Electric flux density vector

In Eq. 2, D is the electric displacement or electric flux density vector, E is the electric field vector, P is the electric polarization vector, and is the permittivity of vacuum. In many isotropic materials the induced polarization is directly proportional to the applied field strength, except for the case of very high fields. We can write [71]... 

Hence the overall electric current density vector j, in A.m", is the product of the overall flux by the elementary charge e, in C, and is expressed by the following equation ... 

The electric flux density or electric displacement, denoted by D, is a vector quantity defined in a vacuum as the product of the electric field strength and the permittivity of vacuum. Its module is expressed in coulombs per square meter (C.m ). It is defined by the following equation ... 

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... 

Just as in the mechanical case, the boundary dA of the dielectric domain A is subdivided to consider two types of boundary conditions. The equilibrium between prescribed charges qsA on the area dAp and the electric flux density can be established with Eq. (3.31). Since these charges are located on the outside, the appearing normal vector e is pointing inward. Thus, for an outward oriented surface normal e = — e on the boundary of the dielectric domain, it may be written as... 

Since the forces fg and charges q A on the boundary are zero apart from their respective working surface, the surface integrals may be summarized. Then the integrands can be collated in vector form, as shown in the last line. Similarly, the virtual work of internal contributions can be formulated, where the vectors of virtual strains 6e and virtual electric field strength SE, as well as the vectors of actual stresses electric flux density D, be merged ... 

The electric flux density D and electric field strength E are vectors, i.e. tensors of first order and therefore may be related via a tensor of second order with nine constants for the three dimensions. Due to the potential property also observed for electrostatic fields, the tensor is symmetric and thus contains six independent entries. The electrostatic constitutive relation can be expressed with the aid of the dielectric permittivity matrix e (to be distinguished from the strains e) or its inverse p ... 

Examining the upper part of Eq. (6.37), the shell internal loads and group-wise electric flux density resultants in the vector L are found to be composed... 

Table 1.1 Conjugate pairs of variables in work terms for the fundamental equation for the internal energy U. Here/is force of elongation, Z is length in the direction of the force, <7 is surface tension, As is surface area, , is the electric potential of the phase containing species i, qi is the contribution of species i to the electric charge of a phase, E is electric field strength, p is the electric dipole moment of the system, B is magnetic field strength (magnetic flux density), and m is the magnetic moment of the system. The dots indicate scalar products of vectors.

We want to introduce the properties of the crystal and of the X-rays and solve f or the electric displacement or flux density, D. Hart gives a careful discussion of the polarisability of a crystal, showing that a sufficient model of the crystal for X-ray scattering is a Fourier sum of either the electron density or the electric susceptibility over all the reciprocal lattice vectors h. Thus the crystal is represented as... 

To solve the electrical problem at the electrodes, two variables need to be found (i.e., the electrical potential (scalar) and the current density (vector)). By analogy, two variables are also needed to find the ionic flux and the potential distribution. The vector relationship is given by Ohm s law (3.10), while the scalar relation is provided by expression (3.5), which can be re-written as ... 

Extended nonequilibrium thermodynamics is not based on the local equilibrium hypothesis, and uses the conserved variables and nonconserved dissipative fluxes as the independent variables to establish evolution equations for the dissipative fluxes satisfying the second law of thermodynamics. For conservation laws in hydrodynamic systems, the independent variables are the mass density, p, velocity, v, and specific internal energy, u, while the nonconserved variables are the heat flux, shear and bulk viscous pressure, diffusion flux, and electrical flux. For the generalized entropy with the properties of additivity and convex function considered, extended nonequilibrium thermodynamics formulations provide a more complete formulation of transport and rate processes beyond local equilibrium. The formulations can relate microscopic phenomena to a macroscopic thermodynamic interpretation by deriving the generalized transport laws expressed in terms of the generalized frequency and wave-vector-dependent transport coefficients. 

It is obvious here that the electricity flux (vector) corresponds to the elec trie current density... 

Both the electric Held (r,t) and the magnetic flux density B(rj) may be derived from the vector potential A(r.r). Using the so-called Coulomb gauge one obtains... 

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... 

In the equations presented in the following sections, is the electric potential, E is the electric field, D is the electric displacement, c,-, z, and Ni are the concentration, valence, and flux of ion species i, respectively, p, u, and Vp are the pressure, velocity vector, and translational velocity of the nanoparticle, correspondingly, nip is the mass of the nanoparticle, t represents time. cr and Op are the surface electric charge densities on the walls of the nanochaimel and the surface of the nanoparticle, respectively. The constants include permittivity (eoE ), medium density (p), the Faraday number (F), fluid viscosity (rj), valence number (z,), diffusion coefficient (D,), and mobility (p,) of ion species i. 

A magnetic field is produced by the motion of electric charge. It is characterized by two vectors B, the magnetic induction or magnetic flux density, and H, the magnetic field strength (also called intensity). 

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