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The image-method certificate for a point charge above a grounded plane

Physical scope

The ideal electrostatic problem of a point charge held above an infinite grounded conducting plane, read in the framework's SI conventions: on the vacuum half-space the image potential is harmonic off the source, has the point-charge distributional source, vanishes on the plane, decays like |x|^-2, and is unique in the stated continuous-boundary and decay class; the plane's induced density is the idealization's bookkeeping normal-derivative reading. The item asserts no finite-plane edge effects, no conductor thickness, no dynamics, no self-force and no measurement, and the half-space conductor and noncompact sheet are explicitly outside the bounded-conductor model and the declared source types. The induced density is explicitly smooth on the whole plane, has absolutely convergent total charge $- q$ , and the full-space point-plus-plane divergence identity is verified with local tests as a separate mathematical idealization. Matching singularities allow any remainder of order $o (∣ x - p ∣^{- 1})$ .

Statement

Assume the Axiom of Countable Choice ( ${A C}_{ω}$ ) inherited from the Newtonian, polar and distribution interfaces (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let

$H : = {x \in R^{3} : x_{3} > 0}, Π : = {x_{3} = 0}, p : = a e_{3} (a > 0),$

Here $e_{3} : = (0, 0, 1)$ is the third coordinate unit vector, $a$ carries metres, and the coordinates are SI numerical readings in the named frame. Let $q \in R$ be a charge parameter (coulombs) and $ε_{0} > 0$ the framework constant (Framework-local SI units, the vacuum constants and the exact/measured status of their values). Define, with arbitrary values on the exceptional sets,

$φ_{I} (x) : = \frac{q}{4 π ε_{0}} (\frac{1}{∣ x - p ∣} - \frac{1}{∣ x + a e_{3} ∣}) (x \in H ∖ {p}), E_{I} : = - \nabla φ_{I} .$

The map $E_{I}$ is a spatial field of The electric and magnetic fields as fixed-time spatial vector fields in the fixed-time reading, with the units of Framework-local SI units, the vacuum constants and the exact/measured status of their values. Then:

(a) [Candidate field] $φ_{I} \in C^{\infty} (H ∖ {p})$ and $Δ φ_{I} = 0$ pointwise on $H ∖ {p}$ : the first term is the Coulomb potential of $p$ , whose $n = 3$ fundamental solution is harmonic off its pole (The Laplace fundamental solution is harmonic off its pole, The localized Newtonian potential and the ideal point-charge Coulomb field(d)), and the second term is $C^{\infty}$ and harmonic on the whole of $H$ because its pole $- a e_{3}$ lies outside $H$ ; $E_{I}$ is $C^{\infty}$ on $H ∖ {p}$ .

(b) [Distributional point source] $φ_{I} \in L_{l o c}^{1} (H)$ and

$- Δ T_{φ_{I}} = \frac{q}{ε_{0}} δ_{p} in D^{'} (H),$

where $T_{φ_{I}}$ is the regular distribution of $φ_{I}$ (Regular distribution from a locally integrable function, Locally integrable functions embed in distributions) and $δ_{p}$ the Dirac distribution at $p$ (Dirac delta and its derivatives); moreover the field satisfies $div E_{I} = (q / ε_{0}) δ_{p}$ and $curl E_{I} = 0$ in $D^{'} (H)$ , where the divergence and curl of $E_{I}$ are the distributional derivatives of its components (Distributional derivative). Thus $(E_{I}, 0)$ is a pure electrostatic model on $H$ with the fixed-time point source $q δ_{p}$ , the static reading of the stationary point charge of Distributional Maxwell equations with point and stationary-sheet sources(b) (The static and pure electrostatic regimes and their fixed-time reading(e)).

(c) [Boundary trace and decay] $φ_{I}$ extends continuously to $\overline{H} ∖ {p}$ with $φ_{I} = 0$ on $Π$ (the two denominators are equal there), and for $∣ x ∣ \geq 2 a$ the cancellation bound holds:

$∣ φ_{I} (x) ∣ \leq \frac{2 ∣ q ∣ a}{π ε_{0} ∣ x ∣^{2}} .$

The bound follows by applying the mean value theorem along the segment from $- a e_{3}$ to $a e_{3}$ to the function $y \mapsto ∣ x - y ∣^{- 1}$ , whose gradient is bounded by $(∣ x ∣ - a)^{- 2}$ there (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ , The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Continuity and derivatives of positive-base real powers, For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ). In particular $φ_{I} (x) \to 0$ as $∣ x ∣ \to \infty$ in $H$ .

