This chapter introduces the language and structure of classical electromagnetism as a field theory coupled to matter. It begins by motivating excitations of the electromagnetic field and their interactions with charged matter. Maxwell’s equations are presented in differential form and related to the charge and current densities that act as sources. The chapter develops a continuum description of charge and current, derives the continuity equation, and obtains equations of motion for charged matter in both continuum and particle form. Recasting Maxwell’s equations as wave equations emphasizes propagation of electromagnetic disturbances. Point-like electric and magnetic dipole sources are introduced, leading to the macroscopic electric displacement and auxiliary magnetic fields and the form of Maxwell’s equations in polarizable, magnetizable media. Finally, the chapter derives the conservation laws for electromagnetic energy, momentum, and angular momentum, connects them to simple continuum mechanics of charged matter, and closes with problems that reinforce these physical foundations.

Chapter 1 presents a summary of the brief history of the geometric optics method for light-scattering computation. Then, basic physical variables and constants of quantifying light scattering by a particle are introduced. In particular, Maxwell’s equations are briefly discussed to introduce the Poynting vector, which is used in later discussions of electromagnetic energy conservation concerning extinction, scattering, and absorption. We also introduce the concepts of the Stokes parameters, amplitude scattering matrix, optical theorem, phase matrix, and scattering/absorption/extinction cross sections. The simplification of the scattering matrix is discussed with respect to particle orientations in both the particle and laboratory coordinate systems. We also discuss chirality and mirror symmetry associated with light scattering by nonspherical particles. In addition, we discuss the extinction matrix associated with a specifically oriented particle. The remaining portion of this chapter is devoted to the discussion of Snell’s law, Fresnel equations, and Fraunhofer diffraction. The scalar and vector Kirchhoff diffraction formulas are discussed in detail. Furthermore, we discuss the applicability of Babinet’s principle to the diffraction computation of a particle.

This chapter reviews the main concepts of electromagnetic theory relevant for the understanding of this textbook. Based on Maxwell’s equations, we derive the wave equation and discuss homogeneous solutions, such as plane waves and evanescent waves. We derive the boundary conditions at interfaces between homogeneous media and the Fresnel reflection and transmission coefficients. We discuss energy conservation, causality, and reciprocity of electromagnetic fields. Point response functions are introduced (Green functions) in order to derive the inhomogeneous solution of the wave equation. The chapter concludes with the angular spectrum representation, a framework that allows arbitrary fields to be described as a superposition of plane and evanescent waves.

To this point in our discussions, we have dealt solely with static fields. We started with static electric fields, in which all charges are stationary. The electric fields produced by these charges are stationary as well. With electric fields, we developed the notion of the electric potential, the energy stored by electric fields, and the capacitance of a configuration of conductors. We then moved on to introduce static magnetic fields, which are produced by stationary currents. For magnetic fields, we have also introduced potential functions, one a vector function, the other a scalar, but we have not yet discussed the energy stored by a magnetic field, or the inductance of a configuration of current-carrying wires. We will, of course, treat these important topics, but before we do so, we find it useful to take a first look at some time-varying effects. In particular, we will develop a law known as Faraday’s Law, which is the basis for circuit elements such as inductors and transformers, as well as electrical generators and many other useful devices. After we have mastered Faraday’s Law, we will be in a much better position to discuss the energy stored in magnetic fields and inductances, and so we will return to these topics at that time.

As we have been discussing electric and magnetic effects throughout this text, we have been developing a set of equations, known collectively as Maxwell’s Equations, that describe the properties of these fields in a very general sense. These equations are named after James Clerk Maxwell, whose contributions are discussed in Biographical Note 7.1. The development of Maxwell’s Equations has been critical to our understanding and application of electromagnetic effects, as they govern such diverse effects as are present in capacitors, transformers, and electric generators, which we have already examined, and free-wave propagation, transmission lines, waveguides, and antennas, which we have not yet discussed. Before we can undertake our study of these new topics, we must first complete the development of Maxwell’s Equations, which are not quite finished. As we will show shortly, there is an inconsistency in these equations as they stand to this point, an inconsistency that can be rectified by introducing a new term, known as the displacement current, to Ampère’s Law. This additional term is the final piece of the puzzle, and with its inclusion Maxwell’s Equations can be used to describe wave propagation, allowing us to understand (at an overview level, at least) the principles that govern our wireless routers, microwave ovens, and cable and satellite TV systems. In this chapter, we will introduce the displacement current, redefine the potential functions for time-varying fields, and re-examine the boundary conditions that must be satisfied at the interface between two different materials.

Readers who do not have strong schooling in physics can consult this book chapter for an introduction to key concepts such as ion fluxes, electric fields, electric potentials, and electric currents as well as for definitions of the ohmic, electrodiffusive, and capacitive currents that govern the electrodynamics of brain tissue. Building on the biophysical principles and approximations introduced here, we explain how the electric potential surrounding neurons can be computed based on the principles of current conservation and electroneutrality, and wegive a brief overview of modeling schemes designed to perform such computations on computers. Towards the end of the chapter, we show how the standard theory for computing extracellular potentials relates to Maxwell’s equations and list the approximations that we typically make when we apply these equations in a complex medium like brain tissue.

Propagation of electromagnetic waves inside a GRIN medium is studied in this chapter. Section 2.1 starts with Maxwell’s equations and uses them to derive a wave equation in the frequency domain. A mode based technique is used in Section 2.2 for solving the wave equation for a GRIN device fabricated with a parabolic index profile. The properties of both the Hermite’Gauss and the Laguerre-Gauss modes are discussed. Section 2.3 is devoted to other power-law index profiles and employs the Wentzel-Kramers Brillouin method to discuss the properties of modes supported by them. We discuss in Section 2.4 the relative efficiency with which different modes are excited by an optical beam incident on a GRIN medium. The intermodal dispersive effects that become important for pulsed beams are also covered. Section 2.5 describes several non-modal techniques that can be used for studying wave propagation in GRIN media.

After the failures of 1858 and 1865, the Atlantic was finally spanned by a submarine cable in 1866. A boom in cable laying ensued as British firms built a global cable network would remain a bulwark of British imperial and commercial power well into the twentieth century. The surging cable industry created a demand for electrical knowledge that stimulated the emergence of physics teaching laboratories in Britain. These laboratories turned out scientists, engineers, and teachers trained in precision electrical measurement—essentially cable testing room techniques. The cable enterprise also set the direction of British electrical research in the late nineteenth century, including the reception and articulation of Maxwell’s field theory. In the early 1880s a circle of young “Maxwellians” emerged in Britain, among them Oliver Heaviside, a former cable engineer who had taken up Maxwell’s theory as a tool to address signalling problems. Guided by ideas about energy flow and signal propagation, in 1884 Heaviside recast the long list of equations Maxwell had given in his Treatise into the compact set now universally known as “Maxwell’s equations.” The form of Maxwell’s field theory that passed into textbooks in the 1890s was rooted in important ways in the global cable network.

Maxwell's discovery of the equations for the electromagnetic field was one of the greatest achievements of nineteenth-century physics. The laws of electrostatics, magnetostatics and the magnetic fields produced by steady currents were established by the 1820s. Faraday's discovery of electromagnetic induction in 1831 led to his concept of magnetic field lines and the visualisation of the laws of induction as being determined by the rate of cutting of the field lines. Maxwell mathematised Faraday's model of magnetic field lines by adopting a mechanical model of the aether. His theoretical investigations led to the introduction of the displacement current and the discovery that light is electromagnetic radiation.

In this chapter Maxwell’s equations are described and common ways to solve them analytically are discussed. The equations imply certain properties of matter with which it interacts and full solutions that describes this behavior analytically are provided from first principles. The chapter shows specifically that one can derive very fundamental properties with simple calculations. Furthermore, the concept of inductance and capacitance are highlighted by reference to their duality. Various high-speed phenomena are studied in some detail with particular attention to the current distributions induced by the magnetic field.