 Turn in rotation 852.11 Potential energy of interaction with a magnetic field: magnetic dipole 862.11.1 Magnetic force and moment acting on the loop 882.12 Electrostatic/magnetostatic analogy 89Chapter 3 Magnetic Media 913.1 Introduction: orbital and spin magnetic moments 913.2 Experimental studies 933.3 Microscopic origins of magnetism: basic concepts 953.3.1 Diamagnetism 953.3.2 Paramagnetism 963.3.3 Ferromagnetism 983.4 Macroscopic appearance; magnetization intensity 1003.4.1 Diamagnetic and paramagnetic materials 1013.5 Determining the magnetic field created by a magnetized medium 1013.5.1 Vector potential of a closed circuit, at a point in the vacuum 1033.6 Macroscopic aspects; magnetization currents 1043.6.1 Total magnetic field in the presence of magnetic media 1083.6.2 General equations of magnetostatics in the presence of magnetized media 1093.7 Generalized Ampère’s theorem: magnetic excitation 1103.7.1 Transit relationships 1113.8 Perfect magnetic media or HLI media – homogeneous, linear, isotropic (Figure 3.21) 1133.8.1 Definition 1133.9 Magnetic field equations for perfect materials and vacuum 1163.9.1 Hysteresis loop 1183.9.2 Applications 121Chapter 4 Induction 1234.1 Introduction: variable regimes 1234.2 Properties of electrical induction and magnetic field 1274.3 Phenomenon of electromagnetic induction 1274.3.1 Faraday–Lenz law 1304.3.2 Terminology and classification of induction phenomena 1324.3.3 Static or Neumann induction and motional or Lorentz induction 1364.3.4 Motional or Lorentz induction 1404.4 Different inductions 1494.4.1 Auto-induction electromotive force 1494.4.2 Mutual inductance – coupling coefficient 1504.5 Applications 1544.6 Electromechanical conversion; moving bar in a uniform B-field 1544.6.1 We place ourselves in the laboratory repository 1544.6.2 We place ourselves in the frame of reference to the bar 1564.7 Vector potential and quantum mechanics 1574.8 Appendix: another example of an induction problem 1704.8.1 Coil with tube-shaped conductive core 170Chapter 5 Propagation: Special Relativity 1795.1 Introduction 1795.1.1. Potential of a moving charge: general solution by Liénard and Wiercherts 1805.1.2 Spherical waves 1825.2 Light and electromagnetic waves 1845.2.1 Spherical wave from a point source 1865.2.2 Paradox of advanced actions 1895.3 Relativity 1955.3.1 Galileo’s relativity 1955.3.2 Special relativity 1965.3.3 Charges in motion: from “Coulomb” to “Ampère” 2175.3.4 Note on Lorentz equations 225Conclusion 227Appendices 229Appendix 1 Ampère/Laplace Magnetic Actions Undergone by a Current Loop Placed in an External Magnetic Field 231Appendix 2 Magnetostatic Potential Energy of a Current System (Perfect Media) 241Appendix 3 Operator Expressions in Cartesian Coordinates 249Appendix 4 Some Key Players in Electromagnetism and Special Relativity 261References 277Index 279

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.

