Attenuation of the Electric Potential and Field in Disordered Systems

TL;DR

This paper investigates how disordered, electroneutral charge distributions in different dimensions influence electric potential and field, revealing that potential can be large even when fields are small, especially in higher dimensions.

Contribution

It provides a probabilistic analysis of electric potential and field in totally disordered, electroneutral systems across one, two, and three dimensions, highlighting dimension-dependent behaviors.

Findings

Electric field remains small in all dimensions for disordered systems.

Electric potential can be very large in 2D and 3D disordered systems.

Local electroneutrality leads to small potentials across all dimensions.

Abstract

We study the electric potential and field produced by disordered distributions of charge to see why clumps of charge do not produce large potentials or fields. The question is answered by evaluating the probability distribution of the electric potential and field in a totally disordered system that is overall electroneutral. An infinite system of point charges is called totally disordered if the locations of the points and the values of the charges are random. It is called electroneutral if the mean charge is zero. In one dimension, we show that the electric field is always small, of the order of the field of a single charge, and the spatial variations in potential are what can be produced by a single charge. In two and three dimensions, the electric field in similarly disordered electroneutral systems is usually small, with small variations. Interestingly, in two and three dimensional systems, the electric potential is usually very large, even though the electric field is not: large amounts of energy are needed to put together a typical disordered configuration of charges in two and three dimensions, but not in one dimension. If the system is locally electroneutral--as well as globally electroneutral--the potential is usually small in all dimensions. The properties considered here arise from the superposition of electric fields of quasi-static distributions of charge, as in nonmetallic solids or ionic solutions. These properties are found in distributions of charge far from equilibrium.