Poisson’s equation of electrostatics

A second relation between p(r) and 4>(r) may be obtained by noting that the sources of 0(r) are the point charge Ze of the nucleus, located at the origin and the charge distribution due to the N electrons. Treating the charge density —ep(r) of the electrons as continuous, Poisson s equation of electrostatics may be used to write... 

Now, we also assume that, even on a microscopic scale, Poisson s equation of electrostatics is valid ... 

In continuum notation, this relation would constitute one form of Poisson s equation of electrostatics. The continuum forms of E(x) and V (x) are valid if the charge density planes are so close together that over small regions of space the charge density can be viewed as a continuous function p(x) of position x. [The local space charge density p(x) has units of Coulombs m-3]. In such cases, the sums in eqns. (37) and (40) for E (x) and V (x) can be approximated by integrals to give... 

This latter relation constitutes the integral form of Poisson s equation of electrostatics. 

The Debye-Hiickel theory focuses on and by using Poisson s equation of electrostatics finds an explicit equation for yfrj, firom which the potential, at the surface of the central j ion... 

Poisson s equation of electrostatics relates pe to the variation of i/>with distance to the charged surface, x, in the form... 

The simplest model of charge shielding and colloidal stability against aggregation was developed around 1910 independently by L-G Gouy (18.34-1926), a French physicist, and DL Chapman (1869-1958), a British chemist. They combined Poisson s equation of electrostatics with the Boltzmann distribution law. 

As mentioned before, the ESP has been a quantity of great significance in quantum biochemistry. Using Poisson s equation of classical electrostatics, as applied to an atom, one can write... 

The mathematical relationship between O(r) and p(r) is given by Poisson s equation of classical electrostatics ... 

The usual derivation of the Poison-Boltzmann equation, which can be found in various texts and monographs [3,4], begins with Poisson s equation for electrostatics in a dielectric medium... 

These early results of Coulomb and his contemporaries led to the full development of classical electrostatics and electrodynamics in the nineteenth cenmry, culminating with Maxwell s equations. We do not consider electrodynamics at all in this chapter, and our discussion of electrostatics is necessarily brief. However, we need to introduce Gauss law and Poisson s equation, which are consequences of Coulomb s law. 

Although the continuum model of the ion could be analyzed by Gauss law together with spherical symmetry, in order to treat more general continuum models of electrostatics such as solvated proteins we need to consider media that have a position-specific permittivity e(r). For these a more general variant of Poisson s equation holds ... 

The profile of the electrostatic potential V in an MS junction can be calculated by solving Poisson s equation... 

The convenience of Eq. (6) is realizable only in the rather unrealistic situation where the charge distribution exhibits cylindrical or spherical symmetry. For storage silos, blenders, fluidized bed reactors, and other real vessel geometries, integral solutions are usually not possible, necessitating an alternate problem formulation. Poisson s equation serves this need, relating the volume charge distribution to the electrostatic potential. 

The charge density of dust transported through ducts and the resultant electric fields at the duct Inner walls was monitored by a Monroe Electronics Inc., Model 171 electric fieldmeter. All the electrostatic sampling In the field was performed In circular cross-section ducts. Thus, the electrostatic field Intensity, for this geometry, can be determined from Poisson s equation using the cylindrical coordinate system. 

Continuum models are rooted in classical electrostatics, and its applications in the analysis of the dielectric constants of polar liquids. A key relationship is Poisson s equation,... 

An early continuum treatment of solvation, associated with Born,17 comes out of the analysis of the electrostatic work involved in building up a charge Q on a conducting sphere of radius R in a medium with dielectric constant e. From Poisson s equation, it follows that the potential outside of the sphere is Q/eR. Thus the work of charging is the result of each additional element dq interacting with the charge q already present 87... 

An important equation of electrostatics, which follows directly from Maxwell s equations (Jackson 1975) is Poisson s equation. It relates the divergence of the gradient of the potential charge density at that point ... 

The charge density x on any electrolyte lamina parallel to the electrode and a distance x from it can be obtained by the application of electrostatics (Poisson s equation) and the Boltzmann distribution. Similarly, one can write for the intrinsic semiconductor, Poisson s equation... 

In an exact calculation of the distribution of the electrostatic potential, the carrier densities and their currents, (4.81)-(4.87) are solved simultaneously, bearing in mind that only the sum of the diffusion and drift currents has physical significance. Due to the complexity of the above relations and in particular due to the coupling of electron and hole concentrations by Poisson s equation, analytical solutions exist only for a few, very specific conditions. Generally, the determination of local carrier concentrations, current densities, recombination rates, etc., requires extensive numerical procedures. This is especially true if they vary with time, but even in the steady state context. 

In the literature sometimes the statement is made that the Poisson-Boltzmann equation is only compatible with electrostatics if linearized, which is not correct. The argument refers to the superposition principle which relies on the presupposed linearity of Poisson s equation. Note, however, that Poisson s equation is not linear if the charge density depends on f itself in a non-linear way as it is the case here. 

The Debye-Hiickel approach is an excellent example of electrochemical theory. Electrostatics is introduced into the problem in the form of Poisson s equation, and the chemistry is contained in the Boltzmann distribution law and the concept of true electrolytes (Section 3.2). The union of the electrostatic and chemical modes of... 

In order to solve for xp it is necessary to have another relationship between p and and this may be obtained by introducing Poisson s equation, which is equivalent to assuming that Coulomb s law of force between electrostatic charges also holds good for ions. This equation in rectangular coordinates is... 

The Poisson Equation From classical electrostatics, the free charge density p(r)—that is, the charge density due to the solute as opposed to the polarization charges in the solvent—in a continuous medium of homogeneous dielectric constant (relative permittivity) e, where r denotes the position in space, is related to the electrostatic potential, ( )(r), by Poisson s equation, which takes the following form, in this case in Gaussian units ... 

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