(d) [Uniqueness in the declared class] let $ψ : H ∖ {p} \to R$ be $C^{2}$ and harmonic on $H ∖ {p}$ , continuous on $\overline{H} ∖ {p}$ , with $ψ = 0$ on $Π$ , $∣ x - p ∣ (ψ (x) - φ_{I} (x)) \to 0$ as $x \to p$ and $ψ (x) \to 0$ as $∣ x ∣ \to \infty$ in $H$ . Then $ψ = φ_{I}$ on $H ∖ {p}$ . Equivalently, the Dirichlet problem $Δ ψ = 0$ on $H ∖ {p}$ , $ψ = 0$ on $Π$ , with the leading singularity of the point charge at $p$ and decay at infinity, has exactly one solution in the stated class.

(e) [Physics reading: ideal grounded plane in the fixed-time reading of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields] the ideal grounded conducting half-space ${x_{3} < 0}$ with the field $E_{I}$ on $H$ , $E \equiv 0$ below $Π$ and the induced density $σ : = ε_{0} n \cdot E_{+} ∣_{Π} = - ε_{0} \partial_{3} φ_{I} ∣_{Π}$ (using the normal $n = e_{3}$ from the conductor into the vacuum; $σ \in C^{\infty} (Π)$ ) is a model of the same equilibrium structure as The ideal electrostatic conductor model: it is a pure electrostatic field on $H$ with the point source at $p$ and the equipotential boundary condition $φ_{I} = 0$ on $Π$ ; the pair $(E_{I}, 0)$ satisfies the static electrostatic equations with source $q δ_{p}$ on $H$ , and the induced density is explicitly $σ (x_{1}, x_{2}) = - q a / [2 π (x_{1}^{2} + x_{2}^{2} + a^{2})^{3 / 2}]$ , is absolutely integrable over the plane, and has total charge $\int_{Π} σ d S = - q$ . However, the half-space conductor is not a compact conductor body: the plane is not a compact embedded hypersurface, $Π$ and ${x_{3} < 0}$ are not among the geometric objects of The ideal electrostatic conductor model, the density $σ$ is not a declared stationary-sheet source of Distributional Maxwell equations with point and stationary-sheet sources(c), and the noncompact plane pairing is not an adopted compact-sheet source of that definition. As a separately specified mathematical idealization, extending $E_{I}$ by zero below the plane gives $div \tilde{E} = (q δ_{p} + σ δ_{Π}) / ε_{0}$ and $curl \tilde{E} = 0$ on $R^{3}$ , where $(σ δ_{Π}) (η) = \int_{R^{2}} σ (x^{'}) η (x^{'}, 0) d x^{'}$ . This verified identity does not extend the framework postulate to arbitrary noncompact sheets. The clause is the explicitly declared idealization whose exact mathematical content is (a)-(d).

(f) [Scope] the statement concerns the ideal point charge above an ideal grounded plane in the declared decay class; it excludes finite-plane edge effects, conductor thickness, transients, moving charges, radiation, self-force and any measurement; no uniqueness is asserted outside the class of (d), and the point-source singularity is never evaluated.

Facts & Assumptions

Given: The half-space $H$ , the plane $Π$ , the point $p = a e_{3}$ , $a > 0$ , the charge parameter $q$ , the constant $ε_{0} > 0$ , and the candidate $φ_{I}$ and field $E_{I}$ of the statement.

[F1]

The $n = 3$ fundamental solution is $Φ (z) = 1 / (4 π ∣ z ∣)$ , smooth and harmonic on $R^{3} ∖ {0}$ ; the Coulomb point-charge branch of The localized Newtonian potential and the ideal point-charge Coulomb field(d) identifies the point-charge potential $φ_{q}^{p} = (q / ε_{0}) Φ (\cdot - p)$ as locally integrable with distributional Laplacian $- Δ T_{φ_{q}^{p}} = (q / ε_{0}) δ_{p}$ on $R^{3}$ and field $E_{q}^{p} = - \nabla φ_{q}^{p}$ satisfying $div E_{q}^{p} = (q / ε_{0}) δ_{p}$ , $curl E_{q}^{p} = 0$ in $D^{'} (R^{3})$ (The Laplace fundamental solution is harmonic off its pole, The localized Newtonian potential and the ideal point-charge Coulomb field, The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, Dirac delta and its derivatives).

