The ideal electrostatic conductor model

Under the Countable Choice convention of the stated boundary and sheet-distribution interfaces (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), adopt as a physical model of ideal electrostatic conductors, for the classical laboratory framework of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields with the units and constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values: a conductor configuration consists of the following data, and every model of this postulate is required to satisfy the clauses below. No experiment is reported and no empirical value is inferred; this is an explicitly adopted equilibrium idealization together with its geometric and source conventions.

(a) [Geometry] a laboratory region $\mathrm{\Omega }\times I$ with $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ an arbitrary open ambient region, a finite family $D_1,\dots ,D_m$ ($m\ge 1$) of bounded $C^{\mathrm{\infty }}$ domains with pairwise disjoint closures in the sense of Bounded C^k domains and boundary charts with closures strictly inside $\mathrm{\Omega }$ (the conductor bodies), the conductor $D:=D_1\cup \cdots \cup D_m$, its surface $\mathrm{\partial }D$, and the vacuum region ${\mathrm{\Omega }}_v:=\mathrm{\Omega }\setminus \bar{D}$. By The boundary of a bounded smooth domain is a compact embedded hypersurface, $\mathrm{\partial }D=\mathrm{\partial }{D}_1\cup \cdots \cup \mathrm{\partial }{D}_m$ is a disjoint union of compact embedded $C^{\mathrm{\infty }}$ hypersurfaces. The vacuum ${\mathrm{\Omega }}_v$ is open, and near each conductor surface it is locally the smooth exterior side of that conductor. No boundedness or regularity of the ambient boundary is inferred from openness. In the additional bounded smooth ambient branch, assume that $\mathrm{\Omega }$ is a bounded $C^{\mathrm{\infty }}$ domain; only then The vacuum region around compact smooth conductors is a bounded smooth domain gives that ${\mathrm{\Omega }}_v$ is a bounded $C^{\mathrm{\infty }}$ domain with the specified outer/conductor boundary charts; we additionally require ${\mathrm{\Omega }}_v$ to be connected (configurations whose vacuum region is separated by the conductors, for instance by an enclosing closed shell, are outside this model as formulated here).

(b) [Equilibrium and sources] the configuration is in the pure electrostatic regime of The static and pure electrostatic regimes and their fixed-time reading on the vacuum region: $B=0$, $J=0$, $E$ is the fixed-time electric field of The electric and magnetic fields as fixed-time spatial vector fields, the volume source density ${\rho }_v$ is a continuous real map on $\overline{{\mathrm{\Omega }}_v}$ (C m${}^{-3}$, Charge parameter, frame charge and current densities, volume charge and flux, and the four-current) extended by zero to $D$; the conductor charge is a stationary surface charge, i.e. a continuous density $\sigma :\mathrm{\partial }D\to \mathbb{R}$ (C m${}^{-2}$) whose pairing with test functions is the stationary-sheet source of Distributional Maxwell equations with point and stationary-sheet sources(c) with tangential current $K=0$ and $S=\mathrm{\partial }D$; and the field pair $(E,0)$ together with the source pair $({\rho }_v+\sigma {\delta }_{\mathrm{\partial }D},0)$ satisfies the adopted distributional vacuum Maxwell equations Distributional vacuum Maxwell equations for the declared point and stationary-sheet sources. Inside the conductor the field vanishes: $E\equiv 0$ on $D$ (the equilibrium condition); equivalently the conductor interior is field-free and carries no volume charge.

(c) [Potential and equipotential convention] there is a function $\phi \in C^0(\mathrm{\Omega })\cap C^2({\mathrm{\Omega }}_v)$ with $E=-\mathrm{\nabla }\phi$ on ${\mathrm{\Omega }}_v$ and with $\phi$ constant on each conductor body, $\phi {\mathrm{\mid }}_{D_j}\equiv V_j$ for real constants $V_j$ (volts, Framework-local SI units, the vacuum constants and the exact/measured status of their values); by continuity of $\phi$ the trace of $\phi$ on each $\mathrm{\partial }{D}_j$ from the vacuum side equals $V_j$ as well. This is the single-function convention of an ideal conductor with no double layer: the model reads the same potential value on both sides of $\mathrm{\partial }{D}_j$, while the normal component of $E$ may jump.

