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EngineeringElectrodynamics House of Maxwell’s Electrodynamics

For Alla, my wife, mylove, and tirelesssupporterFor Inna and Greg,for their great help Vladimir

EngineeringElectrodynamicsHouse of Maxwell’sElectrodynamicsVladimir I. VolmanAndrzej Jeziorski

ContentsPrefaceCHAPTER 1BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICSIntroduction1.1 MACROSCOPIC ELECTRODYNAMICS1.1.1 Duality of Electromagnetic Waves1.1.2 Vector and Scalar Fields1.2 FUNDAMENTAL PRINCIPLES OF ELECTRODYNAMICS1.2.1 Symmetry in Nature and Conservation Laws1.2.2 Conservation Laws1.2.3 Nothing Exists Until It Is Measured1.3 INTERNATIONAL METRIC SYSTEM OF UNITS (SI)1.3.1 Nothing Exists Until It Is Defined and Measured in Units1.3.2 Derived SI Units1.3.3 Dimensional Analysis and Unit Law1.3.4 Table of Mathematical Operators in Use1.4 EM FIELD SENSORS1.4.1 Electric Monopole, Dipole, and Current Element as Field Sensors1.4.2 Electric Current and its Volume Density1.4.3 Charge Volume Density1.4.4 Magnetic Sensors1.5 HOUSE OF MAXWELL’S ELECTRODYNAMICS1.5.1 Introduction1.5.2 Lorentz’s Force Equation (Axiom #1)1.5.3 Maxwell’s House of Electrodynamics1.6 ELECTRIC AND MAGNETIC FIELD VECTORS1.6.1 Vector of Electric Field Strength1.6.2 Electric Potential1.6.3 Line of Force1.6.4 Gauss’s Law for Electric Fields (Axiom #2) and Coulomb’s Law1.6.5 Is The Inverse-Square Relation Imperative?1.6.6 How Much Is One Coulomb (C)?1.6.7 Electric Field Reality1.6.8 Displacement Vector D. 3rd Maxwell’s Equation1.6.9 Why Do We Need Extra D-Vector Describing E-Fields?1.6.10 Electric Charge Conservation Law in Differential Form1.6.11 Lorentz’s Force Equation and 1st Maxwell’s Equation1.6.12 Is Magnetic Inductance Real and Can Be Measured?

CONTENTS1.6.13 Gauss’s Law for Magnetic Field (Axiom #3). 4th Maxwell’s Equation1.6.14 Magnetic Lines of Force1.6.15 Vector of Magnetic Field Strength. 2nd Maxwell’s Equation1.6.16 Net Current Continuity and 2nd Maxwell’s Equation1.6.17 Electric and Magnetic Energy1.7 ELECTRIC CURRENT AND CHARGES AS SOURCES OF EM FIELDS1.7.1 Currents and Charges as Sources of EM Fields1.7.2 Convection Charges and Currents1.8 MAXWELL’s EQUATIONS IN PHASOR FORM1.8.1 Time and Frequency Domain1.8.2 Maxwell’s Equations in Phasor FormREFERENCESCHAPTER 2NEOCLASSICAL THEORY OF INTERACTION OF ELECTRICAND MAGNETIC FIELDS WITH MATERIAL MEDIAIntroduction2.1 TORQUE EXERTED BY ELECTRIC AND MAGNETIC FIELD2.1.1 Mechanical Torque Examples2.1.2 Torque Exerted by Magnetic Field2.1.3 Torque Exerted by Electric Field2.2 PHENOMENA OF ELECTRIC AND MAGNETIC POLARIZATION. ELECTRICAL CONDUCTANCE2.2.1 Phenomena of Electric Polarization2.2.2 Polarization Vector. Permittivity and Dielectric Constant2.2.3 Dielectric Constant of Composite Materials2.2.4 Dielectric Constant of Anisotropic Materials2.2.5 Phenomena of Magnetic Polarization2.2.6 Phenomena of Electric Conductance2.2.7 Polarized Conductive Body in Electric Fields2.2.8 Ohm’s Law2.2.9 Per Square Resistance2.3 BOUNDARY CONDITIONS2.3.1 Introduction2.3.2 Boundary Conditions on Normal Components of Field2.3.3 Boundary Conditions for Tangential Components of Field2.3.4 Dielectric-Dielectric Interface2.3.5 Dielectric-Perfect Electrical Conductor (PEC) Interface2.3.6 Superconductors2.4 CLASSIFICATION OF MATERIALS BASED ON THEIR2.4.1 ELECTRICAL AND MAGNETIC PROPERTY Complex-Valued Dielectric and Magnetic Constant

CONTENTS2.4.2 Classification of Materials Based on Their Electrical Property2.4.3 Linearity and Nonlinearity2.5 BROADBAND COMPLEX-VALUED MATERIAL PARAMETERS2.5.12.5.2 Introduction2.5.3 Drude-Lorentz’s Model of Metal Dielectric Constant2.5.4 Ionospheric Plasma with Negative Dielectric Constant Broadband Complex Constant ������������������������(������������) of Dielectrics2.6 FERRO-MATERIALS2.6.1 Introduction2.6.2 Basic Description of Ferro-Materials2.6.3 Ferromagnetics2.6.4 Ferrimagnetic and Ferrites2.6.5 Complex Magnetic Constant of Non-Magnetized Ferrite2.6.6 Ferroelectrics2.7 EM FIELDS IN MAGNETIZED FERRITES2.7.1 Introduction2.7.2 Free Precession in Fully Magnetized Ferrite2.7.3 Force Precession in Fully Magnetized Ferrite2.7.4 Permeability of Fully Magnetized Ferrite2.8 METAMATERIALS2.8.1 Introduction2.8.2 Negative Permittivity and Permeability2.9 GRAPHENE2.9.1 Introduction2.9.2 Graphene as a Conductor2.9.3 Conductivity of Magnetically Biased Graphene2.9.4 Graphene as Shielding Material2.9.5 Conductive Graphene NanoRibbon (GNR) Thin Film as Deicing Coating2.9.6 Graphene-Based Electromechanical Switch2.10 SOME ADDITIONAL PROPERTIES OF MATERIALS2.10.1 Kramers-Kronig (K-K) Relations2.10.2 Eliminating Negative Frequencies in the K-K Relations2.10.3 Remote Sensing and K-K Relations2.10.4 Eddy Current2.10.5 Eddy Current in Power Transformer CoreREFERENCECHAPTER 3POYNTING’s THEOREMIntroduction3.1 ELECTROMAGNETIC FIELD CONSERVATION LAWS

CONTENTS3.1.1 Conservation of Energy in Space-Time Domain3.1.2 Power Delivered by Excitation Currents3.1.3 Voltage, Current and Power Loss3.1.4 Power Stored in Electromagnetic Fields3.1.5 Electromagnetic Power Flux and Poynting Vector3.1.6 Velocity of EM Waves Energy Transportation3.1.7 Linear Momentum of EM Fields. Radiation Pressure and Solar Sailing3.1.8 Angular Momentum of EM Fields. Polarization. Twisted EM Waves3.1.9 Collecting the Results3.1.10 Poynting Theorem and Circuit Analysis3.1.11 Concept of Capacitance3.1.12 Concept of Inductance3.1.13 Parasitic Parameters3.1.14 Self - Resonances in Capacitor and Solenoid3.1.15 Why did We Pay so Much Attention to the Lumped Circuit Elements?3.1.16 Poynting Theorem in Space-Frequency Domain3.2 UNIQUENESS THEOREM FOR INTERIOR ELECTROMAGNETICS PROBLEMS3.2.1 Necessary of Uniqueness Theorem3.2.2 Uniqueness Theorem in Space-Time Domain3.2.3 Uniqueness Theorem in Space-Frequency Domain3.2.4 Cavity Resonators3.2.5 Quality Factor Q of Cavity Resonator3.3 UNIQUENESS THEOREM FOR EXTERIOR ELECTROMAGNETICS PROBLEMS3.3.1 Radiation Condition3.3.2 Edge Boundary Condition3.3.3 Influence of Conductive Surface Curvature on Electric Charge and Current Distribution3.3.4 Field Electron Emission3.3.5 How to Treat Problem of Field Singularities in Numerical Simulation?3.4 REFLECTION CONCEPT. LORENTZ’s RECIPROCITY THEOREM3.4.1 Concept of Reflection and Impedance3.4.2 Foster's Reactance Theorem3.4.3 Lorentz’s Reciprocity Theorem3.4.4 Receive-Transmit Antenna Reciprocity3.4.5 Ultra-WideBand (UWB) Antenna Impulse Response (Response in the Time Domain)3.4.6 Reciprocity and Antenna Radiation Pattern MeasurementREFERENCESCHAPTER 4SOLUTION OF BASIC EQUATIONS OF ELECTRODYNAMICSIntroduction

CONTENTS4.1 WAVE AND HELMHOLTZ’s EQUATIONS4.1.1 Wave Equation for Electric and Magnetic Vectors in Space-Time Domain4.1.2 Wave Equation in Space-Frequency Domain. Wavenumber4.1.3 Electrodynamic Potentials in Space-Time Domain4.1.4 Is it Vector and Scalar Potentials are Real?4.1.5 Symmetry of Maxwell’s Equations and Principle of Duality. Electrodynamic Potentials for Magnetic Sources4.1.6 Electrodynamic Potentials in Space-Frequency Domain4.1.7 Green’s Function for Static Fields for Unbounded Space4.1.8 Wave Equation. One-Dimensional Unbounded Space4.1.9 General Solution of Wave Equations and Green’s Function4.1.10 Potentials and Green’s Function in Space-Frequency Domain4.2 RADIATION OF ELECTROMAGNETIC WAVES4.2.1 Introduction4.2.2 Radiation EM Waves by Infinitesimal Current Element4.3 ELEMENTARY RADIATORS4.3.1 Electric and Magnetic Fields Emitted by Infinitesimal Current Element4.3.2 Electrically Small Current Loop4.3.3 Loop Antenna as Magnetic Dipole4.3.4 Huygens' Principle and Huygens’ Radiator4.4 CLASSIFICATION OF MATERIALS BASED ON THEIR ELECTRICAL AND MAGNETIC PROPERTY4.4.1 Skin Effect in Conductive Materials. Impact of Surface Roughness4.4.2 Surface Resistivity4.4.3 ConclusionREFERENCECHAPTER 5ANTENNA BASICSIntroduction5.1 EM WAVE POLARIZATION5.1.1 Classification of Common Polarization5.1.2 Co- and Cross-Polarization5.1.3 Twisted EM Waves5.1.4 How Can Antenna Polarization Be Chosen?5.2 ANTENNA PARAMETERS5.2.1 Introduction5.2.2 Radiation Resistance and Lumped Equivalent Circuit of Antenna5.2.3 Return Loss5.2.4 Antenna Quality Factor (Q factor), Bandpass and Radiation Efficiency5.2.5 Near-field Zone vs. Far-field Zone

CONTENTS5.2.6 Radiation Pattern. Main Beam, Beamwidth, and Sidelobes5.2.7 Sidelobes Specification. Grating Lobes5.2.8 Antenna Noise Temperature5.2.9 TEM Waves in Far-Field Zone5.2.10 Directivity and Gain5.2.11 Antenna Effective Aperture5.2.12 Directivity, Effective Aperture, and HPBW5.2.13 G/T Parameter5.2.14 Antenna Factor5.2.15 Antenna Power Handling5.3 SYSTEM REQUIREMENTS AND ANTENNA GAIN5.3.1 Introduction5.2.2 Path Loss (Friis Transmission Formula) and EIRP5.3.3 Monostatic Radar Equation5.3.4 Bistatic Radar Equation5.3.5 Multiple Input Multiple Output (MIMO) Radar System5.4 DECOMPOSITION or ‘DIVIDE and CONQUER’ TACTIC5.5 ANTENNA ARRAY FACTOR AND MAGNETIC PROPERTY5.5.1 Introduction5.5.2 Basic of Linear Array Analysis5.5.3 Pattern Multiplication5.5.4 Basic of Linear Array Synthesis5.5.5 Phasor-Vector Interpretation of Array Pattern5.5.6 Linear Arrays with Progressive Phase Distribution and Their Feed5.5.7 Radiation of Linear Array with Progressive Phase Excitation5.5.8 Continuous Linear Array5.6 BEAM STEERING TECHNIQUES5.6.1 Introduction5.6.2 Linear Array Beam Steering5.6.3 Grating Lobe vs. Beam Steering5.6.4 True Time Delay (TTD) Steering5.6.5 Frequency Scan5.6.6 Within-Pulse Scan Technique5.6.7 Synthetic Aperture Radar (SAR)5.6.8 Linear Array with Multiple Simultaneous Beams5.7 PLANAR AND CONFORMAL ARRAYS5.7.1 Planar Arrays5.7.2 Conformal Arrays5.7.3 Effect of Beam FocusingREFERENCESCHAPTER 6FEED LINE BASICIntroduction

