Electricity at the macroscale and its microscopic origins

Abstract

This work examines the relationships between electrical structures at the microscale and electrical structures at the macroscale. By structures I mean both physical struc- tures, such as the spatial distributions of charge and potential, and the mathematical structures used to specify physical structures and to relate them to one another. I do not discuss magnetism and what little I say about energetics is incidental. I define the fields that describe electrical macrostructure, and their rates of change, in terms of the microscopic charge density ρ, electric field E, electric potential φ, and their rates of change. To deduce these definitions, I lay some foundations of a general theory of structure homogenization, meaning a theory of how any observable macroscopic field V is related to spatial averages of its microscopic counterpart ν. An integral part of structure homogenization theory is the definition of macroscopic excess fields in terms of microscopic fields. The excess field of V : Rn → R on the boundary ∂Ω of a finite-measure subset Ω of Rn is the field σV : ∂Ω → R to which it is related by the generalized Stokes theorem, ∫ Ω V dω = ∫∂Ω σV ω, where ω and dω are volume forms on ∂Ω and Ω, respectively, and V dω ≡ d (σV ω). For example, the macroscopic volumetric charge density % in a material Ω is related to the areal charge density σ on its surface by∫Ω % d3r = ∫∂Ω σ d2s and by % d3r ≡ d (σ d2s). I derive an expression for σV [ν], which generalizes Finnis’s expression for excess fields at the surfaces of crystals (e.g., surface charge density σ[ρ]) to disordered microstructures. I use homogenization theory to define the macroscopic potential Φ ≡ Φ[φ], electric field E ≡ E[E], and charge density % ≡ %[ρ], and I define the macroscopic current density as J ≡ ˙σ[ ˙ρ]. Using the microscopic theory, or vacuum theory, of electromagnetism as my starting point, I deduce that the relationships between these macroscopic fields are identical in form to the relationships between their microscopic counterparts. Without invoking quantum mechanics, I use the definitions J ≡ ˙σ and σ ≡ σ[ρ] to derive the expressions for so-called polarization current established by the Modern Theory of Polarization. I prove that the bulk-average electric potential, or mean inner potential, Φ, 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 E or 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. Only anisotropy of the macrostructure can bestow directionality at the macroscale. The macroscopic charge density % is isotropic in the bulks of most materials, because it vanishes at every point. This implies that, regardless of the microstructure ρ, a macroscopic electric field cannot emanate from the bulk. I find that all relationships between observable macroscopic fields can be expressed mathematically without introducing the polarization (P) and electric displacement (D) fields, neither of which is observable. Arguments for the existence of P and D, and interpretations of them, have varied since they were introduced in the 19th century. I argue that none of these arguments and interpretatons are valid, and that macroscale isotropy prohibits the existence of P and D fields.