Chapter 2 Toward Paradoxical Inconsistency in Electrostatics of Metallic Conductors Michael Grinfeld, Pavel Grinfeld, and Steven B. Segletes Abstract In a recent report, we drew attention to the paradoxical inconsistency of classical electrostatics with the model of a crystalline conductor. There are different ways to avoid this inconsistency. Mostly, they rely on physical adhoc assumptions associated with the introduction of additional constants and models of charged liquids. In this report, we suggest another approach, which does not require the introduction of any additional physical constants. The proposed theory includes classical electrostatics as a special case. Keywords Electrostatics · Laplace and Poisson equations · Simple layers · Thermodynamics Introduction Electromagnetism is in the background of various military applications, including the problem of protection against shaped charges, which Russian adversaries call “cumulative” jets [1]. One of the many problems faced in the computer implementation of physical models is associated with the fact that certain physical fields experience discontinuities across boundaries and interfaces. Some fields (the models thereof) even become infinitely large when approaching such interfaces. Theorists have accumulated various sophisticated mathematical tools for handling these problems when treating the problems analytically. The problem of infinite charge density in the vicinity of boundaries in metals is further aggravated when using numerical modeling, since the singularities in the underlining physical theories are in violent discord with the inability of finite discretizations to handle sharp gradients. One of the methods for handling this violent discord is based on the replacement of the classical physical theories with modified ones that allow for the avoidance of infinite charge density at the boundaries. We suggested one approach of this sort in a prior report [2] and continue developing the approach in this one. Let us recall the paradoxical inconsistency of classical electrostatics first noted in our prior report [2]. According to classical electrostatics, all excess electric charges, positive or negative, concentrate on the conductor’s boundary with a finite 2-D density. In mathematical physics, these sort of boundaries are known as “simple layers” of charges [3, 4]. This surface density can be either positive or negative, depending on the total excess charge of the conductor. What is the physical meaning of this finite 2-D density of electric charge? Actually, this concept implies that the associated 3-D density of the electric charge is infinite. Of course, this infinity, on its own, is an essential inconsistency of classical electrostatics. However, in many cases, this particular limitation is not very important and classical electrostatics provides researchers with reasonable results [3, 4]. It is not, however, this inconsistency that we discussed in our prior work [2]. The paradoxical inconsistency that we introduced [2] is different. In fact, the density of the easily movable negative charges can grow and reach quite large values (for example, on the surface of a body), in accordance with classical electrostatics. If the positive charges are easily movable and compressible also, like in the case of ionized gas, the same can be said about the gas of positive ions. In this sense, classical electrostatics can be applied to a gaseous plasma. The situation, however, changes dramatically when the positive charges belong to a (near rigid) lattice. In this case, the maximum positive charge density is basically fixed, and we cannot expect that it might assume infinite values, even in an idealized sense. In other words, there cannot be surfaces (or interfaces) with finite positive charges—excess positive charge can only M. Grinfeld ( ) · S. B. Segletes Weapons and Materials Research Directorate, Army Research Laboratory, Aberdeen Proving Ground, MD, USA P. Grinfeld Drexel University, Philadelphia, PA, USA © Society for Experimental Mechanics, Inc. 2020 L. E. Lamberson (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-30021-0_2 5
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