Electrostatic boundaries and conductors — Examples

Assume the Countable Choice convention of The image-method certificate for a point charge above a grounded plane and The localized Newtonian potential and the ideal point-charge Coulomb field (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). In the ideal grounded-plane configuration of The image-method certificate for a point charge above a grounded plane: vacuum half-space $H=\{x_3>0\}$, point charge parameter $q\in \mathbb{R}$ at $p=a{e}_3$ ($a>0$), field $E_I$ given by the image potential, and $E_I$ has the Coulomb singularity $(q\mathrm{/}(4\pi {\epsilon }_0))(x-p)\mathrm{/}\mathrm{\mid }x-p{\mathrm{\mid }}^3$ of the source charge as $x\to p$ (The localized Newtonian potential and the ideal point-charge Coulomb field(d), The electric and magnetic fields as fixed-time spatial vector fields). Split the field near $p$ as

$E_I(x)=E_q(x)+E_{\mathrm{e}\mathrm{x}\mathrm{t}}(x),E_q(x):=\frac{q}{4\pi {\epsilon }_0}\frac{x-p}{\mathrm{\mid }x-p{\mathrm{\mid }}^3}(x\mathrm{\ne }p),$

the self-field $E_q$ of the source charge and the external (mirror) field $E_{\mathrm{e}\mathrm{x}\mathrm{t}}:=E_I-E_q$, which is $C^{\mathrm{\infty }}$ on a neighbourhood of $p$ (it is minus the gradient of the smooth mirror term $-q\mathrm{/}(4\pi {\epsilon }_0)\mathrm{\mid }x+a{e}_3{\mathrm{\mid }}^{-1}$). Then:

(a) [External field at the charge] the external field at the location of the charge is

$E_{\mathrm{e}\mathrm{x}\mathrm{t}}(p)=-\frac{q}{16\pi {\epsilon }_0{a}^2}{e}_3\mathrm{.}$

It is the negative gradient of the mirror term evaluated at $p$ (Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$, The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$, Continuity and derivatives of positive-base real powers, For a natural $n\ge 1$ the function $x\mapsto x^n$ is differentiable everywhere with derivative $\iota (n)x^{n-1}$; for $n=0$ it is the constant $1$, with derivative $0$; for a natural $n\ge 1$ the function $x\mapsto x^{-n}$ is differentiable at every $x\mathrm{\ne }0$ with derivative $-\iota (n)x^{-n-1}$; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$, The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$).

(b) [Force on the held charge] under the adopted external Lorentz coupling The Lorentz coupling: external force on a test charge and the continuum force and power densities(i) at zero velocity, a test particle of charge $q$ held at $p$ experiences the electric force

$F=q{E}_{\mathrm{e}\mathrm{x}\mathrm{t}}(p)=-\frac{q^2}{16\pi {\epsilon }_0{a}^2}{e}_3,$

strictly attractive toward the plane when $q\mathrm{\ne }0$ and zero when $q=0$, of magnitude $q^2\mathrm{/}(16\pi {\epsilon }_0{a}^2)$; the particle's own field is excluded from the coupling by the postulate, so the result is a two-body-style external force of the mirror field and is not a self-force.

(c) [Reading and scope] for either nonzero sign of $q$ the force is attractive (the magnitude depends on $q^2$), while $q=0$ gives zero force; the induced total plane charge is $-q$ and its distribution is concentrated under the charge; the example is the exact ideal-model result for a held point charge, an ideal grounded plane and the adopted static external force law. It asserts no measurement, no dynamics of the charge (no motion or radiation), no finite-plane or finite-conductor correction, no self-force and no claim that the pair of idealisations is realised by an apparatus; the source is the idealised stationary point charge of The image-method certificate for a point charge above a grounded plane(e), not a declared compact-source model. Here the displayed force is the electric contribution at rest. Holding the charge at $p$ requires a balancing support force $F_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}}=-F$; the held trajectory is not asserted to solve the equation with the electric Lorentz force as its sole force.