CONTENTS6.1 FEED LINE CHARACTERISTICS6.1.1 TEM Mode6.1.2 Line Impedance6.1.3 Concept of Cutoff Wavelength / Frequency6.1.4 Power Handling6.1.5 Attenuation6.2 OPEN LINES6.2.1 Open line Definition6.2.2 Wire Lines6.2.3 Strip Lines6.3 FIBER OPTIC LINES6.3.1 Introduction6.3.2 Fiber Optic Line Family6.3.3 Hollow-Core Photonic Crystal Fiber6.3.4 Optical Waveguides6.4 CLOSED LINES6.4.1 Introduction6.4.2 Coaxial Lines6.4.3 Waveguide Rectangular (WR)6.4.4 Waveguide Circular (WC)6.4.5 Waveguide of Ridge Double (WRD)6.4.6 Corrugated Elliptical Waveguide6.4.7 Finline Waveguides6.5 BASIC OF LINE THEORY6.5.1 Lumped Circuit Model of a Transmission Line6.5.2 Wave Equations and Boundary Conditions6.6 MORE INFORMATION ABOUT FEED LINES6.6.1 Introduction6.6.2 Two-Wire Line6.6.3 Coaxial Line6.6.4 Waveguide Rectangular6.6.5 Waveguide Circular6.6.6 Symmetric Stripline6.6.7 Microstrip6.6.8 Slotline6.6.9 Coplanar Waveguide (CPW) and Grounded CPW (GCPW)6.7 FEED TRANSITIONS / INTERCONNECTIONS6.7.1 Introduction6.7.2 Coax-to-Coax Transition6.7.3 Coax-to-Waveguide Transition / Adapter6.7.4 Coax-to-Microstrip Inline Mount Adapter6.7.5 Coax-to-Coplanar Waveguide Inline Adapter6.7.6 Vertically Mounted (Right-Angle) Coaxial Transitions

CONTENTS6.7.7 Rotary Joint6.8 PROPAGATION EM WAVES IN FERRITE LOADED LINES6.8.1 Introduction6.8.2 Faraday Rotation6.8.3 Faraday Rotation Isolator6.8.4 Phase Shifter6.8.5 Resonance Isolators6.8.6 Effect of Field Displacement6.8.7 Y-CirculatorREFERENCESCHAPTER 7DISCONTINUITY IN FEED LINESIntroduction7.1 COAXIAL DISCONTINUES7.1.1 Dielectric Beads Supporting Center Conductor7.1.2 Step Up in Coaxial line7.1.3 Open-Ended Coaxial Line7.1.4 Gap in Center Conductor7.1.5 Coaxial Junction7.1.6 Coaxial Stub Discontinuities7.2 DISCONTINUITIES IN PLANAR LINES7.2.1 Introduction7.2.2 Primary Discontinuities7.2.3 Waveguide Discontinuities7.3 SCATTERING MATRIX AND RF MULTI-PORTS CIRCUIT EVALUATION7.3.1 Introduction7.3.2 Generalized Scattering (S) Matrix7.3.3 Return and Insertion Loss7.3.4 Scattering Transfer T-Matrix7.3.5 Z- and Y-matrix7.3.6 S-Matrix of Complex NetworkREFERENCESCHAPTER 8MORE COMPLICATED ELEMENTS OF FEED LINESIntroduction8.1 IN-LINE RESONATORS8.1.1 Outline8.1.2 Basic of In-line Resonator. Bounce Diagram

CONTENTS8.2 DIRECTIONAL COUPLES AND HYBRIDS8.2.1 Introduction8.2.2 WR Discrete Directional Coupler8.2.3 Continuous Directional Coupler8.2.4 WR Hybrids8.2.5 WR Ring Hybrid8.2.6 WR Short-Slot Hybrid (Riblet Hybrid)8.2.7 Microstrip Branch Hybrid8.2.8 Microstrip Ring Hybrid8.2.9 Wilkinson Power Divider8.3 DIRECTIONAL COUPLER AND HYBRID APPLICATIONS8.3.1 Signal Flow Measurements8.3.2 Calibration and Test Units8.3.3 Power Leveling8.3.4 Distributed Antenna System (DAS)8.3.5 Power Combiner/Splitter Networks8.3.6 Butler Matrix (Beam Forming Network)8.3.7 Monopulse Concept8.3.8 Radar Receiver Protection8.3.9 Frequency Multiplexer8.4 ANALOGUE FILTERS8.4.1 Overview8.4.2 Normalized Low-pass Filter. Frequency Transformation8.4.3 Filter Phase Characteristics. Time Delay8.4.4 K - and J - Immittance Inverters8.4.5 Quarter-Wavelength Section of Feed Line as Inverter8.4.6 Filters with Direct-Coupled Resonators8.4.7 Coupled Line or Distributed Filters8.4.8 Combline and Interdigital Filter8.4.9 Evanescent-Mode Filters8.4.10 Surface Acoustic Wave (SAW) Filters8.4.11 Filter Selection Trade-off8.4.12 Optical FiltersREFERENCESCHAPTER 9APPROACH TO NUMERICAL SOLUTION OFELECTRODYNAMICS PROBLEMSIntroduction9.1 BASIC OF COMPUTER DESIGN9.1.1 Design, Analyze, Build9.1.2 Basic Numerical Methods in Computational Electrodynamics (CEM)9.1.3 Perfectly Matched Layer (PML)

CONTENTS9.2 GRID AND CLOUD COMPUTING9.2.1 Introduction9.2.2 Parallel FDTD Technique9.2.3 Parallel Processing9.2.4 GPU and Cache AccelerationREFERENCESAPPENDIXINDEX

Preface If you can’t explain it simply, you don’t understand it well enough. Albert Einstein Why don’t you write books people can read? Nora Joyce to her husband, James JoycePREFACEWelcome to “House of Maxwell’s Electrodynamics” or even better, “Kingdom” ofElectrodynamics. Like any house, it has a solid foundation made of four “all-inclusive”equations discovered by James Clerk Maxwell in 1861. Over the following 150 years, auniverse was erected on this simple basis. It helped to establish the unique relationship betweenelectric and magnetic fields, electromagnetic field interactions with the surrounding world, andthe possibility to carry energy and information through space. Maxwell’s equations “govern”the electromagnetic processes in our body, “make” modern computers intelligent, “deliver” theelectrical power to our home and elsewhere around the world, and “provide” all satellite, wire,and wireless communications, internet connections, etc. It would not be an exaggeration to saythat the current state of civilization is a broad “solution” of Maxwell’s equations.The primary intent of this book is to help readers reach a deeper understanding of thefundamentals of electrodynamics, show how to use this knowledge in a challenging world ofpractical engineering and teach the corresponding skills. We are not going to provide a fullcourse of modern electrodynamics, as that would take many and many volumes. Our relativelynarrow goal is to give our reader a basic understanding of electrodynamics that makes him/hercapable of solving modern engineering problems for a reasonable period of time.If you are looking for ideas on how to shorten your way to a practical solution to yourengineering problems in a time and money constrained environment or you would simply liketo get the most out of your study in continuous education – you will see that this book for you.How This Book Is OrganizedThis book is written by practicing engineers for practicing engineers and university students ofall levels planning to incorporate electrodynamics into their professional toolbox. Over manyyears in the engineering field and teaching electrodynamics, the authors have observed thatstudents and engineers are very results oriented and prefer to avoid extensive mathematicalmanipulations. And they are basically right because now the numerical modeling ofelectromagnetic problems by commercially available and user-friendly software alleviatesmuch of the need for in-depth-mathematical-proficiency. Following this trend, we decided tolimit the complexity of the mathematics in this book. We are focused primarily on readersachieving an intuitive grasp of the material through physical analogies, unit dimension analysis,simplified models, etc. We ask our readers to give us some credit for it not judging us severely.There is no doubt that electrodynamics and almost everything connected to electromagneticconcepts is challenging for understanding, but be patient and spend some extra time and effortsto succeed.The crucial ideas unifying different elements of this book are Danish physicist Niels Bohr’squotation “Nothing exists until it is measured” and the two fundamental laws of physics: (1)the Charge Conservation Law, which states that charges can neither be created nor destroyedin any isolated system, and (2) the Electromagnetic Energy Conservation law, which that in anisolated system the total amount of any kind of energy can be neither created nor lost. This

Prefaceapproach allows reaching the definition of electromagnetic fields through the energy they carry.In other words, electric and magnetic fields become directly accessible to experimentalobservation. Moreover, it paves the way to Maxwell’s equations with minimal mathematics andsubsequently the presentation of Lorentz’s force equation and Gauss’s conservation laws forelectric and magnetic charges.The Contents at a GlanceWe believe that one can learn more from a well-thought-out example than from reading a dozenpages in a book. To make the most out of our book, we advise you to install on your computer,as a minimum, the student version of MATLAB® and CST STUDIO SUITE® tools.The book is organized into 9 chapters and an appendix with short reference material. It can beconditionally divided into two parts. Part I (Chapter 1, 2, 3, and 4) is preliminary and devotedto the classical and neoclassical theory of electromagnetism. The “raison d’etre1” of the bookis in Part II (Chapter 5, 6, 7, 8, and 9) connecting the theory with engineering applications.Chapter 1, Basic Equations of Macroscopic Electrodynamics, provides an introduction to theworld of macroscopic electrodynamics and its fundamental principles based on the symmetryin nature and conservation laws. Most of this chapter is devoted to exploring Lorentz’s forceequation, all four of Maxwell’s equations, and building so-called House of Maxwell’sElectrodynamics with attic and basement.Chapter 2, Neoclassical Theory of Interaction of Electric and Magnetic Fields with MaterialMedia, covers an important practical aspect of such interactions in the conductive, natural andartificial dielectric, ferro-, and metamaterials, graphene, etc. The analysis is primarily based onneoclassical Drude-Lorentz’s models. Chapter constitutes a discussion of boundary conditions,material classification based on their electrical parameters, Kramers-Kronig (K-K) Relations,connected to them ideas of remote sensing, and Eddy current.Chapter 3, Poynting’s Theorem, is one of the central topics in the book. The objectives of thischapter are to present information you need to formulate electromagnetic problems uniquelyand prepare them for analytical or numerical analysis. Particular attention is paid to theassociation between Poynting’s theorem and traditional circuit analysis that allows convertingthe high theoretical electrodynamics into a powerful and intuitive tool for engineering design.Chapter 4, Solution of Basic Equations of Electrodynamics, is quite a math saturated andintroduces the reader to the world of Maxwell’s equations solution. Don't fear – we will try toavoid unnecessary rigor wherever possible. The central topic of this chapter is the theory ofelectromagnetic potentials in space-time and space-frequency domain that opens the door to theconcept of electromagnetic wave radiation and propagation. The obtained solutions are appliedto the family of such elementary radiators as electric, magnetic, and Huygens’. The electric andmagnetic fields induced by them are illustrated by manifold three-dimensional images that givea clear view of their radiation pattern formation in space. The chapter concludes with adiscussion of the skin effect in highly conductive materials and its engineering aspect like theimpact of surface roughness on the attenuation factor.Chapter 5, Antenna Basic, continues the study of EM wave radiation and devotes to such topicsas EM wave polarization including twisted waves, a broad range of antenna parameters and1 The “raison d’etre” may be translated from French as the most important reason or purpose forexistence (https://en.oxforddictionaries.com/definition/raison_d'%C3%AAtre).

Prefacetheir impact on system advance, pattern analysis and synthesis of a different type of antennas,beam steering, and focusing techniques, planar and conformal arrays. Section 5.3.6 is devotedto currently “hot topic” of massive MIMO for 5G (fifth-generation) wireless network.Chapter 6, Feed Line Basic, emphasizes the engineering aspect of a broad range of open andclosed feed lines starting from their elementary theory and fundamental characteristics. Thediscussion includes the different types of transition between lines, ferrite devices, and manyother topics of engineering importance.Chapter 7, Discontinuity in Feed Lines, and Chapter 8, More Complicated Elements of FeedLines, present the ancillary material necessary for engineering design of a wide variety ofmicrowave components like power combiners/splitters, filters, limiters, multiplexers, etc. Theparticular attention is paid to their applications.Chapter 9, Approach to Numerical Solution of Electrodynamics Problems, is a shortintroduction to the modern world of computer analysis.What You Need to KnowThe book is suitable for first-year graduate or senior undergraduate students, and can also beused by practicing engineers who want a quick review covering most of the basic concepts inelectromagnetics and includes many application examples. This is not a book for beginners. Weassume that the reader had a knowledge of the basics of classical mechanics, fundamentals ofelectric and magnetic phenomena and completed a basic course on linear electrical circuitanalysis. Basics of Calculus with Fourier analysis and ordinary differential equations are alsoessential.What You Can Get from the BookWe hope that after completing this book’s course, our readers will learn fundamentals ofelectrodynamics and how to apply the theory to 1. Solving a wide variety of engineering problems in a cost-, time-, and quality- constrained environment. 2. Creating numerical models with some set of parameters that adequately reproduce physical processes in a real device. 3. Selecting appropriate numerical simulation tools and correctly applying it to solving a given a problem. Checking that all solution uniqueness conditions are in place. 4. Verifying the obtaining numerical results using physical observations in combination with intuition, creativity, and well-known analytical, semi-analytical, or measurement data. 5. And much more depending on the reader background and interests.‘Aha!’ moment in ElectrodynamicsFor the last two centuries, numerous books and almost infinite number of papers have beenwritten on the different aspects of the electromagnetic theory. Electromagnetic theory tests theboundary of our imagination the same way as music, abstract painting and sculpture do.What kind of enjoyment can be found in highly theoretical electromagnetic science?Students studying electromagnetic phenomena at schools and universities are sometimesconfused and overwhelmed. Did I miss or misunderstand something critical and significant?

