Electrostatic scalar potential

Note that Gp 0p of eq. (9) can be written in several equivalent but different looking forms, as is typical of electrostatic quantities in general. For example, it is often convenient to express the results in terms of the electrostatic scalar potential ( )(r) instead of the electric vector field E(r). In the formulation above, the dielectric displacement vector field associated with the solute charge distribution induces an electric vector field, with which it interacts. In the electrostatic... 

With respect to the electrostatic scalar potential, electrodynamicists are familiar with the fact that unlimited energy can be extracted from a potential. The very simple equation... 

As is any potential including the electrostatic scalar potential between the poles of an electric dipole and the magnetostatic scalar potential between the poles of a permanent magnet, the A potential is an ongoing set of longitudinal EM energy flows between the time domain (imaginary plane) and real 3-space... 

This is recognized as the equation that relates the magnetic vector potential to the electrostatic scalar potential ... 

Consider next an electrical force field of constant strength E (volt/m) whose electrostatic scalar potential

potential difference d

molecular ions of the fth species, each having a valence of Z, (the magnitude varies from 1 to 10), are exposed to this field in an aqueous solution in between the electrodes, the force exerted on 1 gmol of such charged molecules in the z-direction is... 

Analogously, a Coulomb field can be expressed as the gradient of a scalar potential that obeys the Laplace equation in a source-free region such as the vacuum in conventional electrostatics. To find the general form of B(3> in a multipole expansion, we therefore solve the Laplace equation for 3>g, and evaluate the gradient of this solution... 

The last stage is now to replace tv, by P, + eA, and to use explicit expressions for the potentials A, and (pi. Our previous expressions for A, and electron interactions have now been derived more naturally by starting with the Breit Hamiltonian and the vector and scalar potentials therefore contain only terms describing external fields or electrostatic interactions involving the nuclear charge. Hence we make the substitutions... 

Airy Stress Eunction and the Biharmonic Equation The biharmonic equation in many instances has an analogous role in continuum mechanics to that of Laplace s equation in electrostatics. In the context of two-dimensional continuum mechanics, the biharmonic equation arises after introduction of a scalar potential known as the Airy stress function f such that... 

This equation combines the two laws of electrostatics Gauss s law and the fact that E is derivable from a scalar potential through Eq. (AII.7). 

We shall approximate the scalar potential of the electrostatic field by the sawtooth curve of Fig. 5 that has the periodicity of the Bom von Karman zone, i.e., of the length of one unit cell times the number of k points that is used in a calculation. We have here assumed that the potential takes both positive and negative values. By doing so, the average potential from the field vanishes and we have therefore an optimal starting point for eliminating effects that are linear in the number of k points of the calculation (i.e., in the length of the Bom von Karman zone). 

So far, all types of external electromagnetic potential affecting the electron have been excluded. However, in all but trivially simple systems, such potentials are present, taking the form of scalar potentials, usually due to electrostatic forces, or vector potentials, which occur if magnetic interactions are also present. In the treatment that follows, we shall neglect the magnetic potential and assume that only external electrostatic potentials affect the system under consideration. 

The potentials (5.15), (5.16), (5.18) are obviously built up in like manner as are the respective electrostatic potentials V(r) in Section 4.3-4.5. The scalar potentials used become the electrostatic potentials u r), v r) if K approaches zero, i.e. when the velocity of light c approaches infinity. 

In most quantum chemical applications the electrons are not free, but are subject to electrostatic potentials in the form of Coulomb interactions with nuclei and other electrons. These are accounted for in the Hamiltonian by adding a term involving the scalar potential and the charge of the electron (—e) ... 

Regarding the electrostatic interactions, the most common approaches are based on the exact solution of the Classical Electrostatic Laplace s equation for the electric (scalar) potential, <1>,... 

Knowledge of the spatial dimensions of a molecule is insufficient to imderstand the details of complex molecular interactions. In fact, molecular properties such as electrostatic potential, hydrophilic/lipophilic properties, and hydrogen bonding ability should be taken into account. These properties can be classified as scalar isosurfaces), vector field, and volumetric properties. 

Besides molecular orbitals, other molecular properties, such as electrostatic potentials or spin density, can be represented by isovalue surfaces. Normally, these scalar properties are mapped onto different surfaces see above). This type of high-dimensional visualization permits fast and easy identification of the relevant molecular regions. 

A field is a function that describes a physical property at points in space. In a scalar field, this physical property is completely described by a single value for each point (e.g. temperature, density, electrostatic potential). For vector fields, both a direction and a magnitude are required for each point (e.g. gravitation,... 

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