Vector Maxwell displacement

If we assume that the dipole layer is uniform and that we can neglect its thickness the corresponding polarization per unit volume is P(z) = p0 J(z), where po is the polarization per unit area and the 2-ax is is directed along the normal to the interface planes from the donor to the acceptor side. In our case, the vector po = (0,0, — po), Po = pn. Using the equation divD(r) = 0 where D(r) is the Maxwell displacement vector, D = E + 4-7tP, and taking into account that all these values depend only on z we have ... 

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)... 

In the absence of free charges the continuity of the electric displacement vector (Maxwell equation) reads ... 

To illustrate the use of the vector operators described in the previous section, consider the equations of Maxwell. In a vacuum they provide the basic description of an electromagnetic field in terms of the vector quantifies the electric field and 9C the magnetic field The definition of the field in a dielectric medium requires the introduction of two additional quantities, the electric displacement SH and the magnetic induction. The macroscopic electromagnetic properties of the medium are then determined by Maxwell s equations, viz. 

According to the macroscopic Maxwell approach, matter is treated as a continuum, and the field in the matter in this case is the direct result of the electric displacement (electric induction) vector D, which is the electric field corrected for polarization [7] ... 

Here E, D, and P represent, respectively, the electric field, electric induction (or displacement), and electric polarization vectors P (D - )/4x. The integration must be carried out over all space penetrated by the electrostatic field. Equation (5.6.1), while correct, is awkward in several respects. First, there is the need to integrate over all space, including the region outside the system of interest. In the presence of a medium, the electric lines of force not only are present within the specimen, but also bulge out in all directions away from the system these effects must be included in (5.6.1). Second, there is a tendency in the literature to associate the first term in (5.6.1b) with the establishment of the electric field in free space, and the second term with the reaction of the medium to the electric field. This is wrong The quantity D is subject to direct experimental control because it is linked by Maxwell s equation to the presence of free charges by contrast, E is in part a reaction field that also includes the... 

As an alternative formulation we introduce a local current density, J r) that responds to a steady vector potential 4.(r). The associated work element is dW = (1/c) fy d r J dA., where c is the velocity of light. We next introduce Maxwell s relation V x = An/c)J, which applies when the electric displacement vector is independent of time and when J is the free current density. Thus, dW = (47t) /d rdA (V x 7i). On using line (g) of Table 1.3.1, the work increment reads... 

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... 

Numerous experiments have shown the appropriateness of selecting the vector X in this form. This quantity was called a displacement current. As follows from the second Maxwell equation, there are two sources for the magnetic field conduction currents and displacement currents. Applying Stoke s theorem, we obtain the integral form of the second Maxwell equation ... 

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... 

The vectors of electric displacement D and polarization P are also coupled by the additional Maxwell equation ... 

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. 

The mathematical operator consisting of the dot product, or scalar product, of the differential vector V with another vector is called the divergence operator. For instance, the charge concentration p in a volume is linked to the electric displacement D through one of Maxwell s relationships ... 

As in previous chapters we work in the continuum limit employing quantities averaged over macroscopically infinitesimal volume elements and disregarding microscopic local variations associated with the molecular structure (see Brown 1956). These considerations will be limited to processes sufficiently slow to restrict the treatment to time independent or quasistatic fields. The validity of Maxwell s equations of electrostatics is presupposed. The basic electric state variables are the electric field strength vector E, the electric flux density (or electric displacement) vector D, and the electric polarization vector P, related by... 

Here we generalize Maxwell s equations and the associated wave equations to describe propagation in anisotropic materials [3]. The generalization follows by recognizing that the displacement vector, D, is not, in general, parallel to the electric field E. Instead, it obeys the relationship... 

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