Laws Poisson equation

Note that the mathematical symbol V stands for the second derivative of a function (in this case with respect to the Cartesian coordinates d fdx + d jdy + d jdz y, therefore the relationship stated in Eq. (41) is a second-order differential equation. Only for a constant dielectric Eq.(41) can be reduced to Coulomb s law. In the more interesting case where the dielectric is not constant within the volume considered, the Poisson equation is modified according to Eq. (42). 

The charge density is simply the distribution of charge throughout the system and has 1 units of Cm . The Poisson equation is thus a second-order differential equation (V the usual abbreviation for (d /dr ) + (f /dx/) + (d /dz )). For a set of point charges in constant dielectric the Poisson equation reduces to Coulomb s law. However, if the dielectr... 

Debye and Huckel applied the Boltzmann statistical distribution law and the Poisson equation for electrostatics in the model above (1,6, 10). In the calculations using the model above they considered one particular ion (the reference ion, or central ion) with... 

With this electric potential Poisson equation (A

el = net charge density) to eventually obtain the concentration of electrons at the film surface (A ). It further follows that Ne(A ) varies with the film layer thickness as A -2. If we now assume that the (catalyzed) rate of dissociation of the adsorbed X2 molecules is proportional to the surface concentration of electrons, and that this dissociation process is rate determining, a cubic rate law for the film growth can be expected (A — At 2 At - t in). In fact, during the oxidation of Ni at temperatures between 250 and 400 °C, an approximately cubic rate law has been experimentally observed. We emphasize, however, that the observed cubic oxidation rate does not prove the validity of the proposed reaction mechanism. Different models and assumptions concerning the atomic reaction mechanism may lead to the same or similar dependences of the growth rate on thickness. 

The volumetric charge density is of interest in the study of ionic solutions, in which one can calculate the charge density around a specific ion. This is done by using the Poisson equation, based on electrostatic electric fields or by Boltzman distribution law of classical statistic mechanics. For the simpler case of dilute solutions this approach yields the expression p =... 

Now, a linear charge density-potential relation is consistent with the law of superposition of potentials, which states that the electrostatic potential at a point due to an assembly of charges is the sum of the potentials due to the individual charges. Thus, when one uses an unlinearized P-B equation, one is assuming the validity of the law of superposition of potentials in the Poisson equation and its invalidity in the Boltzmann equation. This is a basic logical inconsistency which must reveal itself in the predictions that emage from the so-called rigorous solutions. This is indeed the case, as will be shown below. 

The Poisson equation (or Gauss Law) describes the electrostatic potential of a fixed charged density of the solute Psoiuteir). The exterior charge density of the ions in the solution, Pexteriorif) is modeled by assuming a Boltzmann distribution. The Poisson-Boltzmann (PB) equation is commonly applied to molecules in aqueous solution to compute the electrostatic potential of the system. The general form of the PB equation is... 

Equations (2) and (12) may be combined and re-expressed, using Gauss s law, as a partial differential equation, the Poisson Equation... 

As was seen earlier, the Poisson equation, which is a combined statement of Coulomb s and Gauss laws, may be written as... 

This may contradict one of the basic laws of electrostatics, that is the linear superposition of fields. This is a fundamental problem for the theory and to any other theory or development, such as that due to Guggenheim below (Section 10.13.1), which makes use of any such combination. In fact this is one of the big problems in the theory. The other big problem is that the x/tjS in the Poisson equation and in the Maxwell-Boltzmann distribution are different and have a different physical basis (see Section 10.6.5). This is believed by many to be yet another fundamental problem for the theory. 

If we would know any of the fundamental forms valid for the ideal gas, we could readily derive both the ideal gas law and the Poisson equation, namely the adiabatic... 

Electrostatics. Two important laws of electrostatics are directly deduced from this scheme of spatial reduction the Poisson equation and the Gauss theorem (case studies B4 and B5 in this chapter). The mutual influence between two of these poles is modeled within the framework of an electrostatic dipole using Coulomb s law. 

The exact distribution of charge and potential can be derived from electrostatic theory and the laws of electron distribution statistics, solving the Poisson-equation with the appropriate boundary conditions. 

Poisson equation, i.e., the electroneutrality equation is basically the Gauss law, that is to say the Maxwell s equation giving the dependence between the electric induction vector and charge density... 

The Debye-Hiickel equation derives from a combination of the Poisson equation and a statistical-mechanical distribution formula (Debye and Hiickel, 1923). The Poisson equation is a general expression of the Coulomb law of force between charged bodies and can be written as... 

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