Integral Maxwell and constraints — Examples

Let $q\mathrm{\ne }0$ be a real charge parameter (coulombs) and let $U:={\mathbb{R}}^3\times \mathbb{R}$ with the laboratory conventions of Distributional Maxwell equations with point and stationary-sheet sources. Define a distribution $\rho$ on $U$ by

$⟨\rho ,\psi ⟩:=q{\int }_0^{\mathrm{\infty }}\psi (0,t)dt,\psi \in C_c^{\mathrm{\infty }}(U)$

— an explicit switched-history distribution for $z(t)=0$ on $(0,\mathrm{\infty })$, with no compensating current or source contribution at the appearance event; it does not satisfy the proper-graph hypothesis of the zero-endpoint continuity branch of Distributional Maxwell equations with point and stationary-sheet sources(b) — and let $J:=0$ be the zero distribution. Then:

(a) ${\mathrm{\partial }}_t\rho =q{\delta }_{(0,0)}$ and $\mathrm{div}J=0$ (Dirac delta and its derivatives), hence ${\mathrm{\partial }}_t\rho +\mathrm{div}J=q{\delta }_{(0,0)}\mathrm{\ne }0$ in ${\mathcal{D}}^{\mathrm{\prime }}(U)$: for every test function $\psi$, $⟨{\mathrm{\partial }}_t\rho ,\psi ⟩=-q{\int }_0^{\mathrm{\infty }}{\mathrm{\partial }}_t\psi (0,t)dt=q\psi (0,0)$, because $t\mapsto \psi (0,t)$ is smooth with compact support. In particular the continuity identity fails at the appearance event $(0,0)$.

(b) Consequently there are no locally integrable fields $E,B$ on $U$ satisfying the four weak residual identities of Distributional Maxwell equations with point and stationary-sheet sources(e) with the source pair $(\rho ,0)$, the $\rho$-pairing being the explicit pairing displayed above and $J=0$: such fields would satisfy the distributional continuity identity of Charge continuity as a compatibility condition of the vacuum Maxwell equations(b), which applies to every distributional source pair for which the four identities hold, and (a) contradicts it. The source pair $(\rho ,0)$ is therefore incompatible with the formal four residual identities themselves; it is not one of the declared point sources of Distributional Maxwell equations with point and stationary-sheet sources(b), because the properness hypothesis fails here as recorded, and no fields turn it into an admitted singular-source model under Distributional vacuum Maxwell equations for the declared point and stationary-sheet sources.

(c) The failure is exactly the omitted endpoint term of the point-source construction: by the endpoint clause of Point-support spacetime densities: distributions, continuity equation and endpoint fluxes, the same trajectory on a bounded interval $I_0=(a,b)$ has ${\mathrm{\partial }}_t\rho +\mathrm{div}J=q{\delta }_{\gamma (a)}-q{\delta }_{\gamma (b)}$ with $\gamma (t)=(z(t),t)$. A switched point charge with $J=0$ and no endpoint or current term is thus inadmissible; a switched charge must be accompanied by an explicit current or endpoint source term, which the weak equations of Distributional Maxwell equations with point and stationary-sheet sources then require to satisfy continuity.

This is a mathematical incompatibility statement about a declared source model, not a report of a measurement: no apparatus, sampling model, uncertainty or empirical claim is involved.

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$, the convention of the test-geometry and divergence-theorem items cited below (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $U\cong I\times \mathrm{\Omega }$ be the laboratory region of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields with the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values, and let $E,B$ be jointly of class $C^2$ fields of The electric and magnetic fields as fixed-time spatial vector fields and $\rho ,J$ of class $C^1$ source fields of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current satisfying the smooth branch of The microscopic vacuum Maxwell equations: smooth and weak branches on a neighbourhood of the data below.

Suppose given two patch test geometries $(D,r_1)$, $(D,r_2)$ of Fixed volume and patch test geometries, outward normals and induced orientations over the same finite elementary Green region $D$ whose induced boundary chains coincide as oriented paths (so both patches span the same closed loop $\gamma$), and suppose there is a bounded open region $V\subseteq \mathrm{\Omega }$ with a specified finite piecewise $C^1$ presentation whose boundary faces are exactly the two images $r_1[D