Physics of Classical Electromagnetism

Physics of Classical Electromagnetism

Minoru Fujimoto Physics of Classical Electromagnetism

Minoru Fujimoto Department of Physics University of Guelph Guelph, Ontario Canada, N1G 2W1 Library of Congress Control Number: 2007921094 ISBN: 978-0-387-68015-6 e-isbn: 978-0-387-68018-7 Printed on acid-free paper. C 2007 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 987654321 springer.com

Contents Preface... xi 1. Steady Electric Currents... 1 1.1. Introduction... 1 1.2. Standards for Electric Voltages and Current... 2 1.3. Ohm Law s and Heat Energy... 4 1.4. The Kirchhoff Theorem... 8 PART 1. ELECTROSTATICS 13 2. Electrostatic Fields... 15 2.1. Static Charges and Their Interactions... 15 2.2. A Transient Current and Static Charges... 16 2.3. Uniform Electric Field in a Parallel-Plate Condenser... 19 2.3.1. The Electric Field Vector... 19 2.3.2. The Flux Density Vector... 21 2.4. Parallel and Series Connections of Capacitors... 25 2.5. Insulating Materials... 26 3. The Gauss Theorem... 30 3.1. A Spherical Capacitor... 30 3.2. A Cylindrical Capacitor... 33 3.3. The Gauss Theorem... 34 3.4. Boundary Conditions... 39 3.4.1. A Conducting Boundary... 39 3.4.2. A Dielectric Boundary... 40 4. The Laplace Poisson Equations... 43 4.1. The Electrostatic Potential... 43 4.2. The Gauss Theorem in Differential Form... 44 4.3. Curvilinear Coordinates (1)... 46 v

vi Contents 4.4. The Laplace Poisson Equations... 49 4.4.1. Boundary Conditions... 49 4.4.2. Uniqueness Theorem... 50 4.4.3. Green s Function Method... 51 4.5. Simple Examples... 53 4.6. The Coulomb Potential... 55 4.7. Point Charges and the Superposition Principle... 58 4.7.1. An Electric Image... 58 4.7.2. Electric Dipole Moment... 60 4.7.3. The Dipole-Dipole Interaction... 63 5. The Legendre Expansion of Potentials... 64 5.1. The Laplace Equation in Spherical Coordinates... 64 5.2. Series Expansion of the Coulomb Potential... 66 5.3. Legendre s Polynomials... 68 5.4. A Conducting Sphere in a Uniform Field... 69 5.5. A Dielectric Sphere in a Uniform Field... 71 5.6. A Point Charge Near a Grounded Conducting Sphere... 72 5.7. A Simple Quadrupole... 75 5.8. Associated Legendre Polynomials... 76 5.9. Multipole Potentials... 79 PART 2. ELECTROMAGNETISM 83 6. The Ampère Law... 85 6.1. Introduction... 85 6.2. The Ampère Law... 86 6.3. A Long Solenoid... 89 6.4. Stokes Theorem... 91 6.5. Curvilinear Coordinates (2)... 94 6.6. The Ampère Law in Differential Form... 96 6.7. The Rowland Experiment... 98 7. Magnetic Induction... 101 7.1. Laws of Magnetic Induction... 101 7.1.1. The Faraday Law... 101 7.1.2. The Lenz Law... 103 7.1.3. Magnetic Field Vectors... 103 7.2. Differential Law of Induction and the Dynamic Electric Field... 104 7.3. Magnetic Moments... 108 8. Scalar and Vector Potentials... 112 8.1. Magnets... 112 8.2. Pohl s Magnetic Potentiometer... 114