(d) [Data] a model is completed either by the component potentials $V_1,\dots ,V_m$ together with the volume source ${\rho }_v$ and the data on $\mathrm{\partial }\mathrm{\Omega }$ where the laboratory region is bounded, or by the component total charges $Q_j:={\int }_{\mathrm{\partial }{D}_j}\sigma dS$ (C) together with those same ambient data. These are two declared data types; the postulate asserts neither that both data types are simultaneously specifiable nor that either is solvable for arbitrary data.

(e) [Scope] This is a static, frame-relative equilibrium idealization: it excludes transient finite-conductivity dynamics, relaxation to equilibrium, moving surface charge, electrode polarization and contact effects, and it is not derived from a microscopic matter model. It asserts no existence or uniqueness theorem (those are separate conditional results), no force, no energy, no radiation, no measurement and no apparatus realization; the value $V_j$ is a model parameter and the surface density $\sigma$ is an idealization variable determined by the equilibrium problem when it is solvable. The general open ambient branch retains compact conductor bodies in exterior configurations. Its additional bounded smooth ambient branch is distinct from the separately declared grounded-half-space and infinite parallel-plane idealizations, whose conductor bodies are themselves noncompact. Those examples retain the equilibrium/equipotential interpretation but their noncompact bodies and plane distributions are not instances of this compact-body configuration or its compact-sheet predicate. No arbitrary exterior solvability is asserted.

The postulate records a model choice, not a deduction, and its clauses are the exact objects the rest of this page consumes. The field-free interior with all charge on the surface and the constant-potential (equipotential) surface are the standard ideal-conductor clauses of the source treatments (Griffiths §2.5.1 printed pp. 96-98; Tong §2.4 printed pp. 30-31; Chew §32.1-32.2 printed pp. 351-354), presented there for equilibrium configurations and reproduced here as adopted assumptions rather than as derived theorems. The division into two data types in (d) is the fixed-potential versus fixed-charge distinction of Tong §2.4.2 printed pp. 33-35 and Griffiths §3.1.5-3.1.6 printed pp. 116-121; it is a definitional choice about which quantities are prescribed. Nothing in (e) turns the idealization into a universal statement about real metals: finite conductivity, transients, contact potentials, surface forces and material properties are outside the model, and no numerical or empirical value of any material quantity is asserted.

The mathematical content used to name the objects is exact and proved earlier on this page: the surface $\mathrm{\partial }D$ of finitely many pairwise disjoint bounded $C^{\mathrm{\infty }}$ bodies is a disjoint finite union of compact embedded smooth hypersurfaces, and in the bounded smooth ambient branch the vacuum region is again a bounded smooth domain whose boundary pieces are the outer boundary and the conductor surfaces. The stationary-sheet pairing $\sigma {\delta }_{\mathrm{\partial }D}$ is the surface-distribution object of Surface distributions and interfacial jumps as used by Distributional Maxwell equations with point and stationary-sheet sources(c), and the solution predicate for the field-source pair is the separately adopted Distributional vacuum Maxwell equations for the declared point and stationary-sheet sources; the postulate itself neither proves that predicate nor asserts that it is satisfiable for the declared data. The countable-choice convention belongs to those measure and distribution interfaces and is carried here explicitly (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

The model is deliberately silent about configurations whose conductor bodies enclose a cavity of the vacuum: the bounded smooth ambient branch requires ${\mathrm{\Omega }}_v$ connected, so configurations separated by the conductors are outside the model as formulated. The separately declared grounded-plane and ideal parallel-plate examples are noncompact idealizations and are not instances of this compact-body configuration; they retain their own explicit statements and are not theorems about this postulate.