Electromagnetic fields vector wave equations

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)... 

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. 

Electromagnetic wave — Oscillatory propagating electromagnetic field. Maxwell s equations, for free space, can be manipulated into the form of two extremely concise vector equations, V2 = eoEof f and V2E =... 

Besides its appearance in the FFMF equation in plasma physics, as well as associated with time-harmonic fields in chiral media, the chiral Beltrami vector field reveals itself in theoretical models for classical transverse electromagnetic (TEM) waves. Specifically, the existence of a general class of TEM waves has been advanced in which the electric and magnetic field vectors are parallel [59]. Interestingly, it was found that for one representation of this wave type, the magnetic vector potential (A) satisfies a Beltrami equation ... 

In free space, an electromagnetic field of frequency w is governed by the homogeneous vector Helmholtz wave equation... 

Light beams are represented by electromagnetic waves that are described in a medium by four vector fields the electric field E r, t), the magnetic field H r, t), the electric displacement field D r,t), and B r,t) the magnetic induction field (or magnetic flux density). Throughout this chapter we will use bold symbols to denote vector quantities. All field vectors are functions of position and time. In a dielectric medium they satisfy a set of coupled partial differential equations known as Maxwell s equations. In the CGS system of units, they give... 

An arbitrary free classical electromagnetic field is described by the vector potential which obeys the wave equation [14,24,25]... 

Here a denotes the maximum field amplitude, rj is the ellipticity together with the pulse-shape function g(rj), which depends on the phase rj = (ot — k r. The laser beam is characterized by the frequency co and the wave vector k with ck = co. The transversality condition implies k A — 0. For a charged point particle interacting with this external electromagnetic field, the Hamilton-Jacobi equation reads... 

The theory of nonlinear optical processes in crystals is based on the phenomenological Maxwell equations, supplemented by nonlinear material equations. The latter connect the electric induction vector D(r,t) with the electric field vector E(r, t). In general, the relations are both nonlocal and nonlinear. The property of nonlocality leads to the so-called spatial dispersion of the dielectric tensor. The presence of nonlinearity leads to the interaction between normal electromagnetic waves in crystals, i.e. makes conditions for the appearance of nonlinear optical effects. 

The potentials which will be considered here are stepwise constant. In each region of these potentials the time-dependent wavefunction is a linear combination of solutions of the wave equation for a particle interacting with an electromagnetic field of vector potential with A(t) [10] ... 

A large class of molecular properties arise from the interaction of molecules with electromagnetic fields. As emphasized previously, the external fields are treated as perturbations and so one considers only the effect of the fields on the molecule and not the effect of the molecule on the field. The electromagnetic fields introduced into the electronic wave equation is accordingly those of free space. From (79) one observes that in the absence of sources the electric field has zero divergence, and so both the electric and magnetic fields are purely transversal. It follows that the scalar potential is a constant and can be set to zero. In Coulomb gauge the vector potential is found from the equation... 

The scalar potential in the rapidly varying electromagnetic field is supposed to be zero. The vector potential A satisfies the wave equation in free space ... 

Light is a form of electromagnetic radiation, that is, it can be described as an electromagnetic wave [1, 2]. Neglecting lateral boundaries, a collimated beam of monochromatic light can be described as a planar wave. The electric field vector of this wave as a function of position and time is Equation 16.1 ... 

The electric field vector of the electromagnetic wave expressed by Equation (B3) is parallel to the y axis, and the wave propagates in the yz plane. Such an electromagnetic wave is called linearly polarized or plane-polarized radiation. An electromagnetic wave polarized parallel to the x axis also exists, and its electric vector is denoted by E. If a sample for an infrared absorption measurement has any molecular orientation, it generally shows different absorptions for the x-polarized and y-polarized infrared radiations. For example, a stretched (uniaxially oriented) thin polymer film usually shows different absorptions for infrared radiations polarized parallel and perpendicular to the stretching direction. This difference is called infrared dichroism. 

Any electromagnetic wave travelling along the interface can be represented as a superposition of two independently polarized components, namely a transverse magnetic (TM) wave and a transverse electric (TE) wave. Let us choose the x-axis along the wave vector q. Then in a TM wave the electric and magnetic field vectors have components (Ex, 0, Ez) and (0, Hy, 0), respectively. A TE wave is represented by the components (0, Ey, 0) and (Hx, 0, Hz)-We shall seek the solution of Maxwell s equations corresponding to SPs in the form... 

To solve the scattering problem in the framework of the null-field method it is necessary to approximate the internal field by a suitable system of vector functions. For isotropic particles, regular vector spherical wave functions of the interior wave equation are used for internal field approximations. In this section we derive new systems of vector functions for anisotropic and chiral particles by representing the electromagnetic fields (propagating in anisotropic... 

As a result, we obtain the familiar expansions of the electromagnetic fields in terms of vector spherical wave functions of the interior wave equation ... 

Applications of the extinction theorem and Huygens principle yield the null-field equations (2.6) and the integral representations for the scattered field coefficients (2.16). Taking into account that the electromagnetic fields propagating in an isotropic, chiral medium can be expressed as a superposition of vector spherical wave functions of left- and right-handed type (cf. Sect. 1.3), we represent the approximate surface fields as... 

There is a close similarity with planar electromagnetic cavities (H.-J. Stockmann, 1999). The basic equations take the same form and, in particular, the Poynting vector is the analog of the quantum mechanical current. It is therefore possible to experimentally observe currents, nodal points and streamlines in microwave billiards (M. Barth et.al., 2002 Y.-H. Kim et.al., 2003). The microwave measurements have confirmed many of the predictions of the random Gaussian wave fields described above. For example wave function statistics, current flow and... 

In this final section, it is shown that the three magnetic field components of electromagnetic radiation in 0(3) electrodynamics are Beltrami vector fields, illustrating the fact that conventional Maxwell-Heaviside electrodynamics are incomplete. Therefore Beltrami electrodynamics can be regarded as foundational, structuring the vacuum fields of nature, and extending the point of view of Heaviside, who reduced the original Maxwell equations to their presently accepted textbook form. In this section, transverse plane waves are shown to be solenoidal, complex lamellar, and Beltrami, and to obey the Beltrami equation, of which B is an identically nonzero solution. In the Beltrami electrodynamics, therefore, the existence of the transverse 1 = implies that of , as in 0(3) electrodynamics. 

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