Vector wave equation

This wave equation is tire basis of all wave optics and defines tire fimdamental stmcture of electromagnetic tlieory witli tire scalar function U representing any of tire components of tire vector functions E and H. (Note tliat equation (C2.15.5) can be easily derived by taking tire curl of equation (C2.15.1) and equation (C2.15.2) and substituting relations (C2.15.3) and (C2.15.4) into tire results.)... 

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 shall adopt Eqs. (9-510) and (9-511) as the covariant wave equation for the covariant four-vector amplitude 9ttf(a ) describing a photon. The physically realizable amplitudes correspond to positive frequency solutions of Eq. (9-510), which in addition satisfy the subsidiary condition (9-511). In other words the admissible wave functions satisfy... 

These equations (14) and (15) determine the scalar and vector potentials in terms of p and J. When p and J are zero, these equations become wave equations with wave velocity c = y/l/pe. That is, A and are solutions of decoupled equations, where they are related by the wave operator... 

The invariance of the general wave equation (22) is a special consequence of a more general four-vector invariance involving the four-gradient defined as... 

In surface sensing, the adsorbed biomolecules onto the inner wall of a microtube will change the wave vector of a supported mode from A 0 to k and the electric field from E0 to ). These unperturbed (L0) and perturbed (E ) electric fields still satisfied wave equation ... 

A wave is described by a wave function y(f, /), either scalar (as pressure p) or vector (as u or v) at position r and time t. The wave function is the solution of a wave equation that describes the response of the medium to an external stress (see below). 

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)... 

The solution of the vector wave equation can be written in terms of the generating function ij/, which is a solution of the scalar wave equation... 

Let M V X tij/) and N = (V x M)/< with p given by Eq. (53), where r is the radius vector of the spherical coordinate system. Then both M and N satisfy the vector wave equation and both have zero divergence. In addition, M and N satisfy... 

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. 

The choice of generating functions is dictated by whatever symmetry may exist in the problem. In this chapter we are interested in scattering by a sphere therefore, we choose functions ip that satisfy the wave equation in spherical polar coordinates r, 6,

[Pg.84]

The reason for the intractability of the anisotropic sphere scattering problem is the fundamental mismatch between the symmetry of the optical constants and the shape of the particle. For example, the vector wave equation for a uniaxial material is separable in cylindrical coordinates that is, the solutions to the field equations are cylindrical waves. But the bounding surface of the... 

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. 

A second question concerns the existence of longitudinal components of the magnetic field. Maxwell s equations in free space are (completely ) equivalent to two homogeneous uncoupled wave equations for the vector fields E and B. The uncoupled wave equations admit longitudinal components for both fields E and B. However, longitudinal components are prohibited in the conventional interpretation of Maxwell s equations. 

Now, from any function that satisfies (29), can be formed three independent vectors that satisfy the vector wave equation (28) [42], They are traditionally signified by L = grad (v(/), P = curl (v(/a), and T = curl curl (vj/a), where a is an arbitrary constant vector. Thus, to find solutions to (21), we express the vector B as a linear superposition of the three vectors... 

In this application we consider EM fields in free space consequently both E and H are solenoidal and satisfy Trkalian field relations. Thus, taking the curl of (71), both vector fields satisfy Helmholtz vector wave equations ... 

Using the general vector identity, VxVxE = V(V E)-V2E, and that V E = 0 in a space free of charge, the following wave equation results ... 

The standard Schrodinger equation for an electron is solved by complex functions which cannot account for the experimentally observed phenomenon of electron spin. Part of the problem is that the wave equation 8.4 mixes a linear time parameter with a squared space parameter, whereas relativity theory demands that these parameters be of the same degree. In order to linearize both space and time parameters it is necessary to replace their complex coefficients by square matrices. The effect is that the eigenfunction solutions of the wave equation, modified in this way, are no longer complex numbers, but two-dimensinal vectors, known as spinors. This formulation implies that an electron carries intrinsic angular momentum, or spin, of h/2, in line with spectroscopic observation. 

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. 

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