PrefaceWhere is my ‘Aha!’ moment of the extraordinary? Do not worry. The beauty of Maxwell’sequations is that they are simple, and we will help you learn and understand them. In the 20thcentury, the major engineering discoveries and ‘Aha!’ moments shifted to unique solutions forengineering problems, innovations, and design of superior to existing products. Plenty of easy-to-use commercial software packages transforms electrodynamics into a user-friendlyengineering tool for the numerical solution of everyday tasks. Modern computers enormouslyintensify our intellectual power and permit us to check and validate at first glance crazy ideasand bring them to life.How to Use This BookYou can use this book any way that you please. If you choose to read it from cover to cover, beour guest. Experienced readers can skip around, picking up useful tidbits here and there. If youare faced with a challenging task, you might try the contents and index first to see whether thebook specifically addresses your problem.Research has shown that roughly half of the human cerebral cortex that facilitates our learning,creates thought, expression, and behavior is solely devoted to visual processing. We are allvisual learners and to us “seeing is believing” because a picture may tell a thousand words. Soto foster active learning, we extensively used a wide variety of images and diagrams including3D plots for illustration purposes and as a source of new information. The most of this graphicmaterial is original, but we reproduced a lot of images from Internet and are grateful tonumerous individuals and companies for their permission to reproduce certain figuresacknowledged in the book. Please forgive us if we missed some references and do not judge usstrictly.We will be delighted to get your opinion, suggestions, and comments that let improve any futureedition. If you have some queries, do not hesitate and send email through our websitehttp://www.AcknowledgmentWe enjoyed the company and help of many people during the research, writing, and editing ofthis book. Sincere gratitude to Dr. Leo G. Maloratsky; Dr. Victor Katok, UkrTelecom; RudyFuks and Joseph Carson, HUBER+SUHNER Astrolab; Dr. Pablo Soto, Universidad Politecnica deValencia; Dr. Yuriy Shlepnev, Simberian Inc.; Dr. Abdul-Rahman O. Raji, University ofCambridge for their critical examinations of various parts of the book and the many constructivesuggestions to improve the book. Three fine editors somehow made sense of the whole messand to them, the greatest debt of thanks is owed: Alla Volman, Inna Plumb and Greg Plumb.We are indebted to Dr. Martin Timm, Computer Simulation Company (CST, a DassaultSystèmes company), who provided a free license for CST STUDIO SUITE® that helps us torealize the multiple computer-oriented simulations included in the book. We hope to continueour cooperation developing CD-ROM containing multiple projects based on CST STUDIOSUITE® . Such CD-ROM will be sent for free to our readers who bought the book and registeredon our website. Besides, all material and textbook electronic version (eBook) will be put athttp://www.We express our sincere gratitude to Dr. Edward Sedek, Telecommunications Research Institute,Poland and Dr. Yevhen Yashchyshyn, Warsaw University of Technology, Poland who took outtime to review the manuscript with many helpful comments.Have fun reading and a good time while learning!



CHAPTER 1 BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICSIf you want to find the secrets of the universe, think in terms of energy. Nikola Tesla Nothing exists until it is measured. Niels Bohr

Chapter 1 Chapter ContentsIntroduction 1.6.3 Line of Force1.1 MACROSCOPIC 1.6.4 Gauss’s Law for Electric ELECTRODYNAMICS Fields (Axiom #2) and1.1.1 Duality of Electromagnetic Coulomb’s Law 1.6.5 Is The Inverse-Square Fields Relation Imperative?1.1.2 Waves Vector and 1.6.6 How Much Is One Coulomb (C)? Scalar Fields 1.6.7 Electric Field Reality1.2 FUNDAMENTAL 1.6.8 Displacement Vector D. 3rd Maxwell’s Equation PRINCIPLES OF 1.6.9 Why Do We Need Extra ELECTRODYNAMICS D-Vector Describing1.2.1 Symmetry in Nature and E-Fields? 1.6.10 Electric Charge Conservation Laws Conservation Law in1.2.2 Conservation Laws Differential Form1.2.3 Nothing Exists Until 1.6.11 Lorentz’s Force Equation and 1st Maxwell’s Equation It Is Measured 1.6.12 Is Magnetic Inductance Real1.3 INTERNATIONAL and Can Be Measured? 1.6.13 Gauss’s Law for Magnetic METRIC SYSTEM OF Field (Axiom #3). 4th UNITS (SI) Maxwell’s Equation1.3.1 Nothing Exists Until It Is 1.6.14 Magnetic Lines of Force 1.6.15 Vector of Magnetic Field Defined and Measured Strength. 2nd Maxwell’s in Units Equation1.3.2 Derived SI Units 1.6.16 Net Current Continuity and1.3.3 Dimensional Analysis and 2nd Maxwell’s Equation Unit Law 1.6.17 Electric and Magnetic Energy1.3.4 Table of Mathematical 1.7 ELECTRIC CURRENT AND Operators in Use CHARGES AS SOURCES OF1.4 EM FIELD SENSORS EM FIELDS1.4.1 Electric Monopole, Dipole, 1.7.1 Currents and Charges as and Current Element as Sources of EM Fields Field Sensors 1.7.2 Convection Charges and1.4.2 Electric Current and its Currents Volume Density 1.8 MAXWELL’s EQUATIONS IN Charge Volume Density PHASOR FORM1.4.3 Magnetic Sensors 1.8.1 Time and Frequency Domain1.5 MAXWELL’S HOUSE Maxwell’s Equations in Phasor Form1.5.1 Introduction1.5.2 Lorentz’s Force Equation REFERENCES (Axiom #1)1.5.3 House of Maxwell’s Electrodynamics1.6 ELECTRIC AND MAGNETIC FIELD VECTORS1.6.1 Vector of Electric Field Strength1.6.2 Electric Potential

BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICSIntroductionThere are two basic versions of Maxwell’s equations: microscopic or quantum form andmacroscopic or classical form. The first set is more fundamental and describes the microscopicfields while taking into account their quantum nature. The second set is more straightforwardand fun because it averages all charges and fields in macroscopic media and allows us to ignorethe quantum effects while giving us sophisticated enough and closed to reality picture ofsurrounding world. Maxwell’s equations are essential not only for understanding the worldaround us but strikingly successful in explaining and predicting a broad range ofelectromagnetic phenomena. Macroscopic Electrodynamics deals with fields averaged on aspatial and temporal scale that is quite large compared to the interatomic space (in average10−10m) and the time of atomic fluctuations (in average 10−11s).The scale of both values is negligible from engineer’s perspective. In 2014, the IntelCorporation start mass production of new chips using very sophisticated the 14 nm technologyenabling the manufacture of monolithic integrated circuits (IC) with conductive line widths ofa few tenths of nanometers (close to 10−8m ). Even this tiny width is two orders of magnitudehigher than the average space between atoms. It means that, so far, the successful IC circuitryanalysis stays in the range of classic electrodynamics settings. Note that the next just comingstep is the 10 nm process.1.1 MACROSCOPIC ELECTRODYNAMICS1.1.1 Duality of Electromagnetic WavesWe know from quantum physics about electromagnetic wave duality. They are both waves anddiscrete particles, photons. Each photon carries a portion of energy ������������ = ℎ������������, where ℎ =6.6260755 ∙ 10−34 [J∙s] is Planck’s constant and ������������ is the electromagnetic field frequency. Evenat ultra-high frequency of 1 THz (1,000,000,000,000 Hertz) a single photon carries just6.6260755 ∙ 10−22 Joules, an extremely small amount in macroscopic world. For comparison, Figure 1.1.1 Electromagnetic spectrum

Chapter 1approximately 4 Joules are required to raise the temperature of 1 gram water by 1°C. Thereby,the macroscopic electrodynamics considers electric and magnetic fields and all valuesconnected to them might be averaged and continuous. There is similar approach is in thehandling of electrical charges, the main source of electromagnetic waves. Experiments showthat an electrical charge is quantized and the smallest charge portion is only ������������ =1.60217657x10−19 Coulombs which corresponds to the charge of an electron (- e), proton (+e), muon (- e), and several other elementary particles. Note that such elementary particles asquarks and antiquarks carry fractional charges ± 2⁄3������������ or ± 1⁄3������������, i.e. less than that of anelectron. Consequently, in macroscopic electrodynamics charge quantization is disregarded,and the averaged charge distribution is considered continuous. In other words, this book aboutthe continuum electrodynamics that lets us define physical objects of infinitesimal physical sizesand introduce the limits like∆l���i���������m→0 (∆������������������������⁄∆������������), where the charge ∆������������������������ is spread continuouslythroughout a volume ∆������������. As long as possible we will stay inside the borders of classicalelectrodynamics and just go beyond to some extent analyzing the interaction theelectromagnetic waves with matter. Electromagnetic waves of different frequencies are anintegral part of our life. We cannot escape them, our body itself generates them, we used themextensively for communication, broadcasting, energy generation and transportation,visualization, etc. The range of electromagnetic waves ordered by frequency in cycles persecond either wavelength in meters, or energy of single photon in electron volts is called theelectromagnetic spectrum and shown in Figure 1.1.11. The portion of the spectrum that isgenerally out of the reach of classical electrodynamics is marked in rose. Note that outsideMaxwell’s classical equations there remains such phenomena as the radiation and absorptionof electromagnetic waves of ultra-high frequencies (e.g., light), photoelectric effect, single-photon light detector, and many other effects involving quantum phenomena.1.1.2 Vector and Scalar FieldsClassical electrodynamics describes a broad range of electromagnetic phenomenon throughfour vectors of electromagnetic fields (see Table 1.1) depending on time and three Cartesiancoordinates (x, y, z) or one vector ������������ = ������������0������������ + ������������0������������ + ������������0������������, where ������������0, ������������0, ������������0 are the basis vectorsof unit length (see Figure A1 of the Appendix).Field Vector Table 1.1 E(������������, ������������) B(������������, ������������) Denotation D(������������, ������������) Electric field strength, defined by force interaction H(������������, ������������) Magnetic induction strength, determined by force interaction Electric displacement strength Magnetic field strengthFour scalar primary sources of these fields are listed in Table 1.2. Table 1.2Source Denotation������������������������(������������, ������������) Electric charge1 Public Domain Image, source: www.flickr.com/photos/advancedphotonsource/5940581568

BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICS������������������������(������������, ������������) Electrical current������������������������(������������, ������������) Magnetic charge (predicted but not found yet)������������������������(������������, ������������) Magnetic current (predicted but not found yet)In what follows we will clarify the meaning of all these quantities. However, what can be saidabout the nature of vector fields and their sources if they are pretty hard to visualize? We aregoing to build the theoretical and quantitative description by observing and measuring theirinteractions with surrounding physical objects. If such explanation of the phenomenon is quiteaccurate and can be presented in the analytical or numerical form nothing prevents fromapplying this knowledge to our practical problems hoping to get from physicists more profoundexplanation of their nature later.1.2 FUNDAMENTAL PRINCIPLES OF ELECTRODYNAMICS1.2.1 Symmetry in Nature and Conservation LawsThe fundamental laws of all science and engineering disciplines are the conservation laws ofelectric charge, mass, electromagnetic energy, momentum, etc. In physics, a conservation lawstates that a particular measurable property of an isolated physical system does not change asthe system evolves over time. Intuitively we believe in it, and our practical life experienceconvinces us to rely on it. The conservation laws are the most trusted and valuable source ofinformation about the complex, sometimes poorly understood, systems.In 1915 the German mathematician Emmy Noether discovered the deep connection betweenthe symmetries of nature and conservation laws and “…showed on very general mathematicalground that for physical theories of a particular type, every symmetry leads to a correspondingconservation law.” [2] For example, if you place a set of charges in free space, far from anythingthat might affect them (isolated system), it does not make a difference where exactly you putthe charges. There are no preferred sites in free space; all locations are equivalent. Thattranslation in space symmetry leads to the law of conservation of linear momentum, i.e. thetotal product of mass and vector velocity is always conserved. Furthermore, it does not make adifference at what time you start an experiment with the same system of charges in free space.The results will be the same. That symmetry with respect to time shift (translation in time) leadsto the law of conservation of energy, maybe the most important conservation law in physics.1.2.2 Conservation LawsIt is possible to prove (it is beyond the scope of this book) that Maxwell’s equations possess allthe required symmetries noted above. They belong to the group of physical theories that followfundamental laws of electrodynamics and can be considered as axioms [3]:1. Electric charge conservation is linked to the symmetry of scale transformation or gauge. Maxwell’s equations for point-to-point field description are a set of partial differential equations. Since the derivatives of constants are equal to zero, we can add arbitrary constants to their solutions. The actual situation is much more complicated. It can be shown that we have “gauge freedom” or can insert the whole set of functions to the same solution! Consequently, deriving a unique solution of Maxwell’s equations is not possible without some additional conditions, and there is, therefore, no way to compare such an uncertain solutions with indeed unique ElectroMagnetic (EM) fields measured in the real world. In

Chapter 1 other words, Maxwell’s equations solutions must be preserved against such “gauge freedom. ” One of such “ gauge protector” is the electric charge conservation law, which states that the charges can neither be created nor destroyed in an isolated system. Until now no experimental data are challenging this law.2. Electromagnetic energy conservation related to the symmetry under translation in time. We must be sure that the same physical process exhibits the same outcomes regardless of place or time (for example, electromagnetic wave propagation in free space are the same at any time in America on a Monday or in Poland on a Sunday). This law means that the total amount of any energy in an isolated system can be neither created nor lost, though energy keeps the ability to transform from one form to another, to be transferred from one object to another inside the isolated system. Even more importantly, the law of electromagnetic energy conservation in the form of Poynting’s theorem leads to the uniqueness of Maxwell’s equations solutions under certain like boundary conditions.3. Linear momentum conservation related to the symmetry under translation in space. This conservation law is the electrodynamics analog of Newton’s 3rd law of motion - “Every action has an equal and opposite reaction.” Note that linear momentum is associated with Poynting’s vector, which will be discussed later in Chapter 3, and radiation pressure exerted upon physical objects. The reality of such pressure was proved experimentally by Russian physicist Pyotr N. Lebedev in 1900.4. Angular momentum conservation related to the symmetry under rotation in space. This law is the rotation analog of linear momentum conservation law is the electrodynamics analog of Newton's 1st law of motion - \"A body continues in a state of rest or of uniform rotation unless acted by an external torque.\" Note that angular momentum is associated with Poynting’s vector, which will be discussed later in Chapter 3.Why did we pay so much attention to energy consideration? “The study of energy has played apivotal role in understanding the creation of the universe, the origin of life, the evolution ofhuman civilization and culture, economic growth and the rise of living standards, war andgeopolitics, and significant environmental change at local, regional and global scales. Virtuallyevery discipline investigates some aspect of energy, including history, anthropology, publicpolicy, international relations, human and ecosystem health, economics, technology, physics,geology, ecology, business management, environmental science, and engineering.”[1]1.2.3 Nothing Exists Until It Is MeasuredEM processes are mainly invisible. The only way to make it concrete and measurable is toconvert something invisible into readable data for an observer, using special sensors sensitiveto EM processes. Practically, all such sensors are based on an exchange between different formsof energy while taking some energy from the monitored system. For example, a voltmeter asshown in Figure 1.1.22 “seizes” a minute invisible portion of electrical current energy fromconnected batteries and converts it into the kinetic energy of rotating coil in voltmeter andvisible movement of the pointer across a scale. As results, the EM phenomena, as with manyprocesses in physics associated with the storage or propagation of energy that can be measured,becomes a reality through the measurements.2 Public Domain Image, source: http://practicalphysics.org/learning-use-voltmeters.html

BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICS “Whenever a prediction of the EM theory has been subjected to experimental verification, an agreement has been V obtained at any desirable level of accuracy. Within its domain of applicability, no observed EM phenomenon has ever been found to contradict the classical Maxwell- Lorentz theory of electrodynamics.” [13] Figure 1.1.1 Voltmeter connected to the Therefore, we need to define some battery additional quantities, forces, and energy,connecting the theory of electromagnetic fields with the other part of physics science (see Table1.3). Symbol Table 1.3������������������������������������(������������, ������������)������������������������ (������������) Denotation������������������������(������������) Force raised from interaction EM fields with material objects Energy accumulated by electric fields (E-field) Energy collected by magnetic fields (H-field)1.3 INTERNATIONAL METRIC SYSTEM OF UNITS (SI)1.3.1 Nothing Exists Until It Is Defined and Measured in UnitsTo begin deriving a solution to Maxwell’s equations, it is first necessary to set a lot of quantities:sources of electromagnetic fields, boundary, and initial conditions, basic requirements andparameters (dimensions of all objects, their conductivity, permittivity, and permeability, etc.).The additional critical parameters can be cost-of-production and time-of-production, thesurrounding environment influences (rain, hail, snow, wind, lightning impact, etc.). Part or allof these quantities can come from different sciences such as physics, chemistry, etc., can betaken from experiments, theoretical or numerical analysis. The data may often come fromdifferent countries, where scientists may be utilizing different measurement units like meters orinches, so it is essential first to ensure that our input value units are consistent. Mix up withunits can lead not only to an incorrect solution but to quite catastrophic consequences in reallife. In 1999, a $125 million Mars Climate Orbiter of the National Aeronautics and SpaceAdministration (NASA) went off its course and crashed on the Martian surface. The primarycause was that the spacecraft engineers calculated the Orbiter’s rocket thrust in pound force ∙ swhereas the team who built the thrusters assumed that the values were provided in Newton ∙ s.It is not surprising that the orbiter crashed as 1 pound-force is equal to 4.54 newtons!To avoid such kind of confusion, all used quantities must belong to the same system of units.In electrical engineering courses, the International System of Units or SI is mandatory. SIconsists of seven base units depicted in Table 1.4. For the convenience, we also included inTable 1.4 two dimensionless units those are not the part of base units. All other units of measureare derived from these base units. Base Unit name Base Unit symbol Table 1.41 Meter m2 Kilogram kg Quantity name Length Mass

3 Second Chapter 14 Ampere5 Kelvin s Time6 Mole A electric current7 Candela K thermodynamic temperature8 Radian mol amount of substance9 Steradian cd luminous intensity rad dimensionless rad2 Dimensionless1.3.2 Derived SI UnitsThe following set of derived units is open-ended and was restricted by the quantities primarilyused in electrodynamics.Quantity Name In terms of Derived Table 1.5 In terms of base units unit derived1 Area/Surface area square meter m2 A units2 Volume cubic meter m3 V m23 Speed/Velocity meter per second m s−1 ������������ m3 m s−14 Mass density kilogram per kg m−3 g kg m−35 Force cubic meter kg m s−2 Newton (N) F kg m s−2 ������������������������ I6 Volume electric Ampere (A) A current ������������������������������������ I m−27 Volume electric Ampere per A m−2 ������������������������������������ I m−1 current density square meter ������������������������ Is ������������������������������������8 Surface electric current Ampere per A m−1 I s m−3 ������������������������������������density meter I s m−2 ������������������������9 Electric charge Coulomb (C) As V ������������������������������������10 Volume electric charge Coulomb per A s m−3 V m−2 ������������������������������������density cubic meter ������������������������ V m−1 ������������������������������������ Vs11 Surface electric charge Coulomb per A s m−2 ������������������������������������ ������������ s m−3density square meter ������������ P ������������ s m−212 Volume magnetic Volt (V) m2 kg ������������−3 A−1 V Fm current E J s−113 Volume magnetic Volt per square kg ������������−3 A−1 W A−1 current density meter V m−114 Surface magnetic Volt per meter m−1kg ������������−3 A−1 current density Weber (Wb) m2 kg A−1 s−215 Magnetic charge16 Volume magnetic Weber per cubic m−1 kg ������������−2 A−1 charge density meter17 Surface magnetic Weber per kg ������������−2 A−1 charge density square meter m2 kg s−2 Joule (J) m2 kg s−318 Energy/Work Watt (W) m2 kg ������������−3 A−119 Power Volt (V) m kg ������������−3 A−120 Electric potential Volt per meter21 Electric field strength

BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICS22 Electric displacement Coulomb per A s m−2 D I s m−2 strength or Electric square meter m3 kg ������������−3 A−1 ������������������������ Vm flux density H m−1 I Volt meter23 Electric flux B V s m−2 ������������������������ Vs24 Magnetic field strength Ampere per m−1 A C meter L Q V−1 R V s I−125 Magnetic inductance Tesla (T) kg A−1 s−2 G V I−1 strength or Magnetic σ ������������ R−1 flux density Weber (Wb) m2 kg A−1 s−2 c R−1m−126 Magnetic flux Farad (F) m−2 kg−1 s4 A2 ������������−1 = R m27 Capacitance Henry (H) m2 kg s−2 A−2 ������������������������ (������������������������������������0)−1/228 Inductance Ohm (Ω) m2 kg s−3A−229 Resistance Siemens (S) m−2 kg−1 s3A2 ������������0 C m−130 Conductance S m−1 m−3 kg−1 s3A2 f31 Conductivity Ωm m2kg s−3A−2 L m−132 Resistivity e cycles per ������������������������33 Speed of light in free 299 792 458 m s−1 eV second space kg−1 m−3 s4 A2 - meter per second34 Permittivity of free ������������������������ = 1⁄(������������0c2) - space or ������������ ∙ 1V 8.8541878176x 10−1235 Permeability of free Farad per meter m kg s−2 A−2 space 4π ∙ 10−7 Henry per meter36 Frequency Hertz (Hz) s−137 Electron charge 1.60217657x10−1 As absolute value Coulombs kg 9.10938291 m2 kg s−238 Electron mass x10−31 kg Electron volt39 Energy/Work (eV)1.3.3 Dimensional Analysis and Unit Law“The premise of dimensional analysis is that the form of any physically significant equationmust be such that the relationship between the actual physical quantities remains validindependent the magnitudes of the base units.” [5] Dimensional analysis (also called factor-label or unit-factor method) based on Table 1.4 and 1.5 is a powerful way to keep track of unitsin multi-step electrodynamics problem solving and verifying the results of the analytical ornumerical analysis. Consider some examples.Example #1 is to establish an affiliation between magnetic field strength and magneticinductance strength. According to Table 1.5, line 25 and 24, the units3 for magnetic inductancestrength B is [kg ∙ A−1 ∙ s−2] and the units for magnetic field strength H is [m−1 ∙ A]. Therefore,if such relation occurs in the form ������������ = ������������������������3 Here and later in the book the square brackets with units inside imply “the dimension of”.

Chapter 1Treating dimensions as algebraic quantities, one can see their relationship as kg ∙ A−1 ∙ s−2 ~ k ∙ m−1 ∙ AIn the above expression, the symbol “~” represents a proportionality, not an approximation.Solving this relationship as an ordinary equation we have k ~ m ∙ kg ∙ s−2 ∙ A−2According to Table 1.5, line 35 the units for the factor k and the magnetic permeability ������������0coincides and we can expect that ������������ = ������������0������������That is the correct scalar equity for magnetic fields in a vacuum.In the same manner, the affiliation between electric field strength E and electric displacementstrength D can be established in a vacuum as ������������ = ������������0������������ [C∙ m−2]Here ������������0 is the permittivity of free space.Example #2. Now check the unit dimensions of factor k in Lorentz’s equation we will presentlater ������������������������������������ = ������������(������������ + ������������ x ������������)Here ������������������������������������ is force exerted by electric and magnetic fields and ������������ is the speed of some chargedparticle. According to Table 1.5, lines 5, 21, 3, and 25 the units for force ������������������������������������ is [kg ∙ m ∙ s−2],for electric field strength E is [m ∙ kg ∙ ������������−3 ∙ A−1], for speed v is [m ∙ s−1], and for magneticinductance strength B is [kg ∙ A−1 ∙ s−2]. Treating dimensions as algebraic quantities, one cansee their relationship as kg ∙ m ∙ ������������−2 ~ ������������ ∙ {m ∙ kg ∙ ������������−3 ∙ A−1 + (m ∙ s−1) ∙ (kg ∙ A−1 ∙ s−2)}Pay attention that both terms in the curly brackets must be and are of the same unit dimension.Solving this relationship as an ordinary equation we have������������ ~ kg∙m∙������������−2 = A∙s=C m∙kg∙ ������������−3∙ A−1Examining line 9 of Table 1.5 one can come to the conclusion that the factor k must be theelectrical charge ������������������������. Therefore, we can expect that ������������������������������������ = ������������������������(������������ + ������������ x ������������)Example #3. The last example is the unit dimension of factor k in the electric chargeconservation law called Gauss’s law (presented later)� ������������ ∘ ������������������������ = ������������������������������������ ������������

BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICSHere ������������ is the vector of electric field strength, ������������ is the integration area, dA is the infinitesimalelement of this area, and ������������������������ is the electrical charge. According to Table 1.5, lines 21, 1, and 9,the units for electric field strength E is [m ∙ kg ∙ ������������−3 ∙ A−1], for area element dA is [m2], andfor the electric charge ������������������������ is [A ∙ s]. Treating dimensions as algebraic quantities one can see theirrelationship as ������������ = (m ∙ kg ∙ ������������−3 ∙ A−1) ∙ m2 = 1 A∙s ∙ kg−1 ∙ m−3 ������������4 ∙ A2Examining line 34 of Table 1.5 one can come to the conclusion that the factor k must be theinverse value of the free space permittivity ������������0. Therefore, we can expect that Φ������������ = ∯������������ ������������ ∘ ������������������������ = ������������������������ [V∙ m] ������������0Later we will see that this equation is the exact formulation of the integral form of Gauss’s law.Following the same path, we can prove that in Gauss’s law for magnetic flux the factor k = ������������0 Φ������������ = ∯������������ ������������ ∘ ������������������������ = ������������0������������������������ [V ∙ s = Wb]where ������������0 is the permeability of free space.It is worth to note that the factors in Gauss’s law as in all following electrodynamics equationsdepend on the chosen unit system. For example, in Gaussian units, unlike SI units, thedimensional coefficients ������������0 and ������������0 disapper from Gauss’s law formulation.Unfortunately, the dimensional analysis does not include the magnitudes of the based units. Ifso, it can predict the numerical value of factor k only up to a multiplicative constant. Forexample, in the last case, nothing will change in dimensional analysis if ������������ = 2/������������0 or ������������ =1/������������������������0. The additional multiplicative constant usually can be established through themeasurements or other means.These examples demonstrate the significant role that the dimensional analysis, based on havingthe same units on both sides of the equation, can play as a prediction and verifying tool. In fact,we can establish the Unit Law as having different units on the two sides of the equation doesguarantee that the equation is wrong. In conclusion, note that in Gaussian CGS (short forcentimeter-gram-second) unit system the dimensional coefficients ������������0 and ������������0 swap to somecombinations of constant 4������������ and speed of light c.1.3.4 Table of Mathematical Operators in UseThe following table provides the meaning of some mathematical operators for use later. We putit here for the reader’s convenience. See more in Appendix. Table 1.6Symbol Meaning Comments������������ = ������������0������������������������ + ������������0������������������������ + ������������0������������������������ ������������������������, ������������������������, ������������������������ is magnitude of vector function ������������ in x, y, z direction, respectively.������������ = ������������0 ������������ + ������������0 ������������ + ������������0 ������������ Vector differential operator ������������ called del or ������������������������ ������������������������ ������������������������ nubla operator written in Cartesian space.

