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Surface charge and normal field on an ideal conductor

Physical scope

The ideal-conductor boundary conditions of classical electrostatics in a named inertial frame: for a stationary smooth conductor surface in the pure electrostatic regime and fields with the stated one-sided regularity, the surface charge density equals epsilon_0 times the normal vacuum-side field, the vacuum-side field is normal to the surface, the potential trace is constant on each connected piece of the conductor surface with the model's component value, and the total component charges are the outward flux integrals of the vacuum field. The statement is conditional and pointwise; it excludes moving or nonsmooth surfaces, magnetic and current sources, forces, energy, material response and any empirical claim.

Statement

Assume the Axiom of Countable Choice ( ${A C}_{ω}$ ) inherited from the surface and distribution interfaces (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $(Ω; D_{1}, \dots, D_{m}; Ω_{v}; E; ρ_{v}; σ; V_{j})$ be a model of The ideal electrostatic conductor model with surface $S : = \partial D$ and vacuum region $Ω_{v}$ , with the fields of The electric and magnetic fields as fixed-time spatial vector fields, the constants and units of Framework-local SI units, the vacuum constants and the exact/measured status of their values and the pure electrostatic regime of The static and pure electrostatic regimes and their fixed-time reading carried by the model, and suppose additionally that the field $E$ satisfies the joint one-sided $C^{1}$ regularity hypothesis of Stationary vacuum interface jumps and the absence of magnetic sheet sources across $S$ : around every $x_{0} \in S$ each side of $S$ is the restriction of a $C^{1}$ field on an open neighbourhood of the corresponding closed side of a smaller graph chart. Let $n$ denote the unit normal on $S$ pointing from the conductor side into the vacuum side (equivalently: $n$ is the inward normal of $Ω_{v}$ and the outward normal of $D$ on $S$ , by the local exterior conductor chart; in the bounded smooth ambient branch this is the sign convention of The vacuum region around compact smooth conductors is a bounded smooth domain(c)), and let $E_{+}$ denote the vacuum-side trace of $E$ (the field is zero on the conductor side, so $[E] = E_{+}$ ). Then, pointwise on $S$ :

(a) [Normal sheet charge] $n \cdot E_{+} = σ / ε_{0}$ and $n \times E_{+} = 0$ ; equivalently $σ = ε_{0} n \cdot E_{+}$ and the tangential part of the vacuum-side field vanishes: $E_{+}$ has no component tangent to $S$ . In particular a conductor surface with $σ = 0$ at a point has $E_{+} = 0$ there (both traces vanish), and the sign of $σ$ is the sign of the outward normal field component.

(b) [Normal derivative of the potential] with $φ$ the model's potential and $\partial φ / \partial n : = \nabla φ_{+} \cdot n$ the one-sided normal derivative in the direction $n$ from the conductor into the vacuum (Classical normal derivative), one has

$\frac{\partial φ}{\partial n} = - \frac{σ}{ε_{0}}, equivalently σ = - ε_{0} \frac{\partial φ}{\partial n} .$

(c) [Equipotential boundary traces] the trace $φ_{+}$ of $φ$ on $S$ is constant on each connected component of $S$ , with value $V_{j}$ on every connected component contained in $\partial D_{j}$ ; consequently the boundary value problem in the vacuum region is a Dirichlet problem whose data are the constants $V_{j}$ on the conductor surfaces.

(d) [Total charges] the total charge of conductor $j$ is

$Q_{j} = \int_{\partial D_{j}} σ d S = ε_{0} \int_{\partial D_{j}} n \cdot E_{+} d S = ε_{0} \int_{\partial D_{j}} \frac{\partial φ}{\partial ν} d S,$

where $ν$ is the local outward normal of the vacuum along the smooth conductor interface $S$ , without asserting regularity of a general ambient outer boundary (so $ν = - n$ on $\partial D$ and the last equality uses $\partial φ / \partial ν = - \partial φ / \partial n$ ); the surface integrals exist as integrals of continuous functions over the compact embedded hypersurfaces $\partial D_{j}$ (Surface integration on compact C1 hypersurfaces, The boundary of a bounded smooth domain is a compact embedded hypersurface).