Contents vii 8.3. Scalar Potentials of Magnets... 116 8.3.1. A Laboratory Magnet... 116 8.3.2. A Uniformly Magnetized Sphere... 118 8.4. Vector Potentials... 119 8.5. Examples of Steady Magnetic Fields... 121 8.6. Vector and Scalar Potentials of a Magnetic Moment... 126 8.7. Magnetism of a Bohr s Atom... 128 9. Inductances and Magnetic Energies... 132 9.1. Inductances... 132 9.2. Self- and Mutual Inductances... 135 9.3. Mutual Interaction Force Between Currents... 138 9.4. Examples of Mutual Induction... 139 9.4.1. Parallel Currents... 139 9.4.2. Two Ring Currents... 140 10. Time-Dependent Currents... 142 10.1. Continuity of Charge and Current... 142 10.2. Alternating Currents... 143 10.3. Impedances... 145 10.4. Complex Vector Diagrams... 147 10.5. Resonances... 149 10.5.1. A Free LC Oscillation... 149 10.5.2. Series Resonance... 150 10.5.3. Parallel Resonance... 151 10.6. Four-Terminal Networks... 152 10.6.1. RC Network... 153 10.6.2. Loaded Transformer... 155 10.6.3. An Input-Output Relation in a Series RCL Circuit... 156 10.6.4. Free Oscillation in an RCL Circuit... 157 PART 3. ELECTROMAGNETIC WAVES 159 11. Transmission Lines... 161 11.1. Self-Sustained Oscillators... 161 11.2. Transmission Lines... 163 11.3. Fourier Transforms... 165 11.4. Reflection and Standing Waves... 167 11.5. The Smith Chart... 170 12. The Maxwell Equations... 172 12.1. The Maxwell Equations... 172 12.2. Electromagnetic Energy and the Poynting Theorem... 175 12.3. Vector and Scalar Potentials... 176

viii Contents 12.4. Retarded Potentials... 177 12.5. Multipole Expansion... 180 13. Electromagnetic Radiation... 184 13.1. Dipole Antenna... 184 13.2. Electric Dipole Radiation... 184 13.3. The Hertz Vector... 188 13.4. A Half-Wave Antenna... 192 13.5. A Loop Antenna... 193 13.6. Plane Waves in Free Space... 195 14. The Special Theory of Relativity... 199 14.1. Newton s Laws of Mechanics... 199 14.2. The Michelson Morley Experiment... 200 14.3. The Lorentz Transformation... 202 14.4. Velocity and Acceleration in Four-Dimensional Space... 204 14.5. Relativistic Equation of Motion... 206 14.6. The Electromagnetic Field in Four-Dimensional Space... 208 15. Waves and Boundary Problems... 214 15.1. Skin Depths... 214 15.2. Plane Electromagnetic Waves in a Conducting Medium... 216 15.3. Boundary Conditions for Propagating Waves... 218 15.4. Reflection from a Conducting Boundary... 219 15.5. Dielectric Boundaries... 221 15.6. The Fresnel Formula... 223 16. Guided Waves... 226 16.1. Propagation Between Parallel Conducting Plates... 226 16.2. Uniform Waveguides... 229 16.2.1. Transversal Modes of Propagation (TE and TM Modes)... 229 16.2.2. Transversal Electric-Magnetic Modes (TEM)... 232 16.3. Examples of Waveguides... 233 PART 4. COHERENT WAVES AND RADIATION QUANTA 241 17. Waveguide Transmission... 243 17.1. Orthogonality Relations of Waveguide Modes... 243 17.2. Impedances... 245 17.3. Power Transmission Through a Waveguide... 249 17.4. Multiple Reflections in a Waveguide... 250 18. Resonant Cavities... 253 18.1. Slater s Theory of Normal Modes... 253 18.2. The Maxwell Equations in a Cavity... 256