Chapter 1 ������������ ∘ ������������ = ������������������������������������ + ������������������������������������ + ������������������������������������ Divergence operator, div (������������), applied to ������������������������ ������������������������ ������������������������ vector field only. It is calculated as formal scalar product of vectors ������������ and ������������ and a ������������0 ������������0 ������������0 measure of how much vector field ������������ ������������ ������������ ������������������������������������� spreads out (diverges) from the source. ������������ x ������������ = ������������������������� ������������������������ Curl operator, curl (������������), applied to vector field only. Calculated as formal vector ������������������������ ������������������������ ������������������������ product of vectors ������������ and ������������. Curl (������������) is a measure of how much the vector field ������������������������ or ������������������������ Partial derivative on time curls around source likes water in������������������������ whirlpool/vortex. Time rate operator applied to vector and scalar field and is a measure of how fast field changes in the time domain. ������������������������ Vector element of infinitesimal path length tangential to path L. ������������������������ Vector element of surface area ������������, with infinitesimally small magnitude and direction normal to surface area. � ������������ ∘ ������������������������ Line integral of vector field Integral typically defines the work done by ������������ ������������ along path L force ������������ on an object moving along L. � ������������ ∘ ������������������������ Line integral of vector field Integral typically defines the total work ������������ along closed path L done by force ������������ on an object moving along ������������ L. Surface integral of vector� ������������ ∘ ������������������������ field G over the unclosed Integral defines the flux of vector field G area ������������ through the unclosed surface area ������������ ������������ constrained by closed contour L. Surface integral of vector� ������������ ∘ ������������������������ field G over the closed Integral defines the total flux of vector field surface area ������������ G through the closed surface area ������������ ������������ Volume integral of scalar � ������������������������������������ field G over some volume ������������ ������������1.4 EM FIELD SENSORS1.4.1 Electric Monopole, Dipole, and Current Element as Field SensorsBefore undertaking the general analysis of Maxwell’s equations, we need to define someuniversal objects we can use as field sensors or test elements. They must transform the energyof mainly invisible electromagnetic fields into some measurable form of energy, for example,into mechanical energy of moving objects, heat, etc. There are plenty of field sensors based ondifferent kind of phenomena [6, 7]. We focus on three simplest of them just hypothetical butsufficient to build the set of Maxwell’s equations.

BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICSFirst of all, note that EM fields can only control the behavior of objects that carry the electriccharges. Moreover, as we discussed above, any field measurement is based on the extractingsome portion of EM energy from the measured fields. Therefore, the measured fields shouldslightly differ from the original ones. Also note that in reality any charged object, at rest ormoving, would generate its own EM fields, which would alter the electromagnetic force that itexperiences. Moreover, the net force must include gravity and any other forces aside from theelectromagnetic force. To minimize all such kind of effects, we assume that the hypotheticalsensor has infinitesimally small mass and charge.Field sensor #1 can be realized as a motionless positively4 charged monopole of infinitesimalphysical sizes. We denoted such point-like sensor by symbol ∆������������������������ where ∆������������������������ is the monopolecharge value. Note that almost ideal natural sensor of this type is a free electron of mass ������������������������ = 9.10938291x10−31 kg carrying the negative charge ������������ = −1.60217657x10−19 Coulombs. d Field sensor #2 is an electric dipole (ball-and-stick model) defined as an assembly of two monopoles ±∆������������������������ shifted in spaceFigure 1.4.1 Electric at short distance d from each other, as shown in Figure 1.4.1. dipole. Note that the vector d points out in the direction of positive charge. We assume that this monopole duo under the influence of EM fields can spin as a solid assembly around their center(blue arrows). Eventually, such element is vectorial in nature and characterized by its dipolepolarization electric moment ������������������������ = ∆������������������������������������ [C ∙ m] (1.1)Almost any atom and molecule in solid, liquid or gas material can behave and serve as thenatural sensor of this kind. If so, its introduction greatly simplifiesthe study of EM field inside such materials as dielectrics andsemiconductors.Field sensor #3 is the electric monopole ∆������������������������ contained insidesome small domain ∆������������ and moving there with speed ������������, as shownin Figure 1.4.2.1.4.2 Electric Current and its Volume Density Figure 1.4.2 PositiveAs we know, the stream of freely moving electric charges is called charge moving withan electric current ������������������������ and defined as the variation ∆������������������������ in charge speed ������������per variation ∆������������ in time at the given point ������������������������ = − ∆l���i���������m→0 ∆������������������������⁄∆������������ = − ������������������������������������⁄������������������������ [C ∙ s−1] (1.2)Therefore, the sensor #3 can be called a current sensor. The scalar definition (1.2) came intoelectrodynamics from the lumped circuit theory giving us the information about the currentmagnitude only and telling nothing about the direction of charge stream. In the circuit theory,such information is irrelevant because the direction of current and wire carrying it alwayscoincides.4 common agreement

Chapter 1Meanwhile, we know very well that the beams of charged particles can propagate in free spacewithout any wire support. As an example, we can bring up the solar wind and lightning bolt,particle beam accelerators and weapon systems, vacuum tubes, and many other naturalphenomena and human-made devices. To include all these phenomena we can associate theflux of charges with the speed vector ������������ pointing in the direction of charge movement and definethe vector ������������������������������������ as the volume current density ������������������������������������ = ∆������������������������������������⁄∆������������ [(A∙ s ∙ m−3) ∙ (m ∙ s−1) = A ∙ m−2] (1.3) Here the new subscription V reflects the fact that this current arouses from the motion of electric charge in volume and the volume current by its nature in slight contradiction with unit dimensions. Looking at Figure 1.4.3 we can find the total electric current ������������������������ by means of positive charge steam ������������������������ = ∯������������ ������������������������������������ ∘ ������������������������ [A] (1.4)Figure 1.4.3 Positive charge Here, by collective agreement, the element ������������������������ is the moving with speed ������������ outward-pointing normal vector to area ������������ as Figure 1.4.3depicts. Therefore, the positive electric current is directed outward.Notice that the sensor #3 converts into the sensor #1 when ������������ = 0. Eventually, the natural modelof this current sensor is the short section of conductive wire with the infinitesimal cross section.1.4.3 Charge Volume DensityAs it was highlighted in section 1.1, the macroscopic electrodynamics disregards the chargequantization. Therefore, side by side with a point-like charge we can consider the charge ∆������������������������that is spread throughout a volume ∆������������ with a volume density ������������������������������������ defined as a limit ������������������������������������ = ∆l���i���������m→0 (∆������������������������⁄∆������������) = ������������������������������������⁄������������������������ (1.5)Then we obtain from (1.3) ������������������������������������ = ������������������������������������ ������������ (1.6)It is worth to note that the definition (1.2) and (1.4) can be extended to any continuousdistribution of electric charges ������������������������ as������������������������������������ = ������������������������������������⁄������������������������ , ������������������������ = ∫������������ ������������������������������������������������������������, ������������������������ = −������������������������������������⁄������������������������ (1.7)Now, it becomes clear why we put the sign minus on the right-hand side of (1.2) and (1.7). Thepositive current ������������������������ > 0 according to (1.4) corresponds the charges leaving the domain Vmeaning that the newt charge ������������������������ in this domain diminishes and ������������������������������������⁄������������������������ < 0 . Therefore, thenegative value of negative the derivative is the positive current.Typically, the physical processes forming the dipoles or currents and describing the interaction

BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICSbetween monopoles are quantum by nature, and we shall not pursue this topic here. The readerinterested in more details should consult [8, 9].1.4.4 Magnetic SensorsLater with better understanding of magnetic fields, we will introduce in a similar manner suchsensors as a magnetic monopole Δ������������������������, magnetic dipole moment ������������������������, and volume density ofmagnetic current ������������������������������������ ������������������������ = ∆������������������������������������ [V ∙ s ∙ m] � (1.8) ������������������������������������ = (∆������������������������⁄∆������������)������������ [V ∙ m−2]Then ������������������������ = ∯������������ ������������������������������������ ∘ ������������������������ [V]� (1.9) ������������������������ = ∫������������ ������������������������������������������������������������ [V ∙ s]Here ������������������������ = −������������������������������������⁄������������������������ [V] [V ∙ s ∙ m−3]� (1.10) ������������������������������������ = ∆l���i���������m→0 (∆������������������������⁄∆������������) = ������������������������������������⁄������������������������There are some problems with all these magnetic quantities. Wikipedia describes a magneticmonopole as “…a hypothetical elementary particle in particle physics that is an isolated magnetwith only one magnetic pole.” It means the unseen presence of a north pole without a southpole and vice versa. Such particles were predicted to exist by English physicist Paul Dirac in1931 - and have never been seen in nature. Nonetheless, an international group of scientists inthe journal Nature reported on January 30th, 2014 about “…controlled creation of Dirac’smonopoles in the synthetic magnetic field” at the temperature of few billionths of a degreeabove absolute zero of −273.15°C. Eventually, it is currently difficult to talk about any practicalapplications of this discovery.A remarkable fact is that the magnetic current magnitude is measured in Volts (see 1.10)) andthus physically equivalent to the voltage source like a battery. Consequently, it will be perfectlyvalid without any mystics to interpret some sources of electromagnetic fields and field sensorsas magnetic charges, dipoles, and currents if the fields created by them and magnetic sourcesare identical.In fact, there is the critical obstacle to making such definitions because the magnetic monopole,if it exists, must be an elementary particle similar to electron and proton, not a piece of materialwith the magnetic current. Then one might wonder why we started this discussion at all. Firstof all, we will show in the following sections that the introduction of magnetic charges andcurrents makes Maxwell’s equations highly symmetrical. In physics as in most arts, thesymmetry is a sign of beauty, elegance, and validity. However, the practical reason is that someman-made sources of electromagnetic fields act in the same way as magnetic charges andcurrents. In other words, this abstract concept of magnetic charge and current substantiallysimplifies the process of electromagnetic field solutions and significantly broadens theirdiversity. That is why the “nonexistent” magnetic sources became the part of electromagneticsalmost from its birthday.

Chapter 11.5 HOUSE OF MAXWELL’S ELECTRODYNAMICS1.5.1 IntroductionLet us move to the central goal of this chapter obtaining the set of macroscopic Maxwell’sequations first in a vacuum and later in materials and making them understandable, withoutunnecessary complexity. Before turning to such a task, we would like to notice that there aremany different ways to carry it out but all of them are based on a particular set of axioms. Thecentral question is how to choose such set. There are three main approaches:1. Theoretical derivation established on symmetry principles outlined in section 1.2. This pathway is the most conclusive and can proceed without assuming any results of an experimental nature. But it is quite tight requiring a good knowledge of high-power mathematics chapters that are not common in engineering practice.2. Historical derivation based mainly on experimentally-justified equations and the chain Coulomb’s law → Ampere’s law → Faraday’s law → Maxwell’s displacement current. Excellent path but the transition from Coulomb’s and Ampere’s law describing static fields to time-varying fields is not so understandable and requires plenty of additional explanations.3. Engineering derivation based on electric and magnetic charge conservation laws in the form of Gauss’s law and Lorentz’s force equation. Lorentz’s force equation establishes the total force exerted by EM field on charged particles and connects as well Maxwell’s equations to classical mechanics governing particle movements. If so, we can quantify the force and then the strength of EM field through the measurement of moving particle energy (a well-established procedure in physics and engineering practice). This fact makes the mainly invisible electromagnetic fields measurable and applicable from engineer’s perspective.We found that the engineering method is the best-suited for our purpose allowing to get thedefinition of all fields through the energy they carry. In other words, electric and magnetic fieldsbecome directly accessible to experimental observation. Moreover, it paves the way toMaxwell’s equations with a minimum of mathematics. We are going to use as much as possiblethe intuitive approach and dimensional analysis to avoid unnecessary rigor wherever practicablein the hope that the readers of our book remember or can quickly refresh the rudiments of EMfield from any introductory physics course [14].1.5.2 Lorentz’s Force Equation (Axiom #1)Suppose that EM field could occur in the domain ∆������������ of free space and we are willing to detectits existence and then count it at some given point O within ∆������������. To do so we can put at thispoint the sensor #3. In 1892 German scientist Leonard Lorentz established that the combinationof electric E and magnetic B fields exerts the force [12, 17] on this sensor that is equal to ������������������������������������ = Δ������������������������������������ + Δ������������������������������������ x ������������ [N] (1.11)In order to provide the field measurement in the given point only, we have to shrink thedomain ∆������������ around the point O and take the limit ∆������������ → 0 in both parts of equation (1.11)

BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICS ������������������������������������ = Δ������������������������ ������������ + �Δ∆������������������������������������ ������������� x ������������ (1.12) ∆������������ ∆������������According to (1.5) and (1.6) we obtain ������������������������������������ = ������������������������������������������������ + ������������������������������������ x ������������ [N/m3] (1.13)Here ������������������������������������ = ∆li���������m���→0(������������������������������������⁄∆������������) is the macroscopic volume density of force at point O. Eventually,equations (1.11) and (1.13) can be used to measure a given electromagnetic field by observingthe motion of charged particles.According to (1.11) the total exerted force ������������������������������������ consists of two parts. The first term ������������������������ = Δ������������������������������������is called the electric force, while the second one ������������������������ = Δ������������������������������������ x ������������ is deduced as the magneticforce. Eventually, in the bounds of macroscopic electrodynamics the limit should exist Δl������i���������m���������→0(������������������������������������⁄Δ������������������������) = ������������ + ������������ x ������������ [V/m] (1.14)As we have demonstrated above in Example #2 of Section 1.3.3, both terms on the right side of(1.14) have the same unit dimension [V/m]. It tells us that the second vectorial term is must besome kind of the electric field ������������������������ = ������������ x ������������ (1.15)induced by magnetic B field. This remarkable fact that the time-varying magnetic field cancause the electric field was discovered by British scientist Michael Faraday in 1831.We will return to Lorentz’s equation later. Now, let us move to the construction business ofbuilding, so called, House of Maxwell’s Electrodynamics. [4]1.5.3 House of Maxwell’s Electrodynamics (Maxwell’s House)Loosely speaking, our Maxwell’s House is no more than the classification diagramsummarizing Maxwell’s equations graphically, helping to memorize and visualizing them asthe items in \"memory palace.\" We intend to translate Maxwell’s equations into images that arethen placed on the palace walls, floor, attic, and basement. So we can mentally navigateourselves through that space and pick up those images we left there and translate them back towhat we memorized. Fully furnished Maxwell’s House [4] is shown in Figure 1.5.1a and b. Theone can read off each of Maxwell’s equations in differential form by just adding all incomingthrough the black arrow quantities at each node and setting it equal to the amount at the node.For the sake of drawing simplicity, we denote the operator ������������ by the symbol ������������������������ from Table 1.6 ������������������������ ������������������������ ≡ ������������ (1.16) ������������������������and omitted the front symbol ∆ and subscript e for charge q in Figure 1.5.1 to make the picturemore readable. It turns out that Maxwell’s House consists of three levels: basement, livingroom, and attic. The left wall in the living room of 1.5.1a can be called as Faraday-Lorentz’swall, the right wall of the same room can be called Ampere-Maxwell’s wall, the attic belongsto electric charges and currents, while the magnetic charges and currents go to the basement asnot found yet but widely used in computational electrodynamics. It is worth to note that in thisdiagram the vectors ������������������������������������, E and B defining the force exerted by electromagnetic fields arelocated on the Faraday-Lorentz’s wall while all the derivative vectors D and H flock on theAmpere-Maxwell’s wall. In the case of electrostatic and steady magnetic fields the time