(e) [Data reading] the two data types of the model are related through (d): if the component potentials and ambient data are fixed and the equilibrium solution is given, the total charges are determined by (d); prescribing the total charges instead determines the potentials only through the (separate) equilibrium problem, and this theorem asserts no existence or uniqueness for charge data.

(f) [Scope] The statement is pointwise and stationary: it assumes the one-sided joint regularity above, a $C^{\infty}$ conductor surface, and the pure electrostatic regime; it excludes moving surface charge, normal sheet currents, magnetic sheet sources, nonsmooth or cornered conductors, fields without one-sided traces, forces, energy and any measurement; no self-force or material response is asserted.

Facts & Assumptions

Given: A model of The ideal electrostatic conductor model with conductor surface $S = \partial D = \partial D_{1} \cup \dots \cup \partial D_{m}$ , vacuum region $Ω_{v}$ , field $E$ with $E \equiv 0$ on $D$ , continuous stationary surface density $σ$ on $S$ , potential $φ \in C^{0} (Ω) \cap C^{2} (Ω_{v})$ with $E = - \nabla φ$ on $Ω_{v}$ and $φ \equiv V_{j}$ on $D_{j}$ , and the joint one-sided $C^{1}$ regularity of the statement across $S$ .

[F1]

The conductor model: $E \equiv 0$ on $D$ ; the pair $(E, 0)$ with the source pair $(ρ_{v} + σ δ_{S}, 0)$ , $K = 0$ , satisfies the adopted distributional vacuum Maxwell equations Distributional vacuum Maxwell equations for the declared point and stationary-sheet sources; $σ$ is the stationary-sheet source of Distributional Maxwell equations with point and stationary-sheet sources(c) under Surface distributions and interfacial jumps; $φ$ is continuous on $Ω$ and $C^{2}$ on $Ω_{v}$ with $E = - \nabla φ$ there, constant on each $D_{j}$ with value $V_{j}$ (The ideal electrostatic conductor model, The static and pure electrostatic regimes and their fixed-time reading, The electric and magnetic fields as fixed-time spatial vector fields, Framework-local SI units, the vacuum constants and the exact/measured status of their values).

[F2]

The fixed-vacuum-jump theorem: under ${A C}_{ω}$ , the stationary compact embedded $C^{2}$ interface hypotheses, the joint one-sided $C^{1}$ field regularity and the weak Maxwell equations with the stationary-sheet source of [F1], one has pointwise on $S$ $n \cdot [E] = σ / ε_{0}$ , $n \times [E] = 0$ , $n \cdot [B] = 0$ , $n \times [B] = μ_{0} K$ , with $[F] = F_{+} - F_{-}$ and $n$ pointing from the $Ω_{-}$ side to the $Ω_{+}$ side (Stationary vacuum interface jumps and the absence of magnetic sheet sources, Surface distributions and interfacial jumps).

[F3]

For a bounded $C^{\infty}$ conductor surface the outward normal of the vacuum region on $\partial D_{j}$ is $n_{j} = - n$ (the negative of the conductor's outward normal) whenever the bounded smooth ambient hypothesis of The vacuum region around compact smooth conductors is a bounded smooth domain(c) is in force; the local exterior conductor chart of the model identifies the vacuum-side normal with the conductor's outward normal in general (The ideal electrostatic conductor model(a), The vacuum region around compact smooth conductors is a bounded smooth domain(c)).

[F4]

Classical one-sided normal derivative: for the vacuum side, $\partial φ / \partial n : = \nabla φ_{+} \cdot n$ is the normal derivative of the $C^{1}$ vacuum-side restriction $φ$ in the direction $n$ , where $φ$ is $C^{2}$ on the vacuum side and its gradient has the continuous trace $- E_{+}$ (Classical normal derivative, $C^{k}$ Euclidean maps and diffeomorphisms).