Assume the Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) as carried by the integral and distribution interfaces (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Declare the ideal parallel-plate model of two infinite parallel charged planes in vacuo, read as equipotential ideal conductor surfaces of the same equilibrium structure as The ideal electrostatic conductor model (the pure electrostatic regime The static and pure electrostatic regimes and their fixed-time reading with fields The electric and magnetic fields as fixed-time spatial vector fields): for a separation parameter $d>0$, half-separation $h=d\mathrm{/}2$, a surface-density parameter $\sigma >0$ (C m${}^{-2}$) and the framework constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values, the model consists of the field $E(x):=\{\begin{array}{ll}-{\displaystyle \frac{\sigma }{{\epsilon }_0}}{e}_3,& \mathrm{\mid }{x}_3\mathrm{\mid }<h,\\ 0,& \mathrm{\mid }{x}_3\mathrm{\mid }>h,\end{array}x=(x_1,x_2,x_3),$ where $e_3:=(0,0,1)$, the coordinates and $d,h$ carry metres, and arbitrary zero values are assigned to $E$ on the two planes for its locally integrable representative; together with the plane densities $+\sigma$ on ${\mathrm{\Pi }}_{+}:=\{x_3=h\}$ and $-\sigma$ on ${\mathrm{\Pi }}_{-}:=\{x_3=-h\}$, and with the piecewise potential $\phi (x):=+{\displaystyle \frac{\sigma }{{\epsilon }_0}}{x}_3$ for $\mathrm{\mid }{x}_3\mathrm{\mid }\le h$, $\phi :=+{\displaystyle \frac{\sigma h}{{\epsilon }_0}}$ for $x_3\ge h$ and $\phi :=-{\displaystyle \frac{\sigma h}{{\epsilon }_0}}$ for $x_3\le -h$. Then: (a) [Direct distributional verification] $E$ and $\phi$ are locally integrable and $C^{\mathrm{\infty }}$ off ${\mathrm{\Pi }}_{+}\cup {\mathrm{\Pi }}_{-}$ with $E=-\mathrm{\nabla }\phi$ there, $\phi$ is continuous on ${\mathbb{R}}^3$ and constant on each plane with the values $\phi {\mathrm{\mid }}_{{\mathrm{\Pi }}_{+}}=+\sigma h\mathrm{/}{\epsilon }_0$, $\phi {\mathrm{\mid }}_{{\mathrm{\Pi }}_{-}}=-\sigma h\mathrm{/}{\epsilon }_0$; and in ${\mathcal{D}}^{\mathrm{\prime }}({\mathbb{R}}^3)$ $\mathrm{div}E=\frac{\sigma }{{\epsilon }_0}({\delta }_{{\mathrm{\Pi }}_{+}}-{\delta }_{{\mathrm{\Pi }}_{-}}),\mathrm{curl}E=0,$ where ${\delta }_{{\mathrm{\Pi }}_{\pm }}(\psi ):={\int }_{{\mathbb{R}}^2}\psi (x_1,x_2,\pm h)d{x}_{1}d{x}_2$ for compactly supported smooth $\psi$ are the plane distributions defined by the iterated Lebesgue integrals. Equivalently, the model realises the static electrostatic equations with the declared pair of plane densities and no other source, and the potential differences read off $\phi$ are $\phi (h)-\phi (-h)=2\sigma h\mathrm{/}{\epsilon }_0=\sigma d\mathrm{/}{\epsilon }_0$. (b) [Uniform field and voltage] the field between the planes has constant magnitude $\mathrm{\mid }E\mathrm{\mid }=\sigma \mathrm{/}{\epsilon }_0$ and is normal to the planes, and the potential difference between the plates is $V:=\phi {\mathrm{\mid }}_{{\mathrm{\Pi }}_{+}}-\phi {\mathrm{\mid }}_{{\mathrm{\Pi }}_{-}}=\sigma d\mathrm{/}{\epsilon }_0$. (c) [Charge, capacitance and energy of the ideal model] for any Lebesgue-measurable region $P\subseteq {\mathbb{R}}^2$ with finite positive area $A={\lambda }_2(P)>0$ the charge on the portion of area $A$ of the positive plate is $Q:=\sigma A$, so $Q=({\epsilon }_{0}A\mathrm{/}d)V$ with the geometric ratio $C:=\frac{Q}{V}=\frac{{\epsilon }_{0}A}{d}>0,$ independent of $\sigma$ and $V$; and the field energy contained in the slab of area $A$ between the plates is $U=\frac{{\epsilon }_0}{2}{\int }_{\text{slab of area }A}\mathrm{\mid }E{\mathrm{\mid }}^{2}d{\lambda }_3=\frac{{\sigma }^{2}Ad}{2{\epsilon }_0}=\frac{1}{2}QV=\frac{Q^2}{2C}=\frac{1}{2}C{V}^2\mathrm{.}$ (d) [Relation to the bounded-conductor theorem and scope] the linearity $Q=CV$, the positivity $C>0$ and the energy identities $U=\frac{1}{2}QV=\frac{Q^2}{2C}=\frac{1}{2}C{V}^2$ have the same scalar form as the conclusions of Capacitance and energy for specified linear conductor geometry(d) for its bounded conductor geometry with only conductor $1$ driven, every other conductor and the outer boundary grounded, and zero volume source, taking $C=C_{11}$, $Q=Q_1$ and $V=V_1$ in that theorem; this example verifies them directly for the ideal model because the infinite planes and the unbounded declared field are outside that theorem's hypotheses (compact conductors in a bounded vacuum region) and outside the declared stationary-sheet source type of Distributional Maxwell equations with point and stationary-sheet sources, exactly as the ideal line source lies outside the compact-source types of the electrostatic page. No approximation theorem from finite plates to the infinite model is asserted; for a finite real plate pair the uniform-field model neglects the fringing field near the edges, which is not computed here, and the idealisation also omits the finite-thickness, material, transient and measurement content of any real capacitor.

