Practical Electrodynamics with Advanced Applications

Description

This textbook covers the advanced application and techniques of electrodynamics. The book begins with an introduction to the topic, with basic notations and equations presented, before moving on to examine various topics such as electromagnetic waves in a vacuum, the theory of relativity (including the Lorentz transformation) and electromagnetic fields in matter. Dispersion and transport are discussed, along with wave interactions in types of plasma and metamaterials, before the problems of electromagnetism in continuous matter are reviewed, and boundary interactions are studied. The second half of the book looks at the more advanced topics, including dielectric guides techniques, further metamaterial and plasma interactions (such as helicoidal phenomena), interactions involving conductivity and X-ray, and magnetic field dynamics. Condensed matter equations are covered along with more general matter relations, and an advanced study of the direct and inverse problems of electrodynamics closes the topic. Finally, advanced exercises are available in the final chapter. This is an excellent learning tool for students studying electrodynamics courses, and serves as a robust resource for anyone involved in the field. Key Features Provides discussions of fundamental principlesIncludes simplified exercises to assist the readerSimplified to bridge the gap between classical and applied mathematics PRELIMS.pdf Preface References Acknowledgement Author biography Sergey Leble Epigraph CH001.pdf Chapter 1 Introduction 1.1 General remarks: units 1.2 Inertial reference frames 1.2.1 Reference body: basic variables in mechanics 1.2.2 Observation of Galileo: relativity 1.2.3 Summarizing: Poincare group 1.3 Tensor fields 1.3.1 Definition of a tensor with respect to a group 1.3.2 Tensor field derivatives 1.3.3 Exercises References CH002.pdf Chapter 2 Basic notions and equations of electrodynamics 2.1 Electrodynamics in vacuum 2.1.1 Definition of electric field vector E⃗ 2.1.2 Lorentz force: definition of magnetic induction field B⃗ 2.2 Maxwell’s equations in integral form 2.2.1 I Maxwell’s equation from Coulomb’s law 2.2.2 II Maxwell equation: absence of magnetic charges 2.2.3 Faraday’s law: III Maxwell equation 2.2.4 Ampère–Maxwell law: IV Maxwell equation 2.2.5 System of Maxwell’s equations in differential form 2.3 Initial-boundary problem for Maxwell system in vacuum 2.4 Vector and scalar potentials 2.4.1 What possibilities give us the gauge choice?—Theoretic investigation 2.4.2 Hertz vector 2.5 Conservation principles: Poynting theorem References CH003.pdf Chapter 3 Electromagnetic waves in vacuum 3.1 Wave equations 3.2 Harmonic plane wave in vacuum without charges 3.3 Wave packets 3.4 Cauchy Problem in 1 + 1 space–time 3.4.1 The initial (Cauchy) problem for plane wave with a fixed polarization 3.4.2 On projection method 3.5 Discussion and exercises 3.6 Inhomogeneous wave equation: wave generation 3.6.1 On the Green function method for wave equations 3.6.2 General formulas for electromagnetic field and wave emission 3.7 Emission of the isolated charged point particle 3.8 Emission of oscillating charged system of particles: multipole expansion 3.8.1 General remarks 3.8.2 Multipole expansion 3.8.3 The emission phenomenon in the dipole approximation 3.8.4 Next approximation: quadrupole momentum 3.8.5 Dipole magnetic moment definition and corresponding wave emission 3.8.6 Minidiscussion and exercises References CH004.pdf Chapter 4 Theory of relativity 4.1 Lorentz transformation 4.2 Space–time geometry 4.3 Relativistic kinematics and four-vectors 4.4 Relativistic mechanics 4.5 Discussion 4.6 Exercises 4.7 A historical note: about a birth of new mechanics (theory of relativity) References CH005.pdf Chapter 5 Electromagnetic field in a matter 5.1 Definition of vectors: polarization, electric induction, magnetization and magnetic field strength—Maxwell’s equations for electromagnetic field in a matter 5.1.1 Implicit definition of the vectors: polarization, electric induction 5.1.2 On implicit definition of magnetization