[F5]

$\partial D$ is a finite disjoint union of compact embedded smooth hypersurfaces carrying a finite surface measure on which continuous functions are integrable (The boundary of a bounded smooth domain is a compact embedded hypersurface, Surface integration on compact C1 hypersurfaces).

[F6]

Euclidean vector decomposition: a vector $v$ with $n \cdot v = 0$ and $n \times v = 0$ for the unit vector $n$ is $v = 0$ , since $v = (n \cdot v) n + n \times (v \times n)$ for orthonormal triples (The electric and magnetic fields as fixed-time spatial vector fields, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The cross product in $R^{3}$ ).

Proof technique: direct.

Proof

1.1F1F2F6given

By [F2] with the conductor side as $Ω_{-}$ and the vacuum side as $Ω_{+}$ (the normal $n$ of the statement points from the conductor into the vacuum, matching [F2]), the jump identities $n \cdot [E] = σ / ε_{0}$ and $n \times [E] = 0$ hold pointwise on $S$ ; the conductor-side trace vanishes because $E \equiv 0$ on $D$ [F1], so $[E] = E_{+} - 0 = E_{+}$ . This gives $n \cdot E_{+} = σ / ε_{0}$ and $σ = ε_{0} n \cdot E_{+}$ , and $n \times E_{+} = 0$ , which is (a); if $σ = 0$ at a point then $n \cdot E_{+} = 0$ and $n \times E_{+} = 0$ there, so $E_{+} = 0$ by [F6], and the sign of $σ$ is the sign of $n \cdot E_{+}$ since $ε_{0} > 0$ .

1.2F1F5given

Equipotential boundary traces. For $x \in \partial D_{j}$ , the boundary definition supplies points of $D_{j}$ arbitrarily close to $x$ . Since $φ$ is continuous on $Ω$ and $φ ∣_{D_{j}} = V_{j}$ [F1], its value and vacuum trace at $x$ equal $V_{j}$ . Thus the trace is constant on all of $\partial D_{j}$ , and in particular on every connected component contained there. The separated boundary pieces partition $S$ [F5], so this proves (c) directly from the model's continuous equipotential convention.

2.1F1F4step 1.1

On the vacuum side $E = - \nabla φ$ [F1], so $\nabla φ_{+} = - E_{+}$ [F4] and $\partial φ / \partial n = \nabla φ_{+} \cdot n = - E_{+} \cdot n = - σ / ε_{0}$ by [step 1.1]; this is (b).

3.1F3F4F5step 1.1step 2.1

Total charges. The surface integrals in $Q_{j} = \int_{\partial D_{j}} σ d S$ exist as integrals of the continuous density $σ$ over the compact embedded hypersurface $\partial D_{j}$ with finite surface measure [F5]. Substituting $σ = ε_{0} n \cdot E_{+}$ from [step 1.1] gives the middle expression. For the last expression, on $\partial D_{j}$ the local outward normal of the vacuum is $ν = - n$ [F3], so $\partial φ / \partial ν = \nabla φ_{+} \cdot ν = - \nabla φ_{+} \cdot n = - \partial φ / \partial n = σ / ε_{0}$ by [step 2.1]; integrating over $\partial D_{j}$ gives $Q_{j} = ε_{0} \int_{\partial D_{j}} \partial φ / \partial ν d S$ . This is (d).

4.1F1F2F5givenstep 3.1∎

Data reading and scope. Clause (e) follows from the definitions in (d): the total charges are determined by the equilibrium solution through the displayed identities, while charge data enter only through a separate equilibrium problem, for which this theorem proves neither existence nor uniqueness [given]. The statement is pointwise on $S$ and stationary: no moving charge, normal sheet current, magnetic sheet source, corner or field without one-sided traces is considered, no product of distributions and no corner equation is formed, and no force, energy, radiation, self-force, material response or measurement is asserted [F1, F2]; the surface integrals exist but no regularity of a general ambient outer boundary is claimed [F5].

Surface charge and normal field on an ideal conductor

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

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