[F2]

A $C^{1}$ function on an open set represents its regular distribution and its distributional first derivatives are the regular distributions of its classical derivatives; distributional differentiation is linear and the Laplacian of a classical harmonic function therefore represents the zero distribution (Regular distribution from a locally integrable function, Locally integrable functions embed in distributions, Distributional derivative, Distribution, Local finite order characterization of distributions, If $u, v$ are differentiable on $[a, b]$ with $u^{'}, v^{'}$ integrable, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ , Distributional differentiation is continuous and commutes, Test function space d of an open set).

[F3]

Local integrability in dimension three: the radial integral of $∣ z ∣^{- 1}$ over a ball is finite by polar coordinates, finite sums and scalar multiples of locally integrable measurable functions are locally integrable, as are their products with locally bounded measurable functions, and continuous functions are Borel measurable (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma, The polar surface set function on the unit sphere, Integral over a measurable subset, Monotonicity and nonnegative homogeneity of the nonnegative integral, Continuous functions on Euclidean spaces are Borel measurable, Agreement with the existing polar sphere measure, The closed form for the volume of the unit $n$ -ball).

[F4]

Mean value theorem and derivative calculus: for $g (y) : = ∣ x - y ∣^{- 1}$ with $x$ fixed, $g$ is $C^{\infty}$ off $x$ and $\nabla_{y} g (y) = (x - y) / ∣ x - y ∣^{3}$ , so $∣ \nabla_{y} g (y) ∣ = ∣ x - y ∣^{- 2}$ ; differentiation with respect to $x$ instead gives $\nabla_{x} ∣ x - y ∣^{- 1} = - (x - y) / ∣ x - y ∣^{3}$ ; the mean value theorem bounds $∣ g (p) - g (- p) ∣$ by $∣ p - (- p) ∣ \sup ∣ \nabla g ∣$ on the segment (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ , The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Continuity and derivatives of positive-base real powers, For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The finite and reverse triangle inequalities for a norm; and for $n \geq 1$ every norm $N$ on $R^{n}$ satisfies $N (x) \leq C ∥ x ∥_{1}$ and is Lipschitz, hence continuous, for $d_{2}$ , $C^{k}$ Euclidean maps and diffeomorphisms).

[F5]

The half-space uniqueness lemma: a function continuous on $\overline{H} ∖ {p}$ , $C^{2}$ and harmonic on $H ∖ {p}$ , zero on $Π$ , sub-Coulomb at $p$ and vanishing at infinity in $H$ is identically zero (Uniqueness for harmonic functions on a punctured half-space with zero trace, decaying at the pole and at infinity).

[F6]

The polar/Fubini calculus on the plane: integrals over $R^{2}$ may be computed in polar coordinates, the product and Euclidean measures agree on Borel sets, the plane is Lebesgue null in $R^{3}$ , and the one-dimensional fundamental theorem computes the radial primitive (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma, Fubini's theorem for L^1 functions on a sigma-finite product, The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n}, On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n}, A box with a degenerate side is Lebesgue null, and so is every coordinate hyperplane in $R^{n}$ , The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , Riemann–Lebesgue comparison for distribution test integrands, Monotone convergence for the integral).

[F7]

Framework readings: the fixed-time reading of the pure electrostatic regime identifies the static fields and the stationary point charge of Distributional Maxwell equations with point and stationary-sheet sources(b) with $q δ_{p}$ ; the units are those of Framework-local SI units, the vacuum constants and the exact/measured status of their values; and the adopted distributional equations are those of Distributional vacuum Maxwell equations for the declared point and stationary-sheet sources (The static and pure electrostatic regimes and their fixed-time reading(e), The electric and magnetic fields as fixed-time spatial vector fields, Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, Distributional Maxwell equations with point and stationary-sheet sources).

[F8]

For $0 < r < R$ there is a smooth function $χ : R^{3} \to [0, 1]$ equal to one on ${\overline{B}}_{r} (p)$ and supported strictly inside $B_{R} (p)$ (Compactly supported scaled Euclidean bumps).

Proof

technique · direct

1.1F1F2

Clause (a). Write $φ_{I} = (q / ε_{0}) Φ (\cdot - p) - (q / ε_{0}) Φ (\cdot + a e_{3})$ [F1]. The first term is smooth and harmonic off its pole $p$ ; the second has its pole at $- a e_{3}$ , which lies outside the closed half-space $\overline{H}$ , so it is $C^{\infty}$ on $H$ and harmonic there by [F1] and linearity of the Laplacian [F2]; hence $φ_{I} \in C^{\infty} (H ∖ {p})$ , $Δ φ_{I} = 0$ there, and $E_{I} = - \nabla φ_{I}$ is $C^{\infty}$ on $H ∖ {p}$ .