vector and magnetic field strength 5.1.3 Maxwell’s equations for electromagnetic field in matter 5.1.4 Explicit definition of vectors: polarization, electric induction 5.1.5 On explicit empiric definition of vectors: magnetic field strength H⃗ and magnetic induction B⃗ 5.1.6 The fields measurements in experimental practice 5.1.7 Material equations as thermodynamic equations of state 5.2 Macroscopic Maxwell’s equations, links to microscopic parameters 5.2.1 Resume, the empiric definition of the fields: macroscopic Maxwell’s equations 5.2.2 A link between polarization vector and dipole momentum 5.2.3 A link between magnetization vector and magnetic dipole momentum 5.2.4 More comments on the definitions 5.2.5 Energy conservation/balance law derivation for matter 5.3 Classification of substances with respect to electric and magnetic properties 5.3.1 Classification of continuum matter on a base of material equations 5.3.2 Dielectrics 5.3.3 Magnetics 5.3.4 Ferromagnetism 5.3.5 Combined action: multiferroics 5.3.6 On dielectrics 5.3.7 Metamaterials 5.3.8 Exercises References CH006.pdf Chapter 6 Dispersion and transport 6.1 Dispersion account, operator material relations 6.1.1 Maxwell’s equations: operators of dielectric permittivity and magnetic permeability 6.1.2 Energy density of wave packets in a dispersive medium 6.1.3 On Maxwell–Lorentz equations 6.2 Discussion 6.2.1 Polarization vector via a potential from coupled point charges 6.3 Dispersion in dielectrics, conductors and plasma 6.3.1 Lorentz model 6.3.2 Drude theory of metals: Ohm’s law 6.4 Back to Ohm’s law: Hall effect 6.4.1 On the DC Ohm law 6.4.2 Magnetic field account: Hall effect 6.4.3 Basic relations for 2D Hall resistance 6.4.4 On quantum electrodynamics manifestation of Hall effect 6.5 EM waves in isotropic conducting matter case 6.5.1 The linear equation and plane wave 6.5.2 The 1D nonlinear model outline 6.5.3 On dynamic projection method in linear problem 6.5.4 Dynamical projecting in a nonlinear problem References CH007.pdf Chapter 7 Plasma 7.1 Plasma types 7.1.1 General remarks: a matter as potential plasma 7.1.2 Atmosphere plasma 7.1.3 Solid state physics: electron–hole plasma 7.1.4 On stability of plasma 7.1.5 Tokamak plasma 7.2 Propagation of waves in a plasma: example of helicoidal waves 7.2.1 General remarks on plasma theoretical description 7.2.2 Simplifications and the linearized system: plasma waves 7.2.3 The initial problem formulation in matrix form: Fourier transform 7.2.4 The initial problem solution by dynamical projecting 7.3 The nonlinear case References CH008.pdf Chapter 8 Metamaterials 8.1 Research on metamaterials 8.1.1 Introduction: on the chapter content 8.2 Statement of problem: dispersion operator 8.2.1 Maxwell’s equations: operators of dielectric permittivity and magnetic permeability 8.2.2 Boundary regime problem 8.3 Projecting operators 8.3.1 Projecting operators approach 8.3.2 Projecting operators construction 8.4 Separated equations and definition for left and right waves 8.5 Nonlinearity account 8.6 Wave propagation in a metamaterial within the lossless Drude dispersion and Kerr nonlinearity 8.6.1 Drude model for dispersion 8.6.2 Interaction of left and right waves with Kerr effect 8.6.3 Stationary problem solutions 8.7 Discussion and conclusion References CH009.pdf Chapter 9 Problems of electromagnetism in a piecewise continuous matter 9.1 Electro- and magneto-statics 9.2 Boundary conditions 9.2.1 Absence of surface charges and currents case 9.2.2 Conditions for magnetic and electric moments 9.2.3 Magnetic moment: spins contribution 9.2.4 Surface charges and currents 9.3 Demagnetization field 9.3.1 Instructive example 9.3.2 General demagnetization 9.4 Stray fields 9.4.1 On definition 9.4.2 Landau–Lifshits–Gilbert (LLG) equation and domain wall (DW) motion model 9.5 Microwire: DW and observations 9.5.1 Wire DW model 9.6 The stray field of the planar DW 9.6.1 Observation of domain walls in ferromagnetic microwires References CH010.pdf Chapter 10 Reflection and refraction of electromagnetic waves at a boundary 10.1 Reflection and transmission of a plane wave on a border 10.1.1 General textbook relations 10.1.2 Ampère–Maxwell equation for a wave with given frequency 10.1.3 Reflection and transmission of wave propagating orthogonally to a boundary 10.2 Problem of a plane wave with fixed frequency refraction 10.3 Boundary conditions impact 10.3.1 Snell law 10.3.2 To Fresnel formulas generalization 10.4 Energy density flux 10.4.1 Preparation: Pointing vector for a matter 10.4.2 Polarization choice 10.4.3 The generalized Snell law 10.5 Discussion References CH011.pdf Chapter 11 New dielectric guides techniques 11.1 Planar waveguides 11.1.1 Novel experiments in dielectric guides 11.1.2 Dielectric slab as a waveguide 11.2 Cylindrical dielectric waveguides 11.2.1 On the problem 11.2.2 Linear problem 11.2.3 Transformation to frequency domain 11.2.4 Projection operators in time domain 11.3 Including nonlinearity 11.3.1 Application of projection operators References CH012.pdf Chapter 12 Propagation of electromagnetic waves in exclusive dispersive media such as metamaterials 12.1 Electromagetic waves in metamaterial 12.1.1 On dispersion in 1D metamaterial 12.2 Directed modes in rectangular waveguides: polarization, dispersion, nonlinearity 12.2.1 Maxwell’s equations for matter inside a waveguide 12.3 Boundary conditions: the transversal waveguide modes evolution 12.3.1 The boundary regime problem formulation for the transversal modes 12.3.2 Projecting operators 12.4 Rectangular waveguide filled with metamaterial: nonlinearity account References CH013.pdf Chapter 13 Plasma basic equations, waveguide formation 13.1 Maxwell-kinetic system 13.1.1 Joint electrodynamics—particles kinetics description 13.1.2 Kinetic equation: Vlasov plasma 13.2 Waves in homogeneous plasma 13.2.1 Cold plasma: general dispersion equation 13.2.2 Maxwell distribution background: Langmuir waves 13.2.3 More roots of the dispersion equation 13.3 Weakly inhomogeneous plasma 13.4 Plasma waveguides 13.4.1 On plasma confinement 13.4.2 The confined plasma perturbations 13.4.3 Hydrodynamic equations approach: flute instability References CH014.pdf Chapter 14 Helicoidal and other plasma wave phenomena 14.1 Helicoidal waves interactions 14.1.1 Basic equations 14.1.2 Introducing the projectors P+ and P− 14.1.3 Model with nonlinear term: the three-waves equation 14.1.4 The three-wave system in 1 + 1 case 14.2 Algebraic method of three-wave systems solution: solitons 14.2.1 Solutions derived using dressing by two-fold Darboux transformation (TfDT) 14.2.2 Solutions plots and discussion 14.3 Interaction of plasma waves 14.3.1 Interaction of Langmuir and ion waves References CH015.pdf Chapter 15 Diffraction in the presence of conductivity, x-rays manipulation and focusing 15.1 General remarks 15.2 Basic equations 15.3 Propagation of x-rays in vacuum 15.4 Approximation of electromagnetic field as a superposition of Gaussian beams 15.4.1 Paraxial equation for Kshevetskii–Wojda beam 15.4.2 Superposition of Gaussian beams 15.4.3 Quasi-exact solution of the Helmholtz equation 15.5 Oriented Gaussian beams method application to x-rays propagation through optical elements 15.6 Study of accuracy and efficiency of Gaussian beam methods 15.6.1 Estimation of convergence rate of solution obtained with superposition of oriented Gaussian beams to electric field described by boundary condition, which is a fast oscillating function 15.6.2 Propagation of x-rays through a lens and its aperture boundaries 15.7 Numerical calculations scheme 15.7.1 Implicit Runge–Kutta scheme 15.7.2 Algorithm: parameters of integration choice 15.8 The numerical simulations 15.8.1 General description 15.8.2 The first case, 33 aluminium lenses with 15 keV x-ray 15.8.3 The choice of space steps and errors 15.9 Results for ideal lenses and the bulk defects influence 15.9.1 Space steps choice and plots 15.9.2 Final remarks References CH016.pdf Chapter 16 Magnetic field dynamics, novel aspects of a theory based on Landau–Lifshitz–Gilbert equations 16.1 An exchange interaction concept 16.2 Heisenberg network dynamics 16.2.1 Heisenberg model: anisotropy 16.2.2 General continuum LLG equations 16.3 Walker theory 16.3.1 