Electric displacement density

From the response of each substrate it is thus possible to calculate the electric displacement density (D) due to the polarization of the material when subjected to E vector. 

Describing anal3dically the piezo effect by the linear state (6.1) and (6.2), the electric displacement density D and the mechanical strain S are combined with the mechanical stress T and the electrical field strength E ... 

The value of the dot product is a measure of the coalignment of two vectors and is independent of the coordinate system. The dot product therefore is a true scalar, the simplest invariant which can be formed from the two vectors. It provides a useful form for expressing many physical properties the work done in moving a body equals the dot product of the force and the displacement the electrical energy density in space is proportional to the dot product of electrical intensity and electrical displacement quantum mechanical observations are dot products of an operator and a state vector the invariants of special relativity are the dot products of four-vectors. The invariants involving a set of quantities may be used to establish if these quantities are the components of a vector. For instance, if AiBi forms an invariant and Bi are the components of a vector, then Az must be the components of another vector. 

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... 

The magnitude of the charge per unit area on either plate, called the electric displacement or flux density. Do, is directly proportional to the field ... 

The definition of electric charge density in Eq. (76) agrees with our opinion that 0 in Maxwell s equations represents charge neutrality (see Section HI) the simplest case is 5+ + S = 0. Also note that X/ defined by Eq. (74) is independent of pe thus allowing for the existence of a displacement current in the absence of electric charge, as also discussed in Section HI. 

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. 

In conventional dielectric measurements, one observes the Maxwell polarization or electric moment density P(t) which is induced by an applied electric field E(t), and from this the complex permittivity e as a function of frequency f = u/2ir of the field. For a system of charges e in volume V with displacement z in the field direction z, P is given by... 

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

Table G Definitions of the Electric Field E, the (Di)electric Polarization P, the Electric Displacement D, the Magnetic Field H, the Magnetization M, the Magnetic induction or flux density B, statement of the Maxwell equations, and of the Lorentz Force Equation in Various Systems of Units rat. = rationalized (no 477-), unrat. = the explicit factor 477- is used in the definition of dielectric polarization and magnetization c = speed of light) (using SI values for e, me, h, c) [J.D. Jackson, Classical Electrodynamics, 3rd edition, Wiley, New York, 1999.]. For Hartree atomic u nits of mag netism, two conventions exist (1) the "Gauss" or wave convention, which requires that E and H have the same magnitude for electromagnetic waves in vacuo (2) the Lorentz convention, which derives the magnetic field from the Lorentz force equation the ratio between these two sets of units is the Sommerfeld fine-structure constant a = 1/137.0359895...

H is the magnetic field vector and D is called the electric induction or displacement field. This equation is known as the Ampere Oersted law and shows that a magnetic field will exist near an electric current density J. The displacement field, D, is necessary to propagate electromagnetic energy through space, /has units charge area-1 t-1... 

Electrical conductivity, 319,335,337,339 Electrical properties, 319 Electric displacement, 348 Electric field, 351 Electric flux density, 348 Electric inductive capacity, 287, 319, 326 Electric permittivity, 287 Electric susceptibility, 348, 349 Electrochemical n-doping, 341 p-doping, 341 Electron acceptor, 333 parameter, 242 Electron donor, 333, 337 parameter, 242... 

Polarization density (of a dielectric medium) — The polarization density P is the difference between the electric displacement in a - dielectric medium Dc and the electric displacement in a vacuum Do- (Note The electric displacement is defined as the product of the electric field strength E and the - permittivity e.) The polarization density of a dielectric medium may also be... 

In the above, E, D, H, B, Jt represent the electric field, electric displacement, magnetic field, magnetic induction, and free current density vectors respectively c is the velocity of light. Form the scalar product of (1.7.1a) with H and of (1.7.1b) with E and subtract to find... 

Magnetic flux density Electric displacement Electric field strength Magnetic field strength Magnetization Polarization (dielectric)... 

E and H being the electric and magnetic field strength vectors, D the electric displacement vector, B the magnetic induction vector, J the electric current density, and p, the electric charge density. 

In chiral vinylcyclopropanes the chromophore can be viewed as inherently chiral, i.e. all the electronic transitions have simultaneously an electric (/i) and a magnetic (m) transition moment. For transoid or cisoid (indicated in broken lines below) arrangements the magnetic moments are generated by a circular displacement of electric charge density around the bond between the cyclopropane and the ethylenic moieties. The helical movement of electron density involves essentially p-atomic orbitals twisted relative to each... 

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... 

It is from the properties of the materials involved that relationships between current density and the electric field, the electric displacement and the electric field, and the magnetic induction and magnetic field are established. In the event that the material through which the electromagnetic radiation is propagating is isotropic, and is moving slowly relative to the speed of light, and the fields involved are time harmonic, then the three material equations become... 

The electric displacement field is related to electric field by D = E, and the magnetic flux density is related to the magnetic field by B = e is the electrical permittivity, which is related to the dielectric constant and p. is the magnetic permeability. 

Inspection of other transformation reveals several other relations. It turns out that a lot of the coefficients in Eq. (15.6) are zero, namely Eq. (15.6) for the electric displacement field and the corresponding equation for the magnetic flux density must be of the form... 

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

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