Contents ix 18.3. Free and Damped Oscillations... 258 18.4. Input Impedance of a Cavity... 260 18.5. Example of a Resonant Cavity... 263 18.6. Measurements of a Cavity Resonance... 265 19. Electronic Excitation of Cavity Oscillations... 268 19.1. Electronic Admittance... 268 19.2. A Klystron Cavity... 270 19.3. Velocity Modulation... 274 19.4. A Reflex Oscillator... 276 20. Dielectric and Magnetic Responses in Resonant Electromagnetic Fields 280 20.1. Introduction... 280 20.2. The Kramers Krönig Formula... 281 20.3. Dielectric Relaxation... 283 20.4. Magnetic Resonance... 288 20.5. The Bloch Theory... 290 20.6. Magnetic Susceptibility Measured by Resonance Experiments.. 292 21. Laser Oscillations, Phase Coherence, and Photons... 294 21.1. Optical Resonators... 294 21.2. Quantum Transitions... 296 21.3. Inverted Population and the Negative Temperature... 299 21.4. Ammonium Maser... 300 21.5. Coherent Light Emission from a Gas Laser... 301 21.6. Phase Coherence and Radiation Quanta... 302 APPENDIX 305 Mathematical Notes... 307 A.1. Orthogonal Vector Space... 307 A.2. Orthogonality of Legendre s Polynomials... 308 A.3. Associated Legendre Polynomials... 310 A.4. Fourier Expansion and Wave Equations... 312 A.5. Bessel s Functions... 314 REFERENCES 317 Index... 319

Preface The Maxwell theory of electromagnetism was well established in the latter nineteenth century, when H. R. Hertz demonstrated the electromagnetic wave. The theory laid the foundation for physical optics, from which the quantum concept emerged for microscopic physics. Einstein realized that the speed of electromagnetic propagation is a universal constant, and thereby recognized the Maxwell equations to compose a fundamental law in all inertial systems of reference. On the other hand, the pressing demand for efficient radar systems during WWII accelerated studies on guided waves, resulting in today s advanced telecommunication technology, in addition to a new radio- and microwave spectroscopy. The studies were further extended to optical frequencies, and laser electronics and sophisticated semi-conducting devices are now familiar in daily life. Owing to these advances, our knowledge of electromagnetic radiation has been significantly upgraded beyond plane waves in free space. Nevertheless, in the learning process the basic theory remains founded upon early empirical rules, and the traditional teaching should therefore be modernized according to priorities in the modern era. In spite of the fact that there are many books available on this well-established theme, I was motivated to write this book, reviewing the laws in terms of contemporary knowledge in order to deal with modern applications. Here I followed two basic guidelines. First, I considered electronic charge and spin as empirical in the description of electromagnetism. This is unlike the view of early physicists, who considered these ideas hypothetical. Today we know they are factual, although still unexplained from first principle. Second, the concept of fields should be in the forefront of discussion, as introduced by Faraday. In these regards I benefited from Professor Pohl s textbook, Elektrizitätslehre, where I found a very stimulating approach. Owing a great deal to him, I was able to write my introductory chapters in a rather untraditional way, an approach I have found very useful in my classes. In addition, in this book I discussed microwave and laser electronics in some depth, areas where coherent radiation plays a significant role for modern telecommunication. I wrote this book primarily for students at upper undergraduate levels, hoping it would serve as a useful reference as well. I emphasized the physics of electromagnetism, leaving mathematical details to writers of books on mathematical xi

xii Preface physics. Thus, I did not include sections for mathematical exercise, but I hope that readers will go through the mathematical details in the text to enhance their understanding of the physical content. In Chapter 21 quantum transitions are discussed to an extent that aims to make it understandable intuitively, although here I deviated from classical theories. Although this topic is necessary for a reader to deal with optical transitions, my intent was to discuss the limits of Maxwell s classical theory that arise from phase coherency in electromagnetic radiation. It is a great pleasure to thank my students and colleagues, who assisted me by taking part in numerous discussions and criticisms. I have benefited especially by comments from S. Jerzak of York University, who took time to read the first draft. I am also grateful to J. Nauheimer who helped me find literature in the German language. My appreciation goes also to Springer-Verlag for permission to use some figures from R. W. Pohl s book Elektrizitätslehre. Finally, I thank my wife Haruko for her encouragement during my writing, without which this book could not have been completed. M. Fujimoto September 2006