Physics of Classical Electromagnetism

Transcription

1 Physics of Classical Electromagnetism

2 Minoru Fujimoto Physics of Classical Electromagnetism

3 Minoru Fujimoto Department of Physics University of Guelph Guelph, Ontario Canada, N1G 2W1 Library of Congress Control Number: ISBN: e-isbn: 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 springer.com

4 Contents Preface... xi 1. Steady Electric Currents Introduction Standards for Electric Voltages and Current Ohm Law s and Heat Energy The Kirchhoff Theorem... 8 PART 1. ELECTROSTATICS Electrostatic Fields Static Charges and Their Interactions A Transient Current and Static Charges Uniform Electric Field in a Parallel-Plate Condenser The Electric Field Vector The Flux Density Vector Parallel and Series Connections of Capacitors Insulating Materials The Gauss Theorem A Spherical Capacitor A Cylindrical Capacitor The Gauss Theorem Boundary Conditions A Conducting Boundary A Dielectric Boundary The Laplace Poisson Equations The Electrostatic Potential The Gauss Theorem in Differential Form Curvilinear Coordinates (1) v

5 vi Contents 4.4. The Laplace Poisson Equations Boundary Conditions Uniqueness Theorem Green s Function Method Simple Examples The Coulomb Potential Point Charges and the Superposition Principle An Electric Image Electric Dipole Moment The Dipole-Dipole Interaction The Legendre Expansion of Potentials The Laplace Equation in Spherical Coordinates Series Expansion of the Coulomb Potential Legendre s Polynomials A Conducting Sphere in a Uniform Field A Dielectric Sphere in a Uniform Field A Point Charge Near a Grounded Conducting Sphere A Simple Quadrupole Associated Legendre Polynomials Multipole Potentials PART 2. ELECTROMAGNETISM The Ampère Law Introduction The Ampère Law A Long Solenoid Stokes Theorem Curvilinear Coordinates (2) The Ampère Law in Differential Form The Rowland Experiment Magnetic Induction Laws of Magnetic Induction The Faraday Law The Lenz Law Magnetic Field Vectors Differential Law of Induction and the Dynamic Electric Field Magnetic Moments Scalar and Vector Potentials Magnets Pohl s Magnetic Potentiometer

6 Contents vii 8.3. Scalar Potentials of Magnets A Laboratory Magnet A Uniformly Magnetized Sphere Vector Potentials Examples of Steady Magnetic Fields Vector and Scalar Potentials of a Magnetic Moment Magnetism of a Bohr s Atom Inductances and Magnetic Energies Inductances Self- and Mutual Inductances Mutual Interaction Force Between Currents Examples of Mutual Induction Parallel Currents Two Ring Currents Time-Dependent Currents Continuity of Charge and Current Alternating Currents Impedances Complex Vector Diagrams Resonances A Free LC Oscillation Series Resonance Parallel Resonance Four-Terminal Networks RC Network Loaded Transformer An Input-Output Relation in a Series RCL Circuit Free Oscillation in an RCL Circuit PART 3. ELECTROMAGNETIC WAVES Transmission Lines Self-Sustained Oscillators Transmission Lines Fourier Transforms Reflection and Standing Waves The Smith Chart The Maxwell Equations The Maxwell Equations Electromagnetic Energy and the Poynting Theorem Vector and Scalar Potentials

7 viii Contents Retarded Potentials Multipole Expansion Electromagnetic Radiation Dipole Antenna Electric Dipole Radiation The Hertz Vector A Half-Wave Antenna A Loop Antenna Plane Waves in Free Space The Special Theory of Relativity Newton s Laws of Mechanics The Michelson Morley Experiment The Lorentz Transformation Velocity and Acceleration in Four-Dimensional Space Relativistic Equation of Motion The Electromagnetic Field in Four-Dimensional Space Waves and Boundary Problems Skin Depths Plane Electromagnetic Waves in a Conducting Medium Boundary Conditions for Propagating Waves Reflection from a Conducting Boundary Dielectric Boundaries The Fresnel Formula Guided Waves Propagation Between Parallel Conducting Plates Uniform Waveguides Transversal Modes of Propagation (TE and TM Modes) Transversal Electric-Magnetic Modes (TEM) Examples of Waveguides PART 4. COHERENT WAVES AND RADIATION QUANTA Waveguide Transmission Orthogonality Relations of Waveguide Modes Impedances Power Transmission Through a Waveguide Multiple Reflections in a Waveguide Resonant Cavities Slater s Theory of Normal Modes The Maxwell Equations in a Cavity

8 Contents ix Free and Damped Oscillations Input Impedance of a Cavity Example of a Resonant Cavity Measurements of a Cavity Resonance Electronic Excitation of Cavity Oscillations Electronic Admittance A Klystron Cavity Velocity Modulation A Reflex Oscillator Dielectric and Magnetic Responses in Resonant Electromagnetic Fields Introduction The Kramers Krönig Formula Dielectric Relaxation Magnetic Resonance The Bloch Theory Magnetic Susceptibility Measured by Resonance Experiments Laser Oscillations, Phase Coherence, and Photons Optical Resonators Quantum Transitions Inverted Population and the Negative Temperature Ammonium Maser Coherent Light Emission from a Gas Laser Phase Coherence and Radiation Quanta APPENDIX 305 Mathematical Notes A.1. Orthogonal Vector Space A.2. Orthogonality of Legendre s Polynomials A.3. Associated Legendre Polynomials A.4. Fourier Expansion and Wave Equations A.5. Bessel s Functions REFERENCES 317 Index

9 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

10 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