The electromagnetic wave equations

For the case of an uncharged infinite, isotropic, nonconducting medium, substitution of the constitutive relations into Maxwell s equations yields 

By taking the curl of equations (2.17) and (2.18) we can transform these first-order differential equations which involve both E and H into second-order equations for either E or H alone. We make use of the vector identity 

In order to present these equations in a more familiar form, we may write the electric field as E = E + E B  

We see then that each cartesian component of the fields E and H satisfies the equation 

It is important to note that equations (2.17) and (2.18) predict that every electric wave will be accompanied by an associated magnetic wave, and vice versa. The waves are therefore more properly termed electromagnetic waves. In 1888, following an experimental search for the magnetic effects of Maxwell s displacement current. Hertz discovered waves which had exactly this electromagnetic character and possessed all the other properties which can be predicted from Maxwell s equations. Our present sophisticated radio and telecommunication systems have all developed from these first primitive experiments. 

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For electric fields E and magnetic fields H, either in a vacuum or in isotropic media, the wave equation for propagation is the electromagnetic wave equation (Problem 2.7.4) ... 

Problem 8.1.5) by using the electromagnetic wave equation and a frequency-dependent complex dielectric constant e(co) defined by... 

Let us now study the nature of the electromagnetic wave as described by Maxwell s equations. The electromagnetic wave equations (without electric charge and current) in the vacuum are ... 

The electromagnetic wave equation may then be written in a shorter form /2nn... 

Optical properties of nanoparticles can be elucidated by solving the electromagnetic wave equations, considering the boundary conditions near the nanoparticles. In 1908, Mie formulated vector wave equations for spherical nanoparticles and... 

We have used a 4 x 4 matrix formulation of the electromagnetic wave equations in stratified media to compute the reflectance and transmittance of single-domain cholesteric liquid crystal films. Our technique is basically equivalent to the 4x4 matrix technique first described by Teitler and Henvis, ) applied by them to... 

Using Eqs. (27c), (29a) and (29b) in evaluating V x (V x E) yields a source-dependent form of the electromagnetic wave equation... 

The free electron resides in a quantized energy well, defined by k (in wave-numbers). This result Ccm be derived from the Schroedinger wave-equation. However, in the presence of a periodic array of electromagnetic potentials arising from the atoms confined in a crystalline lattice, the energies of the electrons from all of the atoms are severely limited in orbit and are restricted to specific allowed energy bands. This potential originates from attraction and repulsion of the electron clouds from the periodic array of atoms in the structure. Solutions to this problem were... 

Thus, M and N have all of the properties required of the electromagnetic field. Furthermore, ij/ satisfies the scalar wave equation in spherical coordinates. 

Therefore, M and N have all the required properties of an electromagnetic field they satisfy the vector wave equation, they are divergence-free, the curl of M is proportional to N, and the curl of N is proportional to M. Thus, the problem of finding solutions to the field equations reduces to the comparatively simpler problem of finding solutions to the scalar wave equation. We shall call the scalar function ip a generating function for the vector harmonics M and N the vector c is sometimes called the guiding or pilot vector. 

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... 

Conventional electromagnetic theory is fully aware of this difficulty, but no attention is paid to the inconsistency. Pragmatically, Jackson simply notes that solutions to the wave equations must also satisfy Maxwell s equations [63, Chap. 7, p. 198], and go on to use Faraday s Eq. (8) as a coupling condition for the two wave equations. We will return to this point in Section V. 

The electromagnetic field is quantized as a set of harmonic oscillators. Maxwell s equations, and the resulting wave equations, are described by partial differential equations that formally have an infinite number of degrees of freedom. Physically this means that the electromagnetic held is described by an infinite number of harmonic oscillators, where one sits at every point in space. The modes of the electromagnetic held are then completely described by this ensemble of harmonic oscillators. 

The galactic redshift could obviously be attributed to the damping of the electromagnetic waves emitted from various galaxies in random motion within a stationary universe. Now, comparison between Hubble relativistic linear law and the logarithmic law that comes out from Maxwell electromagnetic wave equation shows that, in any case, the logarithmic law fits experimental data very well and thus better than linear law. 

Attention must now be paid to the exponential factor, exp( 2nir (n iij)/A), in Equation 6.5, where (n it) is known as the complex refractive index of a substance. It can be seen that the effect of this factor upon the electromagnetic wave increases with the distance Irl that the light travels in that medium. In the general case of an anisotropic medium, n and are referred to as a specific set of axes, usually chosen to coincide with the optical axes of the medium. For example, the axes of maximum and minimum transmittance are selected for anisotropic absorption. The extinction f for an anisotropic medium is related to the extinction coefficient through Equation 6.9. 

In an anisotropic dielectric the phase velocity of an electromagnetic wave generally depends on both its polarization and its direction of propagation. The solutions to Maxwell s electromagnetic wave equations for a plane wave show that it is the vectors D and H which are perpendicular to the wave propagation direction and that, in general, the direction of energy flow does not coincide with this. 

The Helmholtz equation resembles the spatial part of the classical wave equation for matter waves (waves in ocean, sound waves, vibrations of a string, electromagnetic waves in vacuum, etc.) of amplitude F = F(r, f) ... 

As we saw in Chapter 12, a photon s energy is given by E = hv, meaning that the energy, E, is proportional to v, the frequency of the electromagnetic wave. This equation can be combined with the equation for the energy difference between the spin states ... 

Solution In the case of harmonic motion, for which Sj= joiSj, Equation 2.17 implies that attenuation may be accounted for by representing the elastic constants cjj by complex elastic constants cu + jmr u. (This is analogous to accounting for dielectric loss in electromagnetic and optical waveguides by the well-known method of postulating a complex dielectric constant or a complex index of refraction.) Equation 2.13, the lossless wave equation for this shear wave, becomes... 

To determine the scattered radiation spectrum of an oscillating molecule under conditions of resonance excitation, we must consider how the polarizability a varies not only with normal modes of vibration but also with frequency of the incident radiation that excites them. For a molecule in a molecular state ) (initial) perturbed by the electromagnetic wave of frequency vq so that it passes into a molecular state I /) (final) while scattering light of frequency vo r (v = V/ - Vg), the matrix elements of a for the vibrational transition k, [oipa]k, are given by the Kramers-Heisenberg-Dirac (KHD) dispersion equation ... 

The polarization factor arises from partial polarization of the electromagnetic wave after scattering. Considering the orientation of the electric vector, the partially polarized beam can be represented by two components one has its amplitude parallel (Ay) to the goniometer axis and another has the amplitude perpendicular (Ax) to the same axis. The diffracted intensity is proportional to the square of the amplitude and the two projections of the partially polarized beam on the diffracted wavevector are proportional to 1 for (A ) and cos 20 for (Ax). Thus, partial polarization after scattering yields the following overall factor (also see Thomson equation in the footnote on page 140) ... 

Now we shall discuss some features of the electromagnetic wave spectrum in mixed crystalline solutions using the equations for the dielectric constant tensor of the solution derived in the mean polarizability approximation. We know that the effects of concentration broadening of absorption spectra are lost in this approximation. However, Onodera and Toyozawa (16), Dubovsky and Konobeev (17), Hoschen and Jortner (18), and Hong and Robinson (19), who have actually studied the corrections to the mean polarizability approximation, have shown that... 

It is important to note that minimal coupling requires specification of charge. We are interested in electronic solutions and accordingly choose q = e. The positronic solutions are obtained by charge conjugation. We obtain the relativistic wave equation for the electron in the presence of external electromagnetic fields which we shall write as... 

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