Force on a held charge above a grounded plane in the image model

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.