具体描述
Designed to be a text for Jr/Sr/beginning graduate level (4th and 5th year) and a reference for research scientists, "Modern Problems in Classical Electrodynamics" includes materials, such as lasers and nonlinear dynamics that are missing from traditional electrodynamics books. The book begins with relativistic mechanics and field theory, in part because they lend unity and beauty to electrodynamics, and in part because relativistic concepts appear frequently in the rest of the book. Relativity is a natural part of electrodynamics. After that, the book turns to electrostatics and magnetostatics, waves, continuous media, nonlinear optics, diffraction, and radiation by moving particles. Examples and homework exercises throughout the book are taken from condensed-matter physics, particle physics, optics, and atomic physics. Many are experimentally oriented, reflecting the view that classical electrodynamics has a broad importance in modern physics that extends beyond preparing students for quantum mechanics. At the end, the book returns to basics, and discusses the fundamental problems inherent in the classical theory of electrons.
Advanced Electromagnetism: Fields, Waves, and Applications A Comprehensive Exploration of Contemporary Electromagnetic Phenomena and Modern Computational Techniques This volume delves into the intricate world of classical electrodynamics, moving beyond the foundational Maxwell’s equations to explore their advanced applications in contemporary physics and engineering. Focusing on sophisticated mathematical frameworks and their practical manifestations, this text provides a rigorous treatment suitable for advanced graduate students and researchers seeking a deep, nuanced understanding of electromagnetic phenomena in complex, realistic scenarios. Part I: Foundations Revisited and Refined Formalisms The initial section solidifies the bedrock of electrodynamics while immediately pivoting towards higher-order theoretical constructs. Chapter 1: Canonical Electrodynamics and Variational Principles We begin with a meticulous review of Maxwell’s equations in differential and integral forms, emphasizing their covariance under Lorentz transformations. The discussion then transitions to the Lagrangian and Hamiltonian formulations of the electromagnetic field. This section rigorously derives the field equations from the action principle, establishing the fundamental connection between symmetries, conserved quantities (Noether's theorem), and the dynamics of charged particles interacting with the field. Emphasis is placed on gauge invariance and the role of potentials in shaping observable dynamics. Chapter 2: Tensor Analysis and Covariant Electrodynamics This chapter provides an in-depth exploration of the four-vector potential ($A^{mu}$) and the field strength tensor ($F^{mu
u}$). We derive the manifestly covariant form of Maxwell's equations, illustrating how relativistic invariance simplifies the structure of the theory. Topics include the transformation laws for electromagnetic fields under general Lorentz boosts, the construction of the energy-momentum tensor for the electromagnetic field, and the physical interpretation of its divergence. Chapter 3: Green’s Functions and Inhomogeneous Equations A significant portion of this section is dedicated to the formal solutions of the inhomogeneous wave equations governing potentials in the presence of sources. We detail the construction and physical interpretation of fundamental solutions (Green's functions) in both unbounded and bounded media. Specific attention is paid to the Lorenz gauge and Coulomb gauge Green's functions, their retardation properties, and their utilization in formulating causality-respecting solutions for retarded potentials and radiation problems. The discussion includes projection operators necessary for handling constraints imposed by gauge choices. Part II: Radiation and Wave Propagation in Complex Media This section moves into the dynamic aspects of electrodynamics, focusing on how electromagnetic energy propagates, scatters, and interacts with structured matter, often requiring methods beyond simple plane wave expansions. Chapter 4: Advanced Antenna Theory and Diffraction We move beyond elementary dipole and loop radiators to analyze complex, arbitrarily shaped antennas using advanced integral equation techniques. This includes a detailed treatment of the Magnetic Field Integral Equation (MFIE) and the Electric Field Integral Equation (EFIE), outlining the methodologies for their discretization, particularly the Method of Moments (MoM). The chapter concludes with a thorough analysis of diffraction phenomena using the geometrical theory of diffraction (GTD) and the uniform theory of diffraction (UTD) for high-frequency scattering approximations near sharp edges. Chapter 5: Wave Propagation in Anisotropic and Inhomogeneous Media This chapter addresses the propagation of electromagnetic waves through materials where the permittivity ($epsilon$) and permeability ($mu$) are not scalar constants but are tensors or spatially dependent functions. We analyze birefringent crystals, bianisotropic materials (where electric and magnetic polarization are coupled), and layered media. The reflection and transmission coefficients for arbitrarily polarized waves incident upon such interfaces are derived using matrix methods, including the $4 imes4$ transfer matrix formulation for multilayer stacks. Chapter 6: Relativistic Electrodynamics of Moving Media The analysis of electromagnetic fields within media moving at relativistic speeds necessitates a careful application of special relativity. This chapter derives the generalized Maxwell’s equations appropriate for moving frames. Key topics include the transformation of constitutive relations ($mathbf{D}$ and $mathbf{B}$) when crossing boundaries between stationary and moving reference frames, and the resulting polarization currents and magnetization induced by the motion. The phenomenon of aberration and Doppler shift for light propagating in moving media is examined quantitatively. Part III: Advanced Topics in Boundary Value Problems and Numerical Methods The final part focuses on the practical computational tools required to solve analytically intractable electromagnetic problems, particularly those arising in engineering physics. Chapter 7: Boundary Value Problems in Complex Geometries This section tackles sophisticated boundary value problems that demand specialized mathematical techniques beyond separation of variables. We explore the application of Wiener-Hopf methods for semi-infinite structures (e.g., a conducting half-plane). Furthermore, we introduce the dual integral equation formulation for certain boundary conditions, providing analytical insights into problems involving wedges and corners where field singularities are present. Chapter 8: Finite Element Methods (FEM) in Electromagnetics The Finite Element Method provides a powerful framework for modeling electromagnetic fields in structures with arbitrary geometries and non-uniform material properties. This chapter details the weak formulation of Maxwell’s equations suitable for FEM discretization, focusing on the proper handling of boundary conditions (Dirichlet vs. Neumann) and the critical issue of spurious modes arising from $mathbf{H}$ and $mathbf{E}$ field formulations. We discuss advanced interpolation functions (e.g., edge elements or Nédélec elements) necessary for satisfying the divergence constraints ($
abla cdot mathbf{B} = 0$). Chapter 9: Time-Domain Simulations and Dispersive Materials For transient phenomena and scenarios involving frequency-dependent material response, time-domain techniques are essential. This chapter presents the Finite-Difference Time-Domain (FDTD) method, detailing the Yee algorithm and its stability criteria (CFL condition). The complexity introduced by materials exhibiting dispersion (frequency dependence in $epsilon$ or $mu$) is addressed through the incorporation of Debye or Lorentz models for polarization currents into the time-stepping scheme, requiring specialized time-marching procedures. Chapter 10: Magnetohydrodynamics (MHD) and Plasma Waves The final chapter bridges classical electrodynamics with plasma physics. We develop the single-fluid macroscopic model for electrically conducting fluids (MHD), deriving the coupled equations governing fluid motion and magnetic field evolution. Specific attention is given to wave phenomena within plasmas, including Alfvén waves, whistler-mode propagation, and the fundamental concepts underpinning magnetic confinement relevant to contemporary fusion research. The non-linear aspects, such as magnetic reconnection instabilities in highly conductive fluids, are introduced. Intended Audience: Researchers, advanced graduate students in physics, electrical engineering, and applied mathematics. A strong background in vector calculus, partial differential equations, and special relativity is assumed. This text aims to bridge the gap between introductory texts and highly specialized research monographs, providing the rigorous mathematical machinery necessary for tackling the frontier problems in electromagnetism.
