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This chapter presents the basic notions of electromagnetism, chiefly the balance laws in global and local form. The corresponding Maxwell’s differential equations and boundary conditions are then derived. Electromagnetic potentials are introduced along with gauge conditions. They are then applied to determine the electromagnetic field in various cases. Although the standpoint is nonrelativistic, the chapter also exhibits the Lorentz-invariant form of Maxwell’s equations, essential for any relativistic approach.

This chapter presents how MHD equations can be derived from the general conservation equations for fluid mechanics coupled with the Maxwell equations modelling the electromagnetic phenomena. A hierarchy of models is considered, from the most general one (a full time-dependent system consisting in the incompressible Navier-Stokes equations with a Lorentz body force calculated from the Maxwell equations and the Ohm's law) to the most simplified one. Depending on the physical context, one model or the other is appropriate. The most sophisticated model raises unsolved questions of existence and uniqueness, mainly related with the hyperbolic nature of the Maxwell equations, but some simpler models can be fully analysed.

This chapter examines ion transport processes, their description, and the coupling phenomena. It is established that their rate is proportional to the extent of the deviation from the equilibrium, i.e., to the gradients of electrochemical potential and of mechanical pressure. The most common theoretical approaches to describe transport processes are shown: the phenomenological, the Fickian, the Stefan-Maxwell, and the Nernst-Planck approach. Special attention is given to the Nernst-Planck formalism. The assumptions made in the derivation of the Nernst-Planck transport equations are described, and the main ideas of the alternative formulations are outlined.

Nigel Hitchin has played a key role in the exploration of four-dimensional Riemannian geometry, and in particular has made foundational contributions to the theory of self-dual manifolds, four-dimensional Einstein manifolds, spinc structures, and Kähler geometry. In the process, he called attention to the profound mathematical interest of beautiful geometric problems that had previously only been considered by physicists. This chapter focuses on an interesting relationship between the four-dimensional Einstein–Maxwell equations and Kähler geometry, and points out some fascinating open problems that directly impinge on this relationship.

This chapter introduces material relevant to the interaction of atoms and radiation. Maxwell's equations and the electromagnetic wave equation are introduced. The electric and magnetic fields produced by oscillating electric and magnetic dipoles are derived, and the rate at which these dipoles radiate energy and angular-momentum are derived. Multiple fields are considered briefly.

This chapter first provides a brief review of several other theories, all classified as generalized thermoelasticity and due to Green and Naghdi. Next follows a justification of the presence of a material time derivative rather than a partial time derivative in the Maxwell‐Cattaneo equation. Another way to see this is to take the continuum thermodynamics as a starting point for the derivation of constitutive laws. However, the partial time derivative may be employed in a theory focused on solid mechanics in infinitesimal strains. In the sequel, some applications of the L‐S and G‐L theories are given: helices and chiral media, both with homogeneous as well as composite structures; surface waves; and thermoelastic damping in nanomechanical resonators. The chapter culminates with a thermoelasticity with anomalous heat conduction treated via fractional calculus, and a formulation of thermoelasticity of fractal media in the vein of dimensional regularization.

This chapter focuses on the modelling of one-fluid magnetohydrodynamics problems. The crucial point under consideration is the coupling between hydrodynamics phenomena and electromagnetic phenomena. From a mathematical viewpoint, the coupling induces a nonlinearity, additional to the nonlinearities already present in the hydrodynamics. A series of difficult, thus interesting, problems follow. With a reasonable amount of theoretical efforts, these problems can be dealt with. For instance, it can be shown that a system coupling the time-dependent incompressible Navier-Stokes equations with a simplified form of the Maxwell equations (the so-called low-frequency approximation) is well-posed when the electromagnetic equation is taken time-dependent, in parabolic form. In contrast, the same model is likely to be ill-posed when the electromagnetic equation is taken time-independent, in elliptic form.

This chapter considers a basic description of the generation, propagation and scattering of polarised electromagnetic waves. It assumes a starting familiarity with the basic form of Maxwell's equations, and then uses them together with formal matrix methods to develop several key ideas, including the importance of special unitary matrices, the concept of matrix decomposition via the use of the Pauli spin matrices in classical wave problems, and a basic definition of the scattering amplitude matrix. It also includes coverage of the important geometrical concepts of polarimetry such as the Jones vector and related propagation calculus, Stokes parameters, and the Poincaré sphere, as well as considering coordinate issues that arise when using microwave antenna coordinates in radar studies. It concludes with an introduction to the important concept of a scattering vector that forms the basis for studies in later chapters.

This chapter begins with a discussion of Newton's mechanics. It then covers Maxwell's equations, Minkowski spacetime, Poincaré group, Lorentz group, special relativity, Newtonian law, relativistic law, equivalence of mass and energy, and continuous matter.

This chapter considers the theory of classical X-ray wave-fields in free space, taking the Maxwell equations as a starting point. Vacuum wave equations (d’Alembert equations) are developed for X-rays in vacuum. In this respect, the concepts of a spectral decomposition, the complex analytic signal, and the angular spectrum of plane waves are developed. Several diffraction theories are outlined, including the Fraunhofer diffraction formula, the Fresnel diffraction formula, the Kirchhoff integral, and the Rayleigh-Sommerfeld integrals of the first and second kinds. The theory of partially coherent fields is also discussed, including topics such as random processes, the concept of partial coherence, the mutual coherent function, the van Cittert-Zernike theorem, and the Hanbury Brown-Twiss effect.

