Electricity at the macroscale and its microscopic origins

TL;DR

This paper develops a microscopic foundation for macroscopic electrical fields, deriving relationships without quantum mechanics and clarifying the roles of polarization and displacement fields in bulk materials.

Contribution

It introduces a new microscopic basis for macroscopic electric fields, showing that polarization and displacement fields are not necessary for describing observable macroscopic phenomena.

Findings

Macroscopic fields mirror microscopic relationships.

Bulk-average electric potential vanishes in neutral, uniform materials.

Macroscopic E-fields cannot originate from the bulk in charge-neutral materials.

Abstract

I define the fields that describe electrical macrostructure, and their rates of change, in terms of the microscopic charge density, electric field, electric potential, and their rates of change. To deduce these definitions, I lay some new foundations of a theory of how observable macroscopic fields are related to spatial averages of their microscopic counterparts. I find that the relationships between macroscopic fields are identical in form to the relationships between their microscopic counterparts, meaning that the $\vec{P}$ and ${\vec{D}}$ fields do not appear in them. Without invoking quantum mechanics, I derive the expressions for polarization current established by the Modern Theory of Polarization. I prove that the bulk-average electric potential, or mean inner potential, vanishes in a macroscopically-uniform charge-neutral material, and I show that when a crystal lattice lacks inversion symmetry, it does not imply the existence of macroscopic $\vec{E}$ or $\vec{P}$ fields in the crystal's bulk. I point out that symmetry is scale-dependent. Therefore, if anisotropy of the microstructure does not manifest as anisotropy of the macrostructure, it cannot be the origin of a macroscopic vector field. The macroscopic charge density vanishes in a material's bulk. Therefore, regardless of the microstructure, a macroscopic $\vec{E}$ field cannot emanate from the bulk. I find that all relationships between observable macroscopic fields can be expressed mathematically without introducing the polarization ($\vec{P}$) and electric displacement ($\vec{D}$) fields, neither of which is observable. I also show that most `quantum mechanical' aspects of the existing microscopic theory of electricity in materials are compatible with, or required features of, a statistical theory of classical particles whose charges and masses are comparable to those of electrons and nuclei.