Maxwell field vector

For analytical purposes it is convenient to use a parallel-plate capacitor as a measuring cell. In this geometry only the component nij (of niy) that is parallel to the electric field lines between the plates (or to an eventually applied external field creating the Maxwell field E in the dielectric) contributes to the measurable polarization. For capacitor geometry the scalar product of in,E(r) in Eq. (3.97) is given by the parallel component (my) II = my cos 5y and by E, that is, the Maxwell field vector perpendicular to the capacitor plates 5, is the angle between E and my. Therefore, Eq. (3.97) applies in the form... 

TRIFOU is a combined Finite Elements/Boundary Integral formulation code. The BIM formulation in vacuum is suitable for NDT simulation where the probe moves in the air around the test block. The FEM formulation needs more calculation time, but tetrahedral elements enable a large variety of specimens and defect geometries to be modelled. TRIFOU uses a formulation of Maxwell Equations using magnetic field vector h, where h is decomposed as h = hs + hr (hj source field, and hr reaction field). 

The electromagnetic field in free space is described by the electric field vector E and the magnetic field vector H, which in the absence of charges satisfy Maxwell s equations... 

We start from the first pair of Maxwell equations written with account of electric charges moving in vacuum. Let vector J be the electric current density produced by these charges. Combining the above-mentioned equations, we get the second-order differential equation for electric field vector E ... 

In classical electrodynamics, the field equations for the Maxwell field A/( depend only on the antisymmetric tensor which is invariant under a gauge transformation A/l A/l + ticduxix), where x is an arbitrary scalar field in space-time. Thus the vector field A/( is not completely determined by the theory. It is customary to impose an auxiliary gauge condition, such as 9/x/Fx = 0, in order to simplify the field equations. In the presence of an externally determined electric current density 4-vector j11, the Maxwell Lagrangian density is... 

In terms of these vector fields, the Maxwell field Lagrangian density is (1 /8 n) 2 — B2), and the field energy and momentum are... 

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. 

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... 

Let us suppose a medium in which all electromagnetic field vectors F are continuous, bounded, single-valued functions of position and time F = F(x, y, z, t) and incorporate continuous derivatives as well. Then, the differential and integral form of Maxwell s equations can be expressed as... 

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. 

Applied to the potentials of the electromagnetic field the coordinate system is determined only to within an additive gradient, which is the well-known property of the vector potential of the Maxwell field. In common practice it is necessary to assume the gauge invariance, which appears naturally in projective relativity. 

In the Maxwell approach, in which matter is treated as a continuum, we must in many cases ascribe a dipole density to matter. Let us compare the vector fields D and E for the case in which only a dipole density is present. Differences between the values of the field vectors arise from differences in flieir sources. Both the external charges and the dipole density of the sample act as sources of these vectors. The external charges contribute to D and E in the same maimer (2). The electric displacement (electric induction) vector D is defined as... 

From continuity of i(iw) and v(iw), this is also the ratio of i and V in the sample section at x = 0 as determined by the dielectric material of interest, its geometrical configuration, and boundary conditions for the sample section. The dielectric response characteristics can be described macroscopically at frequencies of interest by the relation of Maxwell displacement and field vectors (t,j[) and E(t,r)... 

Electric field strength vector (absolute value). Maxwell field... 

According to Onsager the induced moment term M( ) is determined by the internal or local field whereas the permanent dipole term is related to the directing field E<,ij orienting the permanent dipoles p. The combination of Eqs. (3.10) and (3.2) requires that the two different field vectors must be expressed in terms of the measured Maxwell field. The calculations of the terms and M p as functions of Ej , and E jj usually are approximations. The final expressions may be written in terms of conversion factors (g factors ) which are a function of particle anisotropies as well as of the properties of the medium in which the particles are embedded (polar, nonpolar, gas phase, or fluid phase). 

Thurn-Albrecht et al. gave an informative evaluation of Eq. 2.7 for the simplified picture of an infinitely long, isolated cylinder enclosed in an infinite matrix, with dielectric constants and 2, respectively [32]. Solving Maxwell s equation (V [ (r]E(r]] = Pfree] to find the field distribution and substituting into Eq. 2.7, the difference in free energy per unit volume between states with the long axis parallel [Fn] or perpendicular [Fj ] to the field vector is... 

To find the intensities of the reflected and transmitted beams, we need first to consider the relationship between the amplitudes of these beams. From the viewpoint of the Maxwell theory (1.1.1°), at an interface, the electric field vector of every monochromatic linearly polarized electromagnetic wave has the form described by Eq. (1.5a) ... 

Maxwell current density Mathematical curl of the magnetic field vector H, eqnal to the vector sum of all current densities, which in the atmosphere is usually limited to conduction, convection, diffusion, lightning, and displacement cnrrent terms. 

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