Chapter 1derivative ������������������������ = 0 vanishes and the facade and back walls of House becomes decoupled, asshown in Figure 1.5.1b.Let's say, in 1.5.1a two black arrows connect the ������������������������������������ node, one comes from ������������ node through theoperator (������������ x) and another one comes from ������������ node through the operator (−������������������������). Therefore, the vector ������������������������������������ is the sum of two 0 0 incoming vectors ������������ x ������������ and (−������������������������������������), i.e. ������������������������������������ = ������������������������ q q ������������ x ������������ − ������������������������ that is the 2nd qv - qv Maxwell’s equation and so on. - The entire set of the differential form of Maxwell’s equations + Lorentz’s force equation 0 0 corresponding to Figure 1.5.1 a) b) is shown in the second column of Table 1.7. Some new Figure 1.5.1 Maxwell’s House, a) Time dependable equations and the integral form fields, b) Static and steady fields of Maxwell’s equations in this table will be introduced later. Table 1.7 Integral Form Differential Form Comments ������������������������������������ = ������������������������������������ + ������������������������������������ x ������������ Lorentz’s force ������������������������������������ = ������������������������������������������������ + ������������������������������������ x ������������ equation for electric charge1 −� ������������ ∘ ������������������������ = � ������������������������ ∘ ������������������������ + ������������������������ −������������ x ������������ = ������������������������ + ������������������������������������ 1st Maxwell’s equation ������������������������ ������������������������ or Faraday’s law ������������ ������������ 2nd Maxwell’s equation2 � ������������ ∘ ������������������������ = � ������������������������ ∘ ������������������������ + ������������������������ ������������ x ������������ = ������������������������ + ������������������������������������ or Ampere’s law + ������������������������ ������������������������ Maxwell’s ������������ ������������ displacement current3 � ������������ ∘ ������������������������ = ������������������������ ������������ ∘ ������������ = ������������������������������������ 3rd Maxwell’s equation or Gauss’s law ������������4 � ������������ ∘ ������������������������ = ������������������������ ������������ ∘ ������������ = ������������������������������������ 4th Maxwell’s equation ������������5 � ������������������������������������ ∘ ������������������������ + ������������ ������������������������ (������������) = 0 ������������ ∘ ������������������������������������ + ������������������������������������������������ = 0 Continuity equation or ������������������������ ������������������������ electric charge ������������ conservation law6 � ������������������������������������ ∘ ������������������������ + ������������������������������������ (������������) = 0 ������������ ∘ ������������������������������������ + ������������������������������������������������ = 0 Continuity equation or ������������������������ ������������������������ magnetic charge ������������ conservation law

BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICS7 ������������ = ������������������������������������ ������������ = ������������������������������������ Constitutive relation8 ������������ = ������������0������������ ������������ = ������������0������������ Constitutive relationMaxwell’s equations tell how the electromagnetic fields arise from such sources as charges andcurrents, which are nothing more than moving charges. Electric and magnetic fields are deeplyinterconnected, any variation in any of them leads to a proportional change in another one. Theycan be independent/decoupled if they are produced by the sources independent of time, asshown in Figure 1.5.1b, where Maxwell’s House cuts down to façade and back wall only. In Maxwell’s House, any shift from level up or down is the movement in space and, as expected, the div operator (������������ ∘), curl- operator (������������ x), or vectorial rotation (������������ x) provides such upstairs or downstairs alteration. Any displacement on the same floor level is the time domain movement in parallel to the vector that exerts this movement. Now we can start building Maxwell’s House step-by-step. First, let us put Lorentz’s force in the top left node of the first floor as Figure 1.5.2 shown in Figure 1.5.2. Since the vector, E is connected withVectors ������������������������������������, E, D, H vector ������������������������������������ through the scalar operation, it must be located on the and B in Maxwell’s same level while the vector B connected through the vectorial House operation must be put one level down. For a while, we stop populating Maxwell’s House in order to define the vectors E and B. Note that each component of the electric or magnetic field, charge,current, and matter parameter (if they are not defined differently) in Table 1.7 is the function oftime t and coordinates (x,y,z). It is convenient to use the vector notation for coordinates ������������ =������������0������������ + ������������0������������ + ������������0������������. There and then we assume in Table 1.7 and following text that ������������ =������������(������������, ������������), ������������ = ������������(������������, ������������), ������������������������������������ = ������������������������������������(������������, ������������), ������������������������������������ = ������������������������������������(������������, ������������), and so on.1.6 ELECTRIC AND MAGNETIC FIELD VECTORS1.6.1 Vector of Electric Field StrengthThe Encyclopedia Britannica defines the electric field as “ … an electrical property associatedwith each point in space when the charge is present in any form. The magnitude and direction of the electric field are expressed by the value of E, called electric field strength or electric field intensity or simply the electric field. Knowledge of the value of the electric field at a point, without any explicit knowledge of what produced the field, is all that is needed…” to know. What else can be said about the electric field, which is pretty hard to visualize except in narrow optical window? Practically nothing about its nature but we can detect andFigure 1.6.1 Integration path count it using the monopole sensor #1 carrying tiny charge. According to Lorentz’s equation (1.11) the small

Chapter 1electric force Δ������������������������ pushing or pulling this motionless (������������ = 0) at the starting moment of timesensor is equal to Δ������������������������ = Δ������������������������������������ = Δ������������������������������������ (1.17)where the vector of electric field strength ������������ manifests itself only by the forces exerted upon thesensor. In other words, we can define fields as the way in which forces are spread acrossdistances. In accordance with Newton’s third law (for every action there is an equal andopposite reaction) this action-at-a-distance is reciprocal. The sensor influences on thedistribution of electrical field sources by pushing and pulling them too. To avoid this impactand to be more precise we can redefine the vector ������������ in (1.17) as a limit in macroscopic sense������������ = lim ∆������������������������ ⁄Δ������������������������ = ������������������������������������ [(kg ∙ m ∙ s−2)/(A ∙ s) = V ∙ m−1] (1.18) ������������������������ Δ������������������������→0The symbol Δ������������������������ → 0 means that both the charge and object carrying the charge is lessenedtogether, at the same rate. Let allow the monopole sensor to move freely under electric forceinfluence assuming that the sensor speed ������������ is low enough and thereby the additional forceexerted by magnetic field in (1.11) is negligible. The laws of mechanics [11] tell us that theenergy required for this movement from some starting point 1 to end point 2 along the contourL shown in Figure 1.6.1 is equal to ������������������������ = ∫12 ������������������������ ∘ ������������������������ [(kg ∙ m ∙ s−2) ∙ m = [J = W ∙ s] (1.19)Accordingly, the energy conservation law dictates that all this kinetic energy of movement mustbe delivered by the electric field ������������. Therefore, Δ������������������������ = Δ������������������������ ∫12 ������������ ∘ ������������������������ (1.20)Since the energy Δ������������������������ is the measurable quantity, the expression (1.20) provides the mean forthe electric field strength valuation.1.6.2 Electric PotentialIn order to undo the dependence of measured energy from the sensor charge value Δ������������������������ letintroduce using (1.20) the energy ������������������������ normalized to the test charge quantity defining the electricpotential in volts ������������������������ = Δ������������������������ = ∫12 ������������ ∘ ������������������������ [V] (1.21) Δ������������������������that equal the amount of work done by shifting a unit positive point-like charge down the path.It turns out that the line integral in (1.21) and thus the potential depends on the position of twodifferent points, starting and ending. Therefore, electric potential measurement is alwaysrelative and shows the difference in potential between two distinct points in the same ordifferent regions. One way to fix this problem of uncertainty is to shift the end point to infinitywhere any meaning potential must vanish and define the absolute potential. Theoretically, it isquite acceptable but certainly not practical. So electrical engineers decided to set our Earth’spotential as the equivalent reference available to anyone at any time and any location.Nevertheless, there is some additional uncertainty in (1.21). How to choose the path L? In free

BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICSspace, we can connect two separated points by an infinite number of different ways of differentlength and produce, at first glance, different amount of work. It is true, except the case of electrostatic, i.e. independent of time electric fields. We will come back to this topic later. Note in conclusion// that a unit of energy commonly used in physics is the electron-volt [eV] defined as the energy gained or lost by single electron or proton when it moves through a potential difference of 1 Volt, 1 [eV] = 1.60 ∙ 10−19 [J].Figure 1.6.2 Line of force, electric 1.6.3 Line of Forcevector direction and strength Equation (1.21) tells us that the monopole sensor continuously borrows the maximum energy from agiven electric field and faster increases its kinetic energy if the scalar product ������������ ∘ ������������������������ = ������������ ∙ ������������������������.In other words, the electric field vector must be tangential to the contour L at any point of L.One of this kind contour L, shown in Figure 1.6.2, is called the imaginary line of force if1. The magnitude of electric field is constant (������������ = const.) along the contour L.2. The tangent at any point to it gives the direction of the E-field vector ������������������������ or ������������������������ at the point P or ������������.3. E-field lines begin at positive charges and end at negative charges (see Figure 1.6.3). The number of lines beginning or ending on any particular charge is proportional to the charge value.Note some additional evident properties of lines of force (Figure 1.6.2):1. Two lines never intersect or touch. Think why.2. At every position, the magnitude of the E-field is proportional to the field line density, i.e. they are closer (congested) where E-field is stronger and the lines spread out where the E- field is weaker. Thus, the relative closeness of the lines in some area is the evidence of the higher intensity of fields, i.e. |������������������������| > |������������������������ |. 3. In a uniform field, the lines of force are straight parallel and uniformly spaced.a) b) Figure 1.6.35 illustrates the lines ofFigure 1.6.3 Lines of E-force around a) point-like force nearby the point-like charges.charges, b) dipole The perfectly straight green arrows in Figure 1.6.3a demonstrate thestructure of the electric fields from the single positive charge or monopole, while the yellowarrows in Figure 1.6.3b show the same but for the dipole shown in Figure 1.4.1. The higherdensity of arrows close to the charge corresponds to higher electric field strength there. Bydefinition, the force lines start on positive charges and all finish on infinity if the charge is alone(Figure 1.6.3a) and partially on the negative charge (Figure 1.6.3b) in the case of the dipole.Eventually, the line of force density is maximum nearby the charges. Unfortunately, though5 Public Domain Image, source: a2physicsmontessori.weebly.com/review-electricity-and-magnetism.html, web.ncf.ca/ch865/graphics/EFldPosChargedSphere.jpeg

Chapter 1such images are clear for simple fields and charges distributions they are much messier in morecomplicated cases. 2D- and 3D-diagrams, where the gradients of color variations reflectelectrical field strength, as shown in Figure 1.6.46, are much more informative. In Figure 1.6.4aareas of highest (bright red) and lowest (dark green) strength, are clearly visible and lines offorce can be added if their structure is not too complicated. Quite often, the images like 1.6.4b could be part of a contemporary art exhibition! In conclusion, let us evaluate the integral in (1.20) along the line of force. Since ������������ = const. along the a) b) integration curve LFigure 1.6.4 Electric field strength around electric ������������ = Δ������������������������⁄(Δ������������������������������������) (1.22) dipole If so, measuring the energy acquitted by the sensor #1 we candefine the electrical field strength as the energy of electric field required to move the point-likecharge of 1C at the distance of 1m along the line of force.1.6.4 Gauss’s Law for Electric Fields (Axiom #2) and Coulomb’s LawGauss’s law expresses the total flux of electric fields at any moment of time through the closedsurface area ������������ of any shape (see Figure 1.6.5a) and can be written as ∯������������ ������������(������������, ������������) ∘ ������������������������ = ������������������������(������������)⁄������������0 (1.23) Here ������������������������(������������) is the total charge within some volume V at any current moment of time. Some charges can stay at rest or move in any direction inside V, can leave V or arrive from outside V, as shown a) b) in Figure 1.6.5a. They can beFigure 1.6.5 Volume V with charges positive or negative. The coefficient ������������0 called the permittivity of free space orvacuum is required by the chosen SI unit set as we have demonstrated above in Example #3 ofSection 1.3.3.Assuming that V is the sphere of radius r and holds a single at rest point-like charge ������������������������(������������) = ������������in the sphere center, as shown in Figure 1.6.5b, we can derive Coulomb’s law combiningGauss’s law and Lorentz’s force equation. Using the fact that the geometrical structure in Figure1.6.5b has the complete spherical symmetry, we can suggest that the vector of electrical field isconstant on the sphere surface and can be pulled out of the integral. Then6 Public Domain Image, source: http://sciencewise.blogspot.com/2008/01/exploring-electrostatics.html

BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICS ������������������������ = ∯������������ ������������(������������) ∘ ������������������������ = ������������(������������) ∘ ∯������������ ������������������������ = ������������(������������) ∘ 4������������������������2������������0 � ������������0 ������������0 ������������������������ (1.24) ������������(������������) = 4������������������������0������������2Here ������������0 is the unit vector pointing radially from the charge. The well-known equity (1.24)describing the electric field around a static point-like charge and is usually derived fromCoulomb’s law. Putting the sensor #1 next to the charge Q we can find the exerted electricalforce from (1.11) as (������������ = 0)������������������������ = Δ������������������������ ������������(������������) = Δ������������������������������������������������ ������������0 (1.25) 4������������������������0������������2which is Coulomb’s law. Figure1.6.6a7 demonstrates the reality ofelectrical fields pushing up girl’s hairin full agreement with the vectorequation (1.25) and Figure 1.6.6b8.Do the charges store electric energy? a) b)In fact, they do not store the energyat all. Instead, the energy is stored in Figure 1.6.6 a) A girl’s hair after touching athe electric field surrounding the charged sphere, b) Vector of the electric fieldparticle. generated by positively charged sphere1.6.5 Is The Inverse-Square Relation Imperative?Gauss’s law (1.23) is one of the fundamental theorems in electrodynamics, and we proved thatthe purely empirical Coulomb’s law was followed from it. In particular, Gauss’s law led us tothe exact inverse-square relation between electrical field and distance in (1.24) and (1.25). Notethat Newtonian gravitation law follows the same inverse-square relationship. Is it the law ofnature or something occasional? The best validation test [8] made in 1983 estimated that thedeviation ������������ from Coulomb’s law in the form ������������−2+δ is tremendously small and does not exceedδ ≤ (2.7 ± 3.1) ∙ 10−16. According to Purcell [9], the inverse-square law has beenexperimentally verified over the range of 10−13mm < r < 100 000 km. Why do physicists paysuch great attention to the accuracy of the inverse-square law in Coulomb’s and Newton’sgravitation law? The answer is quite dramatic for humans: if these forces should not be tunedcorrectly, our universe must be very different. Some scientists formulated a very curiousanthropic principal [10] “ … the universe is it is because if it were different we would not behere to observe it.” It means that variation in inverse-square laws can lead, for example, tocatastrophic implications such as the existence of the different kind of universes with noopportunity to produce humans.1.6.6 How Much Is One Coulomb (C)?Quite powerful! Suppose two point-like charges of 1C each are located at a distance 1 km =1000m. The force value can be calculated using (1.1) and the numerical constant ������������������������7 Public Domain Image, source: https://www.flickr.com/photos/57167312@N02/5270904732. Thiswork has been identified as being free of known restrictions under copyright law, including all relatedand neighboring rights.8 Public Domain Image, source: http://inspirehep.net/record/946729/files/CoulombsLaw.png

Chapter 1������������������������ = 4 ∙ 3.1415 1 ∙ 10−12 ∙ 10002 = 8.988 ∙ 109 N ∙ 8.854 1From a mechanical point of view a force (������������������������=ma) can accelerate a heavy truck of m = 50 tons= 50 000 kg with a = 200 km/s2! So in one second the truck will be 100 km from the trafficlights and an hour will reach a speed of 720,000 km/hour, if it could survive such powerfulpush.Significant charges are not uncommon around us. The Earth bears a negative charge of about−4.5 ∙ 105 C [15] and our atmosphere accumulates a roughly equal and opposite charge. It is,therefore, no surprise that nature produces so many fireworks, within average 100 lightningstrikes per second! The immense amount of charge that travels through a lightning bolt canreach up to 350 C or the charge of 2.18 ∙ 1021 electrons with a total mass 2 ∙ 10−12 kg or only6 g per year. The age of the Earth is about 4.56 ∙ 109 years. Thus for the whole Earth historylightning would deliver 27 360 tons of electrons. Meanwhile, a single commercially producedultracapacitor of 10,000 farads for the wind and the solar power generation system connectedto a 2.7-volt battery stores 27,000 C or the equivalent of 77 of the most powerful lightningstrikes! A compact battery of 17 such capacitors is capable of storing more charges than thewhole Earth.1.6.7 Electric Field RealityThe equation (1.20) and (1.24) proves the quite remarkable fact that electrical charges storetheir potential energy in their surrounding electrical fields. That will bring into play theconservation energy law: field potential energy can be released and converted into any otherform of energy such as radiant and heat, motion and sound, chemical, you name them. Theelectrical fields become quite real: they carry energy and can be measured! For example, theexistence of electric potential implies the transformation of electric field energy into kineticenergy of movable charges. In other words, electrical potential or pure voltage can be the sourceof an electrical current in the material where movable charges such as electrons are not boundto atomic nucleus or molecules and are free to respond to outside forces created by electricalfields. Subsequently, the greater voltage means more dominant force, the higher electric currentand more energy to be taken from the electrical sources. However, the movement of a mass ofcharged particles that is too big leads to a high probability of collisions between these particlesand the crystal lattice of conductive material and thus an increase in energy loss. That is why avoltage of 750,000V and higher is used to reduce the transposed mass of charges whiletransferring the bulk of electrical energy from power stations to remote consumers. Thepotential of several hundred million volts between earth surface and thunderstorm clouds causespowerful lightning. Shuffling your feet across synthetic carpets might increase your bodyelectric potential up to 36 000 V. The sparks jumping from your finger are not dangerous butannoying and sometimes rather painful!1.6.8 Displacement Vector D. 3rd Maxwell’s EquationNow, we turn to the electric charge conservation law, one of the fundamental laws of physics,and will demonstrate its association with Gauss’s Law. First of all, let us rewrite Gauss’s lawas the integral form of 3rd Maxwell’s equation (see Table 1.7)

BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICS ∯������������ ������������ ∘ ������������������������ = ������������������������ (1.26)introducing the vector of electric displacement strength or electric flux density ������������ = ������������0������������ [C/m2] (1.27)The equity (1.27) is called the constitutive relation and was put in Table 1.7 as a relationshipsupporting Maxwell’s equation. In order to simplify the notation in (1.27) and the followingequations, the records of time and coordinate dependence are suggested but omitted. From timeto time, we come back to full notation to avoid confusion. The numerical quantity of ������������0 iscalled sometimes the absolute dielectric constant of vacuum.In a vacuum, the only difference between vectors E and D is the unit dimension dictated by thechosen SI units. 3rd Maxwell’s equation in the form of (1.26) describes the integral effect of allelectric fields existing in some volume V bounded by a surface A. Practically, that is not enough.We need more information: namely how electromagnetic fields are dispersed inside the volume.For cell phone customer it is not sufficient to know that the nearest communication tower inorder and radiates electromagnetic energy. We must be sure that this energy is enough at theclient spot to receive signal. Therefore, we need to switch from integral to deferential or topoint-to-point field form of description. To illustrate how to do it let us assume that thedisplacement vector has small z-component only or ������������ = ������������0∆������������������������ and ������������������������ = ������������0������������������������������������������������ = ������������0∆������������∆������������and in (1.26). Then ������������ ∘ ������������������������ = ∆������������������������∆������������∆������������ since ������������0 is the vector or unit length and ������������0 ∘ ������������0 = 1. Itwill be perfectly valid to represent the product ������������ ∘ ������������������������ as������������ ∘ ������������������������ = ∆������������������������ ∆������������∆������������∆������������ = ������������������������������������ ������������������������ (1.28) ∆������������ ������������������������Here ������������������������ is the volume of infinitesimal parallelepiped. Eventually, applying the same transformto all components of displacement vector we obtain������������ ∘ ������������������������ = ������������������������������������������������������������� + ������������������������������������ + ������������������������������������������������������������� ������������������������ (1.29) ������������������������Looking back at Table 1.6 of mathematical operators we have ������������ ∘ ������������������������ = ������������ ∘ ������������������������������������ (1.30)One more step is to transform the right-hand side of (1.26) using the association (1.7) of thecharge ������������������������ with its volume density ������������������������������������ ������������������������ = ∫������������ ������������������������������������������������������������Substituting this equity and (1.30) into (1.26) and putting all terms together we have ∫������������ (������������ ∘ ������������ − ������������������������������������)������������������������ = 0 (1.31)Now, we can say the “magic” words repeating them with some variations many times later:since the volume V in (1.31) is arbitrary and this equity must hold for all of them, the integrandmust be equal to zero that makes this equality right in general. Therefore, ������������ ∘ ������������ = ������������������������������������ (1.32)The equity (1.32) is the differential form of 3rd Maxwell’s equation included in Table 1.7.

Chapter 11.6.9 Why Do We Need Extra D-Vector Describing E-Fields?According to the ancient philosopher Aristotle, “Nature abhors a vacuum.” “Obeying” Aristotlethe world around us is full of matter in different states/phases: solid, liquid, and gases. Underthe special conditions, the matter may be in a state of plasma, superconductivity, and someexotic for engineering practice states like quantum, Bose-Einstein condensate, super fluid andsolid, supercritical liquid, etc. The reader can find further information in the specializedliterature. We will focus on conventional materials widely used in everyday life and devotedChapter 2 to the neoclassical theory of interaction of EM fields with dielectrics, metals, ferrites,etc.Why do EM fields interact with matter in the first place? By the classical atomic model, thebasic units of matter are minute atoms whose estimated nucleus diameter is in order of 9 10-4Å. Meanwhile, the average atom diameter is in the order of several ångströms. Roughlyspeaking, there is a lot of “free space” inside and outside atoms that gives the primary EM fieldsa good chance to penetrate and encounter with the nucleus positively charged protons as wellwith the negatively charged electrons. Lorentz’s force “obliges” these fields to transfer someportion of their energy to the charged particles. The latter start oscillating about a point ofequilibrium, i.e. moving back and forth around their stationary positions, or just shifting slightlyin case of static fields. If we assume that a neutron has a mass of 1, then the relative mass of anelectron is minute and 0.00054386734 only. It means that each electron in almost 2⋅103 timeslighter than a neutron and thus might have a much higher oscillation magnitude. Thereby anycharge movement is equivalent to some electrical current. Consequently, each such movingcharge (most electron) has to induce the secondary EM fields that are added to the primary.Therefore, the total E-fields inside the matter might strongly diverge from the primary in thevacuum. To take into consideration this effect we can adjust the constitutive relation (1.27) as������������ = ������������0������������������������������������ where the dimensionless coefficient ������������������������ called the relative dielectric constant ofmatter. Thereby, we include all additional fields induced by matter bound charges into D-vectorand maintain the flux of D-vector defined by (1.26) dependable on the free electrical charges������������������������ only regardless of the E-fields induced inside the matter by these free charges. It is importantto observe that free charges are those not bound up in atoms and molecules of the matters.Meanwhile, the charges on the surface of conductors (see Section 2.2.7 in Chapter 2) inducedunder influence of the external E-fields must be considered as free charges.1.6.10 Electric Charge Conservation Law in Differential FormBefore settling (1.32) into Maxwell’s House let us formulate the continuity equation for electriccharges. In particular, this equation is one of the possible forms of the electric chargeconservation law that can be expressed as any decrease in the amount of charge in a givenregion of space must be correctly balanced by a simultaneous increase in the quantity of chargein an adjacent region of space. Since any charge, give-or-take a few, means the flow of movingelectric charges or the electric current ������������������������ the describing such movement equation9 1 Å = 10-10 m

BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICS ������������������������ = ∯������������ ������������������������������������ ∘ ������������������������ = − ������������������������������������ (1.33) ������������������������is the integral form of the continuity equation. But remember that this equation describes the movement of positive charges. If the current is defined as the flow0 of electrons, ������������������������ in (1.33) must be replaced by −������������������������. The deferential form of the continuity equation follows from (1.33) if we transformq (see Appendix) the surface integral to volume one applying (1.30) to the current density asqv - ∯������������ ������������������������������������ ∘ ������������������������ = ∫������������ ������������ ∘ ������������������������������������������������������������ (1.34) Replacing in (1.33) ������������������������ = ∫������������ ������������������������������������������������������������ and grouping the terms we obtain Figure 1.6.7 ∫������������ ������������� ∘ ������������������������������������ + ������������������������������������������������������������������������� ������������������������ = 0 (1.35)Populated House of Pronouncing the “magic” words, we finally get the continuity Maxwell’s equation in differential form Electrodynamics ������������ ∘ ������������������������������������ + ������������������������������������������������ =0 (1.36) ������������������������Now, we can populate Maxwell’s House attic with new residents ������������������������������������, ������������������������������������ and ������������0, as shown inFigure 1.6.7.1.6.11 Lorentz’s Force Equation and 1st Maxwell’s EquationLooking back at Lorentz force equation (1.11) we note that the magnetic field exerts the force������������������������ on a moving charge that can be described through the equivalent electric field (1.15)as ������������������������ = ������������ x ������������. That is not something occasional. The most remarkable fact is that magneticand electric fields are in reality just different aspects of the same and inseparable phenomenon— the electromagnetic force. Imagine that a girl shown in Figure 1.6.6a is traveling on a traincar touching a charged metal sphere and having a bunch of tools including her hair to measureelectric and magnetic fields. Taking into account that the girl, the sphere, and all her instrumentsare at rest relative to each other the girl’s instruments can detect only electrical fields.Nevertheless, her friend standing with an identical set of tools on the platform while the trainpasses, will detect not only electric fields but magnetic fields too, because for the friend andhis/her instruments the charged sphere in motion is equivalent to some current and the sourceof magnetic field. Eventually, nothing changes if the train stops and the girl’s friend startsrunning nearby the train car.We will use such duality or principle of relativity sometimes implicitly to bridge the gapbetween static and time-varying fields, static and moving charges, electrical and magneticfields, et cetera. Electric and magnetic fields are just two faces of the same natural phenomenonlike the ancient Roman two-faced god Janus, who looks simultaneously to the future and thepast, symbolizing the transition from one condition to another10. More details about such dualityeffects can be found in the special theory of relativity and are beyond the scope of this book.10 It looks like Romans believing in such god foresaw electromagnetic field propagation long ahead ofMaxwell.