This chapter considers the other great domain of classical physics, the macroscopic and deterministic world of electric charge. Here, the four Maxwell equations unite electricity with magnetism and engender the electromagnetic wave as the fruit of the union. It will be the last stop before quantum mechanics undermines the certainty of the classical mechanical universe.

Tensors in a general coordinate system are introduced. When a tensor is expanded in terms of a set of basis (or inverse basis) vectors, the coefficients of expansion are its contravariant (or covariant) components with respect to this basis. The requirement of metric invariance in Minkowski spacetime leads to a generalized orthogonality condition, from which the Lorentz transformation can be derived. In terms of tensor we can have a manifestly covariant formalism. Maxwell's equations, the Lorentz force law, and the charge conservation equation are presented in their covariant forms. The symmetric energy-momentum tensor of a field system is introduced and the physical meaning of its components is discussed.

This chapter discusses the following: Maxwell's equations and the magnetohydrodynamic approximation; properties of cosmical plasmas; macroscopic equations for a fully ionized gas and the two-fluid model; equations to the flow of the whole gas; generalized Ohm's law; the energy equation of a fully ionized gas; kinematic coupling; dynamical coupling; the three-fluid model; and ‘anomalous’ resistivity.

This chapter illustrates how the foundations of the fluid description are rooted in statistical mechanics and in kinetic theory. This approach, which is appropriate for those systems composed of a very large number of free particles and extending over a length-scale much larger than the inter-particles separation, is first presented in the Newtonian framework and then extended to the relativistic regime. A number of fundamental conceptual steps are taken and treated in detail: the introduction of a distribution function that depends on the positions and on the four-momentum of the constituent particles, the definition of the energy–momentum tensor as the second moment of the distribution function, the discussion of the relativistic Maxwell–Boltzmann equation with the corresponding H-theorem and transport equations. Finally, equations of state are described for all possible cases of relativistic or non-relativistic, degenerate or non-degenerate fluids.

Chapter 3 explores the concept of the field, which is necessary to describe forces without resorting to action at a distance, and uses it to describe electromagnetism, as encapsulated by the Maxwell equations. First, scalar fields and the Klein–Gordon equation are discussed. Vector calculus is introduced. The physical meaning of Maxwell’s equations is explained. The equations are then solved for electrostatic fields. Non-uniform charge distributions and dipole moments are discussed. The vector and scalar potentials are introduced. Electromagnetic wave solutions of Maxwell’s equations are found and the Hertz experiment is described. Magnetostatics is discussed briefly. The Lorentz force is described and used to determine the motion of a charged particle in a cyclotron or synchrotron. The action principle for electromagnetism is described. The energy and momentum carried by the electromagnetic field are calculated. The reaction of a charged particle to its own electromagnetic field is considered.

This chapter gives a theoretical description of the basic properties of electromagnetic radiation. Maxwell's equations are first reviewed; the expressions of electrodynamic potentials in the vacuum and in polarized media are then given. The classic theory of the scattering of X-rays by electrons is described (Thomson scattering). The dielectric susceptibility (polarizability) of matter for X-rays and the Fourier expansions of its real and imaginary parts in a periodic medium (index of refraction, atomic scattering factor, and absorption coefficient) are discussed. A detailed account of Ewald's dispersion theory that is at the base of Ewald's dynamical theory is then presented. The propagation equation of X-rays, which is used throughout the book, is derived from Maxwell's equations according to Laue's basic assumptions. The last part of the chapter is devoted to specular reflection and Fresnel relations.

This chapter first introduces the constitutive relations which are commonly used in electromagnetic theory for the mathematical modelling of complex electromagnetic media. These constitutive relations are to be understood as operators connecting the electric flux density and the magnetic flux density with the electric and the magnetic fields. When they are introduced into the Maxwell equations, this chapter obtains differential equations (PDEs) that govern the evolution of the electromagnetic fields. This chapter also seeks to formulate and discuss the scope of the various problems related to the Maxwell equations that will be treated in this volume. It introduces and formulates in terms of differential equations various problems of interest related to the Maxwell equations: time-harmonic problems, scattering problems, time-domain evolution problems, random and stochastic problems, controllability problems, homogenisation problems, and others.

Chapter 2 reacquaints the reader with electric and magnetic force fields and their interactions with ponderable media through Maxwell’s equations and the accompanying Lorentz force law. Macroscopic quantities of permittivity and permeability are emphasised, and through the constitutive relations, polarisation and magnetisation fields. Dipole radiation, space-propagating and surface-propagating wave solutions to Maxwell’s equations are all fundamental to understanding energy and momentum transport around, and through, atomic scale and nanoscale structured materials. The chapter ends with a development of plane wave propagation, reflection, and transmission in homogenous media and at dielectric and metallic surfaces.

This chapter begins with a review of the semi-classical model of the interaction of charged particles with light, and then proceeds to the full quantum theory by replacing the classical fields with the corresponding field operators. A presentation of the quantum Maxwell equations and their behavior under parity and time reversal transformations is followed by a discussion of stationary density operators. The notion of positive- and negative-frequency parts of field operators is extended to interacting fields and used to define multi-time correlation functions that describe experimental results. The perturbation expansion is formulated by means of the interaction picture and combined with the dipole approximation to calculate the Einstein A- and B-coefficients. The chapter ends with a discussion of spontaneous emission in a cavity and Raman scattering.