Walker solution of 1D LLG 16.3.2 Walker solution instability 16.3.3 Nanowires as Heisenberg chain 16.4 Propagation of domain wall in cylindrical amorphous ferromagnetic microwire 16.4.1 Introductory remarks 16.4.2 The LLG equation for cylindric microwires 16.4.3 LLG transforms: statement of problem 16.4.4 Basic equation in quadratic-linear approximation and its general solution 16.5 Average magnetization fields and DW dynamics 16.5.1 The third order nonlinearity account 16.5.2 Stationary background introduction 16.5.3 A linearization on a nonzero background 16.5.4 Averaging procedure and DW mobility 16.5.5 Velocity and acceleration 16.6 Exact particular solutions of LLG equation 16.6.1 Anisotropy coefficient coordinate dependence impact 16.6.2 The illustrations of DW form in 3D 16.6.3 Velocity of DW propagation: anisotropy constants determination 16.6.4 The field strength and induced magnetization by a coil References CH017.pdf Chapter 17 Condensed matter electrodynamics: equations of state by partition function 17.1 On derivation ab initio of an equation of state 17.1.1 The first law of thermodynamics forms by classical statistical physics 17.1.2 The first law of thermodynamics by quantum statistical physics 17.1.3 The scheme for two subsystems 17.2 Spin system and equations of state 17.2.1 Classical Langevin theory 17.2.2 Brillouin theory: space quantization 17.3 Heisenberg theory 17.3.1 Partition function 17.3.2 On Heisenberg equation solution 17.4 Para-, and ferro-magnetic matter 17.4.1 The magnetization curve for a paramagnetic 17.5 Problem of ferromagnetic state 17.5.1 Towards the Curie law 17.6 Multiferroics 17.6.1 Electric field action: Stark effect 17.6.2 Exchange integrals 17.6.3 Magneto-electric effect: material equation of state 17.6.4 On ferroelectricity 17.7 Fine particles case 17.7.1 Energy distribution 17.7.2 Back to statistical sum 17.7.3 Partition functions for a tiny particle 17.7.4 Numerical estimations and plot References CH018.pdf Chapter 18 More general material relations 18.1 A concept 18.1.1 E–D–B–H relations 18.1.2 Hydrodynamics–electrodynamics material relations 18.1.3 Continuum medium-electrodynamics material relations 18.1.4 Energy balance 18.2 Symmetry and groups 18.2.1 Crystallographic symmetry and groups 18.2.2 Tensorial symmetry with respect to indexes transpositions 18.2.3 Pauli symmetry with respect to electrons permutations 18.3 Euclidean and Lorentz symmetry 18.3.1 Euclidean group covariance 18.3.2 Lorentz group covariance 18.3.3 General tensors relations 18.4 Active dielectrics 18.4.1 Ferroelectrics 18.4.2 Piezoelectricity 18.4.3 Paraelectricity 18.4.4 Magnetoelasticity 18.5 Flexoelectricity 18.6 Ferroelasticity References CH019.pdf Chapter 19 On direct and inverse problems of electrodynamics 19.1 Direct problem of plane wave propagation in a layered medium 19.2 On inverse problem 19.2.1 Remarks 19.2.2 Some details of problem formulation 19.2.3 Equations of the inverse problem 19.3 Data collection methods: examples 19.3.1 Plasma Langmuir probe 19.3.2 Radar 19.3.3 Huygens’ and Kirchhoff’ formulas 19.3.4 Direct and inverse problems for a radar/lidar 19.4 Inverse problems as ill-posed one 19.4.1 The Tikhonov regularization References CH020.pdf Chapter 20 Advanced exercises 20.1 Short list of useful vector and tensor relations 20.2 A few definitions: curves, surfaces, integrals, etc 20.2.1 Curves 20.2.2 Surfaces 20.2.3 Integrals 20.2.4 Dirac delta-function 20.3 Projecting operators 20.4 Dressing method 20.5 Dielectric waveguides 20.5.1 Dielectric slab 20.5.2 Dielectric cylinder—optical fibers 20.5.3 Rectangular waveguide 20.6 Electromagnetic waves in metamaterials 20.6.1 Electromagnetic waves in metal rectangular waveguide, system derivation 20.7 Plasma confinement 20.8 Wave propagation at plasma 20.9 Refraction in presence of conductivity 20.10 Magnetism, a novel aspect 20.11 Condensed matter electrodynamics: equations of state by partition function 20.11.1 Paramagnetics and ferromagnetics 20.11.2 Multiferroics 20.12 General material relations 20.12.1 Piezoelectricity 20.13 Inverse problems of electrodynamics References