1.2F1F2F3F7

Local integrability and distributional identities. $φ_{I}$ is locally integrable on $H$ : $∣ x - p ∣^{- 1}$ is locally integrable in dimension three and $∣ x + a e_{3} ∣^{- 1}$ is continuous on $H$ , and sums and scalar multiples of locally integrable measurable functions are locally integrable [F3]. Its regular distribution is $T_{φ_{I}} = (q / ε_{0}) T_{Φ (\cdot - p)} - (q / ε_{0}) T_{Φ (\cdot + a e_{3})}$ [F2, F3]. On $H$ the second term is $C^{\infty}$ with classically vanishing Laplacian, so its distributional Laplacian is the regular distribution of the zero function [F2]; the first term has $- Δ T = (q / ε_{0}) δ_{p}$ on $R^{3}$ , hence on the open subset $H$ [F1]. Summing gives $- Δ T_{φ_{I}} = (q / ε_{0}) δ_{p}$ in $D^{'} (H)$ . For the field, $E_{I} = E_{q}^{p} - E_{q}^{- p}$ where $E_{q}^{- p} : = - \nabla ((q / ε_{0}) Φ (\cdot + a e_{3}))$ is smooth on $H$ with $div E_{q}^{- p} = 0$ , $curl E_{q}^{- p} = 0$ there [F2]; componentwise $T_{E_{I, i}} = - \partial_{i} T_{φ_{I}}$ on $H$ by the same representation and [F2], so the distributional divergence and curl of $E_{I}$ are those of the sum, giving $div E_{I} = (q / ε_{0}) δ_{p}$ and $curl E_{I} = 0$ in $D^{'} (H)$ [F1, F2]. This is (b); taking it together with [F7] reads $(E_{I}, 0)$ as the pure electrostatic model on $H$ with the fixed-time point source $q δ_{p}$ , as asserted.

1.3F4

Clause (c). On $Π$ one has $∣ x - p ∣ = ∣ x + a e_{3} ∣$ for every $x$ with $x_{3} = 0$ , so the bracket in $φ_{I}$ vanishes and $φ_{I}$ extends continuously (indeed $C^{\infty}$ away from $x = p \notin Π$ ) with value $0$ on $Π$ . For the decay, fix $x$ with $∣ x ∣ \geq 2 a$ and apply the mean value theorem to $g (y) : = ∣ x - y ∣^{- 1}$ on the segment from $- a e_{3}$ to $a e_{3}$ : since every point $y$ of that segment satisfies $∣ x - y ∣ \geq ∣ x ∣ - a \geq a > 0$ , one has $∣ \nabla g (y) ∣ = ∣ x - y ∣^{- 2} \leq (∣ x ∣ - a)^{- 2}$ and therefore $∣ g (p) - g (- p) ∣ \leq 2 a (∣ x ∣ - a)^{- 2}$ [F4]. Multiplying by $∣ q ∣ / (4 π ε_{0})$ gives $∣ φ_{I} (x) ∣ \leq ∣ q ∣ a / (2 π ε_{0} (∣ x ∣ - a)^{2}) \leq 2 ∣ q ∣ a / (π ε_{0} ∣ x ∣^{2})$ because $∣ x ∣ \leq 2 (∣ x ∣ - a)$ for $∣ x ∣ \geq 2 a$ . This is (c).

2.1F5step 1.1step 1.3

Clause (d). Let $ψ$ be as in the statement and put $w : = ψ - φ_{I}$ on $H ∖ {p}$ . Then $w$ is $C^{2}$ and harmonic on $H ∖ {p}$ and continuous on $\overline{H} ∖ {p}$ by (a) [step 1.1]; $w = 0$ on $Π$ because both functions vanish there [step 1.3]; $∣ x - p ∣ w (x) \to 0$ as $x \to p$ by the assumed leading-singularity condition; and $w (x) \to 0$ as $∣ x ∣ \to \infty$ in $H$ because both $ψ$ and $φ_{I}$ do. By [F5] applied to $w$ , $w = 0$ on $H ∖ {p}$ , so $ψ = φ_{I}$ . This is (d).