• Preface ixChapter 1 Magnetic Field 11.1 Overview of history 11.2 Magnetic fields and magnetic forces 61.2.1 First experiments 71.2.2 Topography: invariances and symmetries 81.3 Magnetic fields created by currents 111.3.1 Magnetic field created by a volume current distribution 111.3.2 Magnetic field created by a surface current distribution or by a filiform current element 121.4 Biot–Savart experiment 131.5 From field B to vector potential A 141.6 Symmetry and invariance properties of the magnetic field related to the symmetry and invariances of the current distribution 151.6.1 Distribution of currents having a plane of symmetry 171.6.2 Current distribution and anti-symmetry plane 171.6.3 Invariance 181.7 Calculation of the magnetic field (principle of) 201.7.1 Examples of field calculations 201.8 Circulation properties of B Ampère’s theorem 361.8.1 Integral form of Ampère’s theorem 371.8.2 Local form of Ampère’s theorem 431.9 Magnetic field flux conservation – vector potential 431.9.1 Local relationship 431.9.2 Integral relationship – magnetic flux 441.9.3 Potential vector of the magnetic field 451.10 Transit relationships 521.10.1 Circulation property of B Discontinuity of the tangential component of B 531.10.2 Flow property of B Continuity of the normal component of B 55Chapter 2 Magnetic Forces and their Work 572.1 Introduction: Academy of Sciences 572.2 Action of a magnetic field on a circuit through which a current flows 592.2.1 Ampère/Laplace force 592.3 Current in a conductor subjected to an electromagnetic field 632.3.1 Examples: action of a rectilinear wire, through which a current flows on another rectilinear wire 632.4 Local Ohm’s law 642.5 Hall effect 652.5.1 Hall effect applications (Figure 2.9) 672.6 Ampère/Laplace magnetic forces on a conductor (Figures 2.10 and 2.11) 682.6.1 Ampère definition 722.7 Work of electromagnetic forces 722.7.1 Cut-off flow theorem 732.7.2 Case of a closed circuit through which a constant current I flows: Maxwell’s theorem 732.8 Application to the study of torsor of magnetic forces exerted by an invariable field on a rigid circuit 802.9 Potential energy 812.9.1 Case of a transverse displacement 822.9.2 Case of a rotation 832.10 Example: flux of a turn in a magnetic field 842.10.1 Turn in a transverse displacement 842.10.2 Turn in rotation 852.11 Potential energy of interaction with a magnetic field: magnetic dipole 862.11.1 Magnetic force and moment acting on the loop 882.12 Electrostatic/magnetostatic analogy 89Chapter 3 Magnetic Media 913.1 Introduction: orbital and spin magnetic moments 913.2 Experimental studies 933.3 Microscopic origins of magnetism: basic concepts 953.3.1 Diamagnetism 953.3.2 Paramagnetism 963.3.3 Ferromagnetism 983.4 Macroscopic appearance; magnetization intensity 1003.4.1 Diamagnetic and paramagnetic materials 1013.5 Determining the magnetic field created by a magnetized medium 1013.5.1 Vector potential of a closed circuit, at a point in the vacuum 1033.6 Macroscopic aspects; magnetization currents 1043.6.1 Total magnetic field in the presence of magnetic media 1083.6.2 General equations of magnetostatics in the presence of magnetized media 1093.7 Generalized Ampère’s theorem: magnetic excitation 1103.7.1 Transit relationships 1113.8 Perfect magnetic media or HLI media – homogeneous, linear, isotropic (Figure 3.21) 1133.8.1 Definition 1133.9 Magnetic field equations for perfect materials and vacuum 1163.9.1 Hysteresis loop 1183.9.2 Applications 121Chapter 4 Induction 1234.1 Introduction: variable regimes 1234.2 Properties of electrical induction and magnetic field 1274.3 Phenomenon of electromagnetic induction 1274.3.1 Faraday–Lenz law 1304.3.2 Terminology and classification of induction phenomena 1324.3.3 Static or Neumann induction and motional or Lorentz induction 1364.3.4 Motional or Lorentz induction 1404.4 Different inductions 1494.4.1 Auto-induction electromotive force 1494.4.2 Mutual inductance – coupling coefficient 1504.5 Applications 1544.6 Electromechanical conversion; moving bar in a uniform B-field 1544.6.1 We place ourselves in the laboratory repository 1544.6.2 We place ourselves in the frame of reference to the bar 1564.7 Vector potential and quantum mechanics 1574.8 Appendix: another example of an induction problem 1704.8.1 Coil with tube-shaped conductive core 170Chapter 5 Propagation: Special Relativity 1795.1 Introduction 1795.1.1. Potential of a moving charge: general solution by Liénard and Wiercherts 1805.1.2 Spherical waves 1825.2 Light and electromagnetic waves 1845.2.1 Spherical wave from a point source 1865.2.2 Paradox of advanced actions 1895.3 Relativity 1955.3.1 Galileo’s relativity 1955.3.2 Special relativity 1965.3.3 Charges in motion: from “Coulomb” to “Ampère” 2175.3.4 Note on Lorentz equations 225Conclusion 227Appendices 229Appendix 1 Ampère/Laplace Magnetic Actions Undergone by a Current Loop Placed in an External Magnetic Field 231Appendix 2 Magnetostatic Potential Energy of a Current System (Perfect Media) 241Appendix 3 Operator Expressions in Cartesian Coordinates 249Appendix 4 Some Key Players in Electromagnetism and Special Relativity 261References 277Index 279