Chapter 1Let us start off with the equivalent electric field ������������������������ and calculate the work/potential ������������������������provided by magnetic field accordingly to (1.15) and (1.26)������������������������ = ∮������������ ������������������������ ∘ ������������������������ = ∮������������ (������������ x ������������) ∘ ������������������������ = − ∮������������ (������������ x ������������) ∘ ������������������������ [V] (1.37) supposing that E = 0 in Lorentz’s force equation (1.15) while B-vector is time-independent. The integration curve L in (1.37) is a closed one, as shown in Figure 1.6.8 with the surface area A bounded by thisFigure 1.6.8 Closed curve. Eventually, we can Figure 1.6.9 Scalar boundary curve L build the infinite number of triple product as such kind of surfaces volume bounded by the same curve Lwith the only restraint. The orientation of L should be positive meaning that if you keep yourhead up walking along the curve, the surface must be on your left.Choosing the vectors ������������ and ������������������������ as the base vectors (see Figure 1.6.9) and recognizing that thevector product ������������ ∘ ������������ x ������������������������ = ������������ ∘ ������������������������ is the volume of parallelepiped we can transform (1.37) to(������������ x ������������) ∘ ������������������������ = ������������ ∘ (������������ x ������������������������) = ������������ ∘ ������������������������������������������������� x ������������������������� = ������������ ∘ ������������ (������������ x ������������������������) = ������������ ∘ ������������������������ (1.38) ������������������������ ������������������������Therefore, while our man in Figure 1.6.8 travels down the curve L the element ������������������������ moves withspeed ������������ along the ribbon on the surface area ������������. As soon as the man made his journey along thecurve, the parallelepiped covered the total surface area ������������. Consequently, in (1.37) ∮������������ (������������ x ������������) ∘ ������������������������ = ∬������������ ������������ ∘ ������������������������ (1.39) ������������������������Now recall the two faces of EM field. The moving element ������������������������ can be considered as the traincarrying the current sensor #3 while the static magnetic field ������������ is detected by the observer onthe platform. Invoking the principle of relativity, we can put the sensor at rest and start changingthe magnetic field ������������ at the same rate. Therefore, in (1.39) ∬������������ ������������ ∘ ������������������������ = ∬������������ ������������������������ ∘ ������������������������ (1.40) ������������������������ ������������������������It means that the expression (1.37) can be rewritten as ∮������������ ������������ ∘ ������������������������ + ∬������������ ������������������������ ∘ ������������������������ = 0 (1.41) ������������������������Pay attention that the subscription m in electric field vector ������������������������ was omitted as a pointless. From(1.41) and the energy conservation law immediately follows the existence of a unique physicsphenomenon – time-varying magnetic fields induce time-varying electric fields. No charge orcurrent sources are involved in this process, which forecasts electromagnetic wavespropagation. The equity (1.41) is the 1st Maxwell’s equation in Table 1.7 for the EM fields in

BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICSthe space free of field sources (������������������������ = 0). Note that (1.41) is the well-known law ofelectromagnetic induction ℰ(������������) = − ������������Φ������������(������������) (1.42) ������������������������discovered by Michael Faraday in 1831. Here the ElectroMotive Force (EMF) ℰ(������������) =∮������������ ������������ ∘ ������������������������ in Volts or and the magnetic field flux Φ������������(������������) = ∬������������ ������������ ∘ ������������������������. The minus sign is anindication that the electric potential in such a direction as to produce a current flux. Added tothe original flux, it would reduce the magnitude of the potential. This statement that inducedvoltage acts to produce an opposing flux is known as Lenz’s law.In order to detect and measure the EMF let us put the small wire loop in the magnetic field asa sensor and connect a voltmeter, as shown in Figure 1.6.10a. It is well-known from school and college course of physics the magnetic flux causes the electric current in the loop flowing through the voltmeter and spinning its pointer across the scale thereby displaying the EMF magnitude ℰ(������������). Eventually, we can replace the voltmeterFigure 1.6.10 Faraday’s law, a) Magnetic flux induced with AC voltage source of EMF, b) EMF induced magnetic flux the same magnitude ℰ(������������), as shown in Figure 1.6.10b, and recreate the magnetic flux identically to the presentedin Figure 1.6.10a. Therefore, in this case, EMF became the measured in Volts source of thefield. Looking back in Table 1.5 we see that the only voltage source is a volume magneticcurrent ������������������������ which we can put in the right-hand side of (1.41) as an equivalent voltage source ofelectromagnetic field∮������������ ������������ ∘ ������������������������ + ∬������������ ������������������������ ∘ ������������������������ = −������������������������ (1.43) ������������������������The reader interested in more details and rigorous consideration of the link between Lorentz’sforce equation and Maxwell’s equations can look into [1T, 2T].In order to get the point-to-point field description, we should present all terms in (1.43) as thesurface integrals. Applying Stokes’s theorem to the left-hand linear integral (see Appendix) wehave ∮������������ ������������ ∘ ������������������������ = ∬������������ ������������ x ������������ ∘ ������������������������ and introducing the volume magnetic current density (1.9)������������������������ = ∬������������ ������������������������������������ ∘ ������������������������ we can rewrite equation (1.42) as follows∬������������ (������������ x ������������ + ������������������������ + ������������������������������������ ) ∘ ������������������������ = 0 (1.44) ������������������������Pronouncing our “magic” words about arbitrary surface area A, we finally get 1st Maxwell’sequation in differential form

Chapter 1 −������������ x ������������ − ������������������������ = ������������������������������������ (1.45) ������������������������0 Now, we can populate the left wall of Maxwell’s House with new residents ������������������������������������ as shown in Figure 1.6.11a.q 1.6.12 Is Magnetic Inductance Real and Can Beqv - Measured? Sure. Let us put an infinitesimal sensor #3 in the field to pick up measurable portion of magnetic energy ∆������������������������ from the field. According to (1.21) and (1.37) ∆������������������������ = ∆������������������������������������������������ = ������������ ∘ (∆������������������������������������ x ������������������������) = ������������ ∘ (∆������������������������������������ x ∆������������) [W⋅s] (1.46) As we have mentioned before, the natural model of the Figure 1.6.11a Left wall of current sensor #3 with known current density ∆������������������������������������ isMaxwell’s House population the short section of conductive wire with infinitesimal cross section. If so, the magnetic inductance field transfers its energy ∆������������������������ to the kinetic energy ofmoving wire element. Therefore, appraising the wire movement direction we can define themagnetic inductance vector orientation. Then measuring ∆������������������������ and knowing | ∆������������ | we cancalculate the magnetic inductance strength as a double limit |������������| = Δl���������i���������m������→0 ∆������������������������ (1.47) |∆������������������������������������||∆������������| | ∆������������ |→0Consequently, we can define the magnetic inductance strength as the magnetic energy requiredto move the infinitesimal element carrying the volume current density 1A/m2 at distance 1m.1.6.13 Gauss’s Law for Magnetic Field (Axiom #3). 4th Maxwell’s EquationGauss’s law expresses the total flux of magnetic fields at any moment of time through the closedsurface area ������������ of any shape (see Figure 1.6.5a where the electric charges are replaced withmagnetic monopoles) and can be written as ∯������������ ������������ ∘ ������������������������ = ������������������������ (1.48)The equity (1.48) is the 4th Maxwell’s equation in the integral form in Table 1.7. Here ������������������������(������������) isthe total magnetic charge within some volume V at any current moment of time.If you have some concerns that nobody has ever seen magnetic monopole, and not for lack oflooking, you always can put ������������������������ = 0. Maxwell’s equations will survive but loose some beautyof symmetry. Hence, we would like to keep it. As John Preskill pointed out in [18] that “Thecase for its (magnetic monopole) existence is surely as strong as the case for any otherundiscovered particle.” That is not the single argument but the further discussion is a bit beyondthis book subject. Transforming (1.48) in the same way as (1.26) with the volume density ������������������������������������defined by (1.10) we can get the point-to-point or differential form of 4th Maxwell’s equation

BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICS ������������ ∘ ������������ = ������������������������������������ (1.49) 0 Now, we can populate 3rd Maxwell’s equations as well the new residents ������������������������������������, ������������������������������������ and ������������0 into the basement of Maxwell’s q House, as shown in Figure 1.6.11a. Use the fact that the charge qv - conservation law must be fair for any charges we can get the 0 conservation law and continuity equation for magnetic charges Figure 1.6.11b using (1.8) – (1.10) as Maxwell’s House ������������������������ = ∯������������ ������������������������������������ ∘ ������������������������ = − ������������������������������������ (1.50) ������������������������ (1.51) ������������ ∘ ������������������������������������ + ������������������������������������������������ =0 ������������������������ The relationship (1.51) is incorporated in Maxwell’s House basement, as shown in Figure 1.6.11b.1.6.14 Magnetic Lines of ForceThe images of the magnetic field distributions can be no less useful and beautiful than the a) b) c) Figure 1.6.12 Magnetic inductance field distributioninductance distributions in Figure 1.6.12 like around a) metal rings with currents, b) the Earth,and c) two coplanar bar magnets.1.6.15 Vector of Magnetic Field Strength. 2nd Maxwell’s EquationCome back to Lorentz’s force equation ������������������������������������ = Δ������������������������(������������ + ������������ x ������������) and consider the cross product ������������ x ������������������������������������ = Δ������������������������(������������ x ������������ + ������������ x (������������ x ������������) ) (1.52) For the sake of simplicity, assume that the charge carried by sensor #3 moves with speed ������������ in direction perpendicular to the magnetic vector B, as shown in Figure 1.6.13.Figure 1.6.13 Triple cross It is evident from this figure that ������������ x (������������ x ������������) = −������������2������������. product ������������ x (������������ x ������������) Therefore, the right-hand side of (1.53) can be rewritten as Δ������������������������(������������ x ������������−������������2������������) = Δ������������������������ �������1������0 ������������ x ������������−������������2������������0������������� = Δ������������������������ �������������������������22 ������������ x ������������ − ������������� (1.53) ������������2������������0Here ������������2 = 1 is the speed of light. In order to undo the dependence of the B-vector in (1.53) ������������0������������0from the permeability ������������0 we introduced a new vector H as ������������ = ������������⁄������������0 [(kg A−1 s−2)⁄(m kg A−2 s−2) = A m−1] (1.54)

Chapter 1called magnetic field strength. This vector plays the same role in the magnetic field descriptionas the displacement vector D in electric fields. The equity (1.54) is the second constitutiverelation in Table 1.7. In a vacuum, the only difference between vectors H and B is thedimension dictated by the chosen SI units. We explain the importance of vector H lateranalyzing the interaction of EM fields with magnetic moments of materials. Loosely speaking,this additional vector is not required, and everything can be expressed in term of B-field alone.Nevertheless, it is worthwhile to point out that according to (1.48) B-fields depend on eachmagnetic charge, i.e. on each of the myriad of binding in the matter magnetic moments as wellas the external free ones playing the role of EM field source. If so, it is convenient all theseadditional fields induced by the bound magnetic moments in a matter to count and include intoH-vector the same manner as for E-field, i.e. putting ������������ = ������������0������������������������������������. The coefficients ofproportionality ������������������������ taking into account such secondary magnetic fields is called the relativepermeability of material. We will come back to this subject later in Chapter 2.The factor in the front of parenthesis according to Table 1.5 has the unit dimension [V/s] thatcan be interpreted as time variation of magnetic current Δ������������������������⁄Δ������������ = Δ������������������������⁄(������������2������������0) of the samedimension. Finally, let us multiply both sides of (1.54) by Δ������������ and keep in mind that ������������Δ������������ = Δ������������where Δ������������ is the path that the electric charge Δ������������������������ passes for time Δ������������. Then we haveΔ������������ x ������������������������������������ = −Δ������������������������ ������������� − ������������2 ������������ x ������������� [J] (1.55) ������������2It turns out that all terms in (1.55) according to Table 1.5 has the unit dimension of energy inJoules and thus are measurable. If so, the magnetic field strength | ������������ | [A/m] can be defined asthe limit |������������| = Δl���������i���������m������→0 ∆������������������������⁄ Δ������������������������ (1.56)Here Δ������������������������ = Δ������������������������(Δ������������)3⁄((Δ������������)2������������0) and all the values Δ������������������������, Δ������������ and Δ������������ can be measuredexperimentally. Later we will give more natural definition of | ������������ | through the electric currentinduced by magnetic field in a small loop. Note that for the first time and quite naturally wecould replace the electric charge with some equivalent in action magnetic current in order tosimplify the equation. As we have pointed out before, such use of “no existing” equivalent fieldsources is widely practiced in electrodynamics.Now, look at two terms in parenthesis (1.55). Since the ratio ������������2⁄������������2 is dimensionless both termshave the same units of [A/m]. Therefore, they both are described the magnetic field and (1.55)can be written as ������������ x ������������������������������������ = − Δ������������������������ ������������� + ������������2 ������������������������ � (1.57) Δ������������ ������������2Here ������������������������ is the vector of magnetic field is due to the electric field ������������ existence ������������������������ = −������������ x ������������ (1.58)Eventually, we can define the magnetic potential ������������������������ in [Amps] in the same way as electricpotential������������������������ = ∮������������ ������������������������ ∘ ������������������������ = − ∮������������ (������������ x ������������) ∘ ������������������������ = ∮������������ ( ������������ x ������������ ) ∘ ������������������������ [A] (1.59)


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