Maxwell wave equations

We relate here the scattered field to the polarization induced in the sample by external fields. The radiated field is the solution of the Maxwell wave equation... 

The Maxwell wave equation for a monochromatic probe of frequency a = ck in a medium... 

In this subsection, we will investigate the propagation dynamics of an incident probe pulse in ultracold atoms aroimd a photonic bandgap induced by a time-independent SW coupling. Then it is necessary to resort to the Maxwell wave equations coupled with the density matrix equations, i.e. the cowpled Maxwell-Liouville equations. In particular, when the probe is very weak,... 

The interaction of various optical fields inside a nonlinear medium are described by Maxwell equations. For n-wave mixing processes, there are n coupled Maxwell wave equations. 

The effects associated with these static and grating terms become clearer when we write down the corresponding coupled Maxwell wave equations. From Equation (11.47), ignoring the transverse Laplacian term, the equations for the amplitudes A of these three waves can be derived to give... 

The necessary boundary conditions required for E and //to satisfy Maxwell s equations give rise to tire well known wave equation for tire electromagnetic field ... 

Maxwell s equations can be combined (61) to describe the propagation of light ia free space, yielding the following scalar wave equation ... 

Spin 1, Mass Zero Particles. Photons.—For a mass zero, spin 1 particle, the set of relativistic wave equations describing the particle is Maxwell s equations. We adopt the vector 9(x) and the pseudovector (x) which are positive energy (frequency) solutions of... 

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. 

Invariance of the fields with respect to changes in potential is known as gauge invariance. It is used to simplify Maxwell s equations in regions where there is no free charge. In this case ip itself is a solution of the wave equation, so that it can be adjusted to cancel and eliminate the scalar potential. This means that in (13) V A= 0 and, as before... 

Incorrect conclusion 1 above is sometimes said to derive from the reciprocity principle, which states that light waves in any optical system all could be reversed in direction without altering any paths or intensities and remain consistent with physical reality (because Maxwell s equations are invariant under time reversal). Applying this principle here, one notes that an evanescent wave set up by a supercritical ray undergoing total internal reflection can excite a dipole with a power that decays exponentially with z. Then (by the reciprocity principle) an excited dipole should lead to a supercritical emitted beam intensity that also decays exponentially with z. Although this prediction would be true if the fluorophore were a fixed-amplitude dipole in both cases, it cannot be modeled as such in the latter case. 

Many interesting phenomena can arise in nonlinear periodic structures that possess the Kerr nonlinearity. For analytic description of such effects, the slowly varying amplitude (or envelope) approximation is usually applied. Alternatively, in order to avoid any approximation, we can use various numerical methods that solve Maxwell s equations or the wave equation directly. Examples of these rigorous methods that were applied to the modelling of nonlinear periodical structures are the finite-difference time-domain method, transmission-line modelling and the finite-element frequency-domain method." ... 

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. 

The electromagnetic fields (x,r) and H(x, t) associated with scattering from a microsphere satisfy Maxwell s equations. For a homogeneous, isotropic linear material the time-harmonic electrical held E and the magnetic held H satisfy vector wave equations, which in SI units are (Bohren and Huffman, 1983)... 

Bartlett and Corle [46] proposed modification of Maxwell s equations in the vacuum by assigning a small nonzero electric condictivity to the formalism. As pointed out by Harmuth [47], there was never a satisfactory concept of propagation velocity of signals within the framework of Maxwell s theory. Thus, the equations of the latter fail for waves with nonnegligible relative frequency bandwidth when propagating in a dissipative medium. To resolve this problem, a nonzero electric conductivity ct and a corresponding current density... 

The Maxwell-Heaviside theory of electrodynamics has no explanation for the Sagnac effect [4] because its phase is invariant under 7 as argued already, and because the equations are invariant to rotation in the vacuum. The d Alembert wave equation of U(l) electrodynamics is also 7 -invariant. One of the most telling pieces of evidence against the validity of the U(l) electrodynamics was given experimentally by Pegram [54] who discovered a little known [4] cross-relation between magnetic and electric fields in the vacuum that is denied by Lorentz transformation. 

Equations (573) have overall 0(3) symmetry, and have the same structure as the Maxwell-Heaviside equations with magnetic charge and current [3,4]. From Eqs. (573), we obtain the wave equation... 

Furthermore, the general method presented in this chapter applies directly to solving the full Maxwell equations with currents. It can also be used to construct exact classical solutions of Yang-Mills equations with Higgs fields and their generalizations. Generically, the method developed in this chapter can be efficiently applied to any conformally invariant wave equation, on the solution set of which a covariant representation of the conformal algebra in Eq. (15) is realized. 

These new potentials are solutions of wave equations including inside the sources. To obtain the general solution, one must add a particular solution of the inhomogenous potential equations. Usually, the electromagnetic helds Eo, Eo and the potentials , C are discarded for the following reasons. Either (1), they represent transient solutions of Maxwell s equations that decay rapidly to zero or... 

The electromagnetic field defined above is a solution of Maxwell s equations, provided the new potentials . C are solutions of wave equations and satisfy the Lorenz gauge. 

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. 

This net symmetric regauging operation successfully separates the variables, so that two inhomogeneous wave equations result to yield the new Maxwell... 

Thus the two previously coupled Maxwell equations (1) and (2) (potential form) have been changed to the form given by Eqs. (6) and (7), to leave two much simpler inhomogeneous wave equations, one for and one for A. 

By 1903. llie wave theory of light based oil Maxwell s equations was well established, but certain phenomena would not fit in. It seemed that emission and absorption of hght occur discontinuously. This led Einstein to (lie view that the energy is concentrated in discrete particles. It was a revolutionary idea, very hard to understand, as the successes of the wave theory were undeniable. It seemed that light had to be understood sometimes as waves, sometimes as particles, and physicists had to get used to it, The idea was incorporated into Bohr s theory of the hydrogen atom and forms an essential part of it. 

The derivation of Eq. (218) from Eq. (206) follows from local gauge invariance, and it is always possible to apply a local gauge transform to the vector A, the Maxwell vector potential. The ordinary derivative of the d Alembert wave equation is replaced by an 0(3) covariant derivative. The U(l) equivalent of Eq. (218) in quantum-mechanical (operator) form is Eq. (13), and Eq. (212) is the rigorously correct form of the phenomenological Eq. (25). It can be seen that Eq. (212) is richly structured in the vacuum and must be solved numerically. The vacuum currents present in Eq. (218) can be computed from the right-hand side of the wave equation (212), and these vacuum currents follow from local gauge invariance. 

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