2.2F4F6step 1.2

Clause (e), first part. On the plane, $n = e_{3}$ and $σ = - ε_{0} \partial_{3} φ_{I} ∣_{Π}$ ; differentiating the two Coulomb terms, whose gradients are $- (x \mp p) / ∣ x \mp p ∣^{3}$ [F4], at $x_{3} = 0$ gives $\partial_{3} ∣ x - p ∣^{- 1} = a / (x_{1}^{2} + x_{2}^{2} + a^{2})^{3 / 2}$ and $\partial_{3} ∣ x + a e_{3} ∣^{- 1} = - a / (x_{1}^{2} + x_{2}^{2} + a^{2})^{3 / 2}$ , so $\partial_{3} φ_{I} = q a / [2 π ε_{0} (x_{1}^{2} + x_{2}^{2} + a^{2})^{3 / 2}]$ and hence $σ (x_{1}, x_{2}) = - q a / [2 π (x_{1}^{2} + x_{2}^{2} + a^{2})^{3 / 2}]$ , a smooth bounded function; the boundary condition $φ_{I} = 0$ on $Π$ is the equipotential (zero) trace of the ideal grounded plane, and $(E_{I}, 0)$ satisfies the static equations with source $q δ_{p}$ on $H$ by (b) [step 1.2]. The total induced charge is $\int_{R^{2}} σ d λ_{2} = - \frac{q a}{2 π} \int_{0}^{\infty} \frac{2 π r d r}{(r^{2} + a^{2})^{3 / 2}} = - q a [- \frac{1}{\sqrt{r^{2} + a^{2}}}]_{r = 0}^{\infty} = - q,$ with the polar-coordinate computation and the improper radial primitive justified by [F6]; the identical polar computation with $∣ q ∣$ in place of $- q$ gives $\int_{R^{2}} ∣ σ ∣ d λ_{2} = ∣ q ∣ < \infty$ , including $q = 0$ .

3.1F1F2F6F8step 1.2step 1.3step 2.2

Clause (e), extension identity. Extend $E_{I}$ by zero to ${x_{3} \leq 0}$ , obtaining $\tilde{E} \in L_{l o c}^{1} (R^{3})$ by the point-field integrability in [F1] and smoothness away from $p$ . The plane pairing is a distribution: on tests supported in a compact box its absolute value is bounded by $\sup ∣ σ ∣$ times the finite area of the box's planar section times $\sup ∣ η ∣$ . Choose $0 < r < R < a$ and $χ$ from [F8], so $χ = 1$ near $p$ and its support lies in $H$ . Split an arbitrary scalar test as $η_{1} = χ η$ , $η_{2} = (1 - χ) η$ . For $η_{1}$ the half-space identities of step 1.2 give the divergence pairing $(q / ε_{0}) η (p)$ and zero curl pairing; its plane trace is zero. For $η_{2}$ , all field/test products are smooth away from the pole and absolutely integrable on a compact box. Fubini and one-dimensional integration by parts [F2, F6] therefore give, for every $i, j$ , $- \int_{H} E_{I, j} \partial_{i} η_{2} = \int_{H} (\partial_{i} E_{I, j}) η_{2} + 1_{{i = 3}} \int_{R^{2}} E_{I, j} (x^{'}, 0) η_{2} (x^{'}, 0) d x^{'}$ . Horizontal boundary terms vanish by compact support; the vertical integral over $(0, \infty)$ has precisely the displayed lower-end term. Summing $i = j$ gives only $\int_{Π} E_{I, 3} η_{2} = ε_{0}^{- 1} \int_{Π} σ η$ , since $div E_{I} = 0$ away from $p$ . Taking the antisymmetric derivative combinations gives the sheet term $e_{3} \times E_{+} = 0$ , because the smooth zero plane trace of $φ_{I}$ has zero horizontal derivatives; the classical curl also vanishes away from $p$ by step 1.2. Adding the two test pieces gives $div \tilde{E} = (q δ_{p} + σ δ_{Π}) / ε_{0}$ and $curl \tilde{E} = 0$ in $D^{'} (R^{3})$ . No limit at infinity or unproved divergence theorem on a cornered truncation is used.

4.1F7step 2.1step 3.1∎

Scope. The computations above concern exactly the ideal point charge above the ideal grounded plane in the declared class: the plane is not a compact embedded hypersurface and $Π$ , ${x_{3} < 0}$ and the plane pairing are not among the declared conductor or compact stationary-sheet objects of The ideal electrostatic conductor model and Distributional Maxwell equations with point and stationary-sheet sources(c) [F7]; the extension identity of [step 3.1] is a verified statement about the idealization and does not extend the framework postulate to arbitrary noncompact sheets. No finite-plane edge effect, conductor thickness, transient, moving charge, radiation, self-force or measurement is asserted, the singularity at $p$ is never evaluated, and uniqueness is claimed only in the class of (d) [step 2.1]. This is (f).

The image-method certificate for a point charge above a grounded plane

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources