Electromagnetic wave equations

MAXWELL S EQUATIONS, ELECTROMAGNETIC WAVES, AND STOKES PARAMETERS... 

M.I. Mishchenko, L.D. Travis, Maxwell s equations, electromagnetic waves, and Stokes parameters. In Photopolarimetry in Remote Sensing, ed. by G. Videen, Ya. Yatskiv, M. Mishchenko (Kluwer, Dordrecht, 2004) pp. 1-44... 

Consider the reflection of a normally incident time-harmonic electromagnetic wave from an inhomogeneous layered medium of unknown refractive index n(x). The complex reflection coefficient r(k,x) satisfies the Riccati nonlinear differential equation [2] ... 

Equation (A 1,6.8). along with the definitions (Al.h.S) and (Al.6.6) constitute the central equation for the propagation of electromagnetic waves in free space. The fonn of section A 1,6.4 admits hamionic solutions of the fonn... 

In the course of his research on electromagnetic waves Hertz discovered the photoelectric effect. He showed that for the metals he used as targets, incident radiation in the ultraviolet was required to release negative charges from the metal. Research by Philipp Lenard, Wilhelm Hallwachs, J. J. Thomson, and other physicists finally led Albert Einstein to his famous 1905 equation for the photoelectric effect, which includes the idea that electromagnetic energy is quantized in units of hv, where h is Planck s con-... 

Equation (1) has been set above in the case of the SHG process. This process entails the mixing of two identical frequency electromagnetic waves and therefore constitutes a particular case of the general problem of the mixing of two different frequency waves. The general form of Eq. (1) is therefore [15,22] ... 

We know from Maxwell s equations that whenever a charged particle undergoes acceleration, electromagnetic waves are generated. An electron in a circular orbit experiences an acceleration toward the center of the orbit and as a result emits radiation in an axis perpendicular to the motion. 

This invariance defines the principle of (Galilean) relativity, once thought to be universally valid. However, the situation for electromagnetic waves is different and the form of equation (22) is destroyed3 under a transformation... 

We are trying to discover what waves are generated inside and outside a crystal when it is illuminated by X-rays. The propagation of any electromagnetic waves in any medium is accurately described by Maxwell s equations. These are, in vector notation. 

Ray TtiacU.ng. An electromagnetic wave propagating perpendicular to the gradient of the refractive index of the medium will bend in the direction of this gradient. The bending phenomenon is completely described by the ray equation ... 

Radiation is the rate of heat transfer by electromagnetic waves emitted by matter. Unlike conduction and convection, radiation does not require an intervening medium to propagate. The basic rate of radiation heat-transfer equation between a high temperature (Th) black body and a low temperature Tf) black body is Stefan-Boltzmann s law ... 

As stated in the introductory chapter, we adopt a macroscopic approach to the problem of determining absorption and scattering of electromagnetic waves by particles. Therefore, the logical point of departure is the Maxwell equations for the macroscopic electromagnetic field at interior points in matter, which in SI units may be written ... 

Let us look for plane-wave solutions to the Maxwell equations (2.12)- (2.15). What does this statement mean We know that the electromagnetic field (E, H) cannot be arbitrarily specified. Only certain electromagnetic fields, those that satisfy the Maxwell equations, are physically realizable. Therefore, because of their simple form, we should like to know under what conditions plane electromagnetic waves... 

At the end of the nineteenth century classical physics assumed it had achieved a grand synthesis. The universe was thought of as comprising either matter or radiation as illustrated schematically in Fig. 2.1. The former consisted of point particles which were characterized by their energy E and momentum p and which behaved subject to Newton s laws of motion. The latter consisted of electromagnetic waves which were characterized by their angular frequency and wave vector and which satisfied Maxwell s recently discovered equations, ( = 2nv and — 2njX where v and X are the vibrational frequency... 

Whittaker s early work [27,28] is the precursor [4] to twistor theory and is well developed. Whittaker showed that a scalar potential satisfying the Laplace and d Alembert equations is structured in the vacuum, and can be expanded in terms of plane waves. This means that in the vacuum, there are both propagating and standing waves, and electromagnetic waves are not necessarily transverse. In this section, a straightforward application of Whittaker s work is reviewed, leading to the feasibility of interferometry between scalar potentials in the vacuum, and to a trouble-free method of canonical quantization. 

Therefore, questioning the physical significance of potential is not relevant here. The new formulation of Maxwell s equations [20-23], where potentials are directly coupled to fields clearly indicates that potentials, play a key role in particle behavior. To make a long story short, the difference in nature between potentials and fields stems from the fact that potentials relate to a state of equilibrium of stationary waves in the medium usually nonaccessible to an observer (except when potentials are used in a measurement process of the interferometric kind, at a given instant in time). Conversely, fields illustrate a nonequilibrium state of the medium as an observable progressive electromagnetic wave, since this wave induces the motion of material particles. 

The first attempt to formulate a theory of optical rotation in terms of the general equations of wave motion was made by MacCullagh17). His theory was extensively developed on the basis of Maxwell s electromagnetic theory. Kuhn 18) showed that the molecular parameters of optical rotation were elucidated in terms of molecular polarizability (J connecting the electric moment p of the molecule, the time-derivative of the magnetic radiation field //, and the magnetic moment m with the time-derivative of the electric radiation field E as follows ... 

When a beam of x-rays strikes an electron, some of the energy is momentarily absorbed, displacing the electron from its unperturbed position. This sets the electron in periodic motion with the same frequency as that of the exciting radiation. As a result the electron radiates an electromagnetic wave in all directions with the same frequency as the exciting radiation. This leads to the experimental observation that the incident radiation is scattered by the electron. A theoretical analysis (10, 11) of these events leads to the Thompson scattering equation which relates the intensity of x-rays scattered by a single electron, Ie, to that of the incident non-polarized x-radiation, I0 ... 

Equations (723) and (727) therefore represent a closed, cyclically symmetric, algebra in which all three space-like components are meaningful. The cyclical commutator basis can be used to build a matrix representation of the three spacelike magnetic components of the electromagnetic wave in the vacuum... 

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