Maxwell’s field equations

Under the conditions of a static electric field, the Maxwell s field equations reduce to the Poisson equation [111,112,172,261], given by... 

Widespread disagreement on the understanding of chemical matter can, to a large extent, be traced back to the confusion over the nature of the electron. The intuitive concept of elementary particles completely dominates theoretical thinking, although the ultimate quantum entity, the photon, is a particle in name only. The carrier of the electromagnetic field is defined unequivocally by Maxwell s field equations, of the form... 

The components of the gradient tensor are not independent. They are related by Maxwell s field equations [Schl]... 

Two equivalent forms of Maxwell s field equations in terms of the standard vector formalism are Eq. (5), or (6) with the Lorentz gauge 9 4 = 0. The former is in terms of the antisymmetric second-rank tensor solution F, which is a combination of the electric and magnetic field variables. The latter is in terms of the vector potential, A, shown in Eq. (6) [as well as Eq. (7) in terms of the pseudo vector potential Bassuming that the parameter E, is nonzero]. (Experimental results to this point in time indicate that indeed this parameter is zero to within experimental accuracy [15]—even though the symmetry of relativity theory has no reason to exclude it. Henceforth, we will assume that this parameter is zero.)... 

To avoid further confusion, it is recommended to use the term nonclassical theory instead of the unfortunate quantum mechanics. Nonclassical theory became important after the discovery of the electromagnetic field. The summary of Maxwell s field equations in the form... 

Ionic current density maps can be recorded with the aid of the pulse sequence shown in Figure 2.9.2. The principle of the technique [48-52] is based on Maxwell s fourth equation for stationary electromagnetic fields,... 

The FDTD approach is based on direct numerical solution of the time dependent Maxwell s curl equations. In the 2D TM case the nonzero field components are E Hy and E, the propagation is along the z direction and the transverse field variations are along x. In lossless media. Maxwell s equations... 

This result is inconsistent with the fact that the differential equation developed by Heaviside from Maxwell s original equations describe circular polarization. The root of the inconsistency is that U(l) gauge field theory is made to correspond with Maxwell-Heaviside theory by discarding the commutator Am x A(2). The neglect of the latter results in a reduction to absurdity, because if S3 vanishes, so does the zero order Stokes parameter ... 

This modified equation is just the Schrodinger equation that describes the interaction of a charged particle with the elctromagnetic field. This appearance of interaction with a field is known as the gauge principle. A vector field such as A, introduced to guarantee local phase invariance, is called a gauge field. The local invariance of Schrodinger s equation ensures that quantum mechanics does not conflict with Maxwell s field. 

Although is a field as real as Maxwell s fields, it does not show up immediately as the result of a single measurement, but only in the statistics of many measurements. It is the De Broglie-Bohm variable x that shows up immediately each time. The particle has an equation of motion (T3.2)... 

The problem of studying the interaction of an electromagnetic field with a two-dimensionally structured model reduces to the solution of two far simpler problems, the solution of problems for ii-polarized and /f-polarized fields separately. The simplicity lies in the fact that when i -polarizcd or //-polarized fields are considered, Maxwell s vector equations reduce to scalar differential equations. 

The derivative of the Green s function, which is needed in computing the magnetic field components, is determined similarly. In accord with equation (9.21) and Maxwell s second equation, we can write... 

The spectrum of polaritons can be found by means of Maxwell s macroscopic equations (see Ch. 4), provided that the dielectric tensor of the medium (44) is assumed to be known. Without going into details, we emphasize here that always a gap appears in the polariton spectrum (here we ignore spatial dispersion) in the region of the fundamental dipole-active vibration (C-phonon, exciton, etc.). At present, there is a sufficiently detailed theory for RSL by phonon-polaritons, taking many phonon bands into consideration. With this theory the RSL cross-section can be calculated for various scattering angles provided that the dielectric tensor of the crystal is known, as well as the dependence of the polarizability of the crystal on the displacement of the lattice sites and the electric field generated by this displacement (45). 

The interaction of laser radiation with the medium occurs through the third-order, non-linear, electric susceptibility denoted by and gives rise to an induced polarization field, which acts as a source term in Maxwell s wave equation. On solving the wave equation, one arrives at the following expression for the intensity of the CARS signal ... 

The electromagnetic field is defined by Maxwell s four equations in vector notation ... 

In the derivation of the molecular properties, which give rise to this effect, we have to take the spatial variation of the electric field vector into account and can thus not make the dipole approximation, contrary to the last section. This implies that we have to include a contribution from the interaction with the curl of the time-dependent electric-field, V xS r,t), to the expansion of the induced dipole moment of a molecule in Eq. (7.18). However, Maxwell s third equation, Eq. (2.37) relates the curl of the electric-field vector to the time derivative dB r,t)/dt of the magnetic induction and we can thus alternatively replace the spatial variation and expand the induced dipole moment instead in the electric field and the time derivative of the magnetic induction of a monochromatic wave (Buckingham, 1967) as... 

Outside the source, the electric field is obtained by applying Maxwell s relation equation (2.18), taking into account the sinusoidal nature of the fields ... 

Maxwell s equation are the basis for the calculation of electromagnetic fields. An exact solution of these equations can be given only in special cases, so that numerical approximations are used. If the problem is two-dimensional, a considerable reduction of the computation expenditure can be obtained by the introduction of the magnetic vector potential A =VxB. With the assumption that all field variables are sinusoidal, the time dependence... 

The central equations of electromagnetic theory are elegantly written in the fonn of four coupled equations for the electric and magnetic fields. These are known as Maxwell s equations. In free space, these equations take the fonn ... 

In the previous sections we have described the interaction of the electromagnetic field with matter, that is, tlie way the material is affected by the presence of the field. But there is a second, reciprocal perspective the excitation of the material by the electromagnetic field generates a dipole (polarization) where none existed previously. Over a sample of finite size this dipole is macroscopic, and serves as a new source tenu in Maxwell s equations. For weak fields, the source tenu, P, is linear in the field strength. Thus,... 

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 ... 

Above we described tire nature of Maxwell s equations in free space in a medium, two more vector fields need to be... 

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