Assume the Axiom of Choice for the cited Sobolev trace/weak-solvability interfaces and Countable Choice for the surface-measure interfaces (The Axiom of Choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $B:=B_1(0)=\{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }<1\}$ be the open unit ball, a bounded $C^{\mathrm{\infty }}$ (in particular $C^1$) domain with $\mathrm{\partial }B=S^2$ and outward normal $\nu (x)=x$ (Euclidean balls, annuli and spheres as bounded smooth domains and hypersurfaces, Open ball, closed ball and sphere in a metric space, The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$), and consider the pure electrostatic Neumann problem on $B$ (regime The static and pure electrostatic regimes and their fixed-time reading, fields The electric and magnetic fields as fixed-time spatial vector fields, constants Framework-local SI units, the vacuum constants and the exact/measured status of their values) with zero volume source and unit outward normal derivative: $-\mathrm{\Delta }\phi =0\text{ in }B,{\mathrm{\partial }}_{\nu }\phi =1\text{ on }\mathrm{\partial }B\mathrm{.}$ All displayed coordinates and potentials are numerical SI readings: the ball has radius $1\mathrm{m}$, $\phi$ is read in volts, and ${\mathrm{\partial }}_{\nu }\phi =1$ denotes $1\mathrm{V}{\mathrm{m}}^{-1}$. Thus the corresponding outward electric field is $E\cdot \nu =-1\mathrm{V}{\mathrm{m}}^{-1}$, and $4\pi$ in the derivative-flux integral carries $\mathrm{V}\mathrm{m}$. The companion datum $f=1$ denotes $1\mathrm{V}{\mathrm{m}}^{-2}$ in $-\mathrm{\Delta }\phi =f$; it is not a claim of unit physical charge density.

Then: (a) [No classical solution] there is no real $\phi \in C^2(\bar{B})$ satisfying these equations: if such a $\phi$ existed, Necessary compatibility for the classical Neumann Poisson problem with $f=-\mathrm{\Delta }\phi =0$ and $g={\mathrm{\partial }}_{\nu }\phi \equiv 1$ would give $0={\int }_{B}fd{\lambda }_3=-{\int }_{\mathrm{\partial }B}gdS=-\mathrm{\mid }{S}^2\mathrm{\mid }=-4\pi ,$ because the surface area of the unit sphere is $\mathrm{\mid }{S}^2\mathrm{\mid }=4\pi$ (Agreement with the existing polar sphere measure, The closed form for the volume of the unit $n$-ball); the contradiction is the failure of the necessary flux compatibility of Uniqueness and flux compatibility for the electrostatic boundary value problem(b). (b) [No weak solution] more generally there is no $u\in H^1(B)$ with ${\int }_B\mathrm{\nabla }v\cdot \mathrm{\nabla }ud{\lambda }_3=⟨g,Tv⟩$ for all $v\in H^1(B)$ with $g\equiv 1$ read as an element of the dual trace space (The fractional Sobolev space on a compact $C^1$ boundary): testing $v\equiv 1$ would force $\mathrm{\ell }(1)={\int }_{\mathrm{\partial }B}1dS=4\pi =0$, and the weak Neumann existence lemma states existence exactly when $\mathrm{\ell }(1)=0$ (Weak Neumann and transmission existence with interface charges); equivalently the unit ball's boundary flux budget fails already in the weak formulation. (c) [Physics reading and scope] in the pure electrostatic regime the datum ${\mathrm{\partial }}_{\nu }\phi \equiv 1$ on the whole boundary of the unit ball is inadmissible: no electrostatic potential with zero volume source satisfies it, and the obstruction is the strictly positive total boundary flux $4\pi$, not a smoothness or compactness artefact. Every solvable zero-source Neumann problem must satisfy ${\int }_{\mathrm{\partial }B}gdS=0$ for an integrable derivative datum $g$; this necessary flux budget alone does not state the regularity hypotheses of an existence theorem (sufficiency at the smooth level is the separate content of Existence and regularity for the electrostatic boundary value problem(b) under its hypotheses). The item is a mathematical impossibility statement about the ideal boundary value problem: it reports no measurement, no apparatus and no material property, and the companion failure with unit source and zero boundary field is the same flux obstruction, obtained by evaluating that lemma with $f\equiv 1$ and $g\equiv 0$.