Total electric displacement

Note that D and JF do not appear separately in (8.6) and (8.7) but only in the combination D + /JF/co, which we may interpret as the total electric displacement and assume that D in (8.5) is this quantity. In fact, we could have done this in previous chapters but refrained from doing so because the notion of conductivity is well established. However, it is not possible to determine from macroscopic experiments of the type discussed in this book if the imaginary part of the refractive index originates from free or bound charge currents. Thus, we need not make separate assumptions about the relations between D and E and between JF and E. 

When an electric field E is applied to a polar material the total electric displacement D is given by... 

This electric displacement represents the contribution of the polarization to the total electric displacement, which is written according to relationships J12.8 and J12.9 ... 

When the electronic charge in the optical material is displaced by the electric field (E) of the light and polarization takes place, the total electric field (the "displaced" field, D) within the material becomes ... 

The Poisson-Boltzman (PB) equation relates the electric displacement to the charge density (see Equation (4.28)). The total charge distribution includes the solute charge inside the solute cavity (pint) and that generated by the ion atmosphere outside the cavity (Pext) The external charge density can be represented as shown in Equation (4.30), which leads to the expanded form of the PB equation (Equation (4.31)), which can be simplified for low (Equation (4.32)) and zero (Equation (4.33)) ionic strengths. 

The total charge density uT is equivalent to the magnitude of the electric displacement vector D, so that... 

The total displacement vector D and the total electric field vector E are related through the dielectric constant (Eq. 3.A.1),... 

Another important quantity is the electric displacement, A which corresponds to the total charge density in the surfaces of the electrodes (o+P) ... 

Here E is the total electric field at the position of the Drude particle, r - d, arising from the fixed charges as well as all the induced dipoles (modeled with Drude oscillators). For atomic positions r,the relaxed displacements produce the potential... 

The contribution of the solvent reorganization energy to the total X as considered by Kharkats [8] and later by Marcus [11]. The expressions obtained appeared not entirely consistent, but successive revision established that the key aspect lies in separating the static and optical terms of the integral of the electric displacement vectors over the volume of the two liquids system [37]... 

The dielectric polarization, or simply polarization, within dielectric materials is a vector physical quantity, denoted by P, and its module is expressed in C.m Electric polarization arises due to the existence of atomic and molecular forces and appears whenever electric charges in a material are displaced with respect to one another under the influence of an apphed external electric field strength, E. On the other hand, the electric polarization represents the total electric dipole moment contained per unit volume of the material averaged over the volume of a crystal cell lattice, V, expressed in cubic meters (m ) ... 

The constants of proportionality d and e are called piezoelectric stress and strain coefficients. The stress and strain forces are represented by matrix quantities, and the coefficients are tensor quantities. A tensor mathematically represents the fact that the polarization can depend on the stress or strain in more than one direction. This is also true for the relationship between the stress or strain and the electric field. Many other physical properties in crystals also exhibit this nature, which is called anisotropy. Thus when a property is anisotropic, its value depends on the direction of orientation in the crystal. For the direct piezoelectric effect, the total polarization effect is the sum of these two contributions, an applied electric field and applied mechanical force. Based on the relationship between the electric displacement and the electric polarization it is then possible to write equations that relate the displacement D to the applied stress or strain. Electric displacement is the quantity that is preferred in experiment and engineering. 

The permittivity (e) is a characteristic of a material, which describes how any electric field affects and is affected by a material (a dielectric medium). In nonconducting materials (insulators or dielectrics) charges do not move freely only might be slightly displaced from their equilibrium position (Heaviside, 2007). Permittivity is determined by the ability of a material to polarize in response to the field, and reduce the total electric field inside the material, hence it can be calculated by Equation 1, which gives the electric field of a point charge (Q) at the distance r from Q. 

The thermodynamical derivation of piezoelectricity includes two steps (1) The relevant mechanical or electrical quantities are calculated as partial derivatives of the Gibbs free energy with respect either to one of the two mechanical or to one of the two electrical observables, respectively. (2) The second partial derivative of the Gibbs free energy with respect to the other domain (electrical or mechanical, respectively) yields one of the piezoelectric coefficients. Because there is one intensive (force-like or voltage-like) observable, namely, mechanical stress and electrical field, and one extensive (displacement-like) observable, namely, mechanical strain and electrical displacement, in each of the two domains, we have four possible combinations of one mechanical and one electrical observable in total. Thus, we obtain four different piezoelectric coefficients that are usually abbreviated as d, e, g, and h. As the sequence of the two partial derivations can be reversed, we arrive at two different expressions for each coefficient one for direct piezoelectricity (mechanical stimulus leads to an electrical response) and one for inverse or converse piezoelectricity (electrical stimulus leads to a mechanical response). For example, the piezoelectric d coefficient is given by the two alternative terms ... 

Total true stress and current electric displacement can be easily determined by Eq. 1 and the second part of of Eq. 8. 

D-E Hysteresis Loop and Inversion Current the electric displacement D or the polarization P of the crystal is determined by the total summation of electric dipole moments in a unit volume the lattice. Therefore the change in dipole orientation should reflect on the D (and P) explicitly. The time dependence of elect dipole rcorwtslioo is equivalent to dm time evolution of charges, i.e., the current L A clear D-f hysteresis... 

Poisson s ratio (v). For elastic deformation, the negative ratio of lateral and axial strains that result from an applied axial stress, polarization (P). The total electric dipole moment per unit volume of dielectric material. Also, a measure of the contribution to the total dielectric displacement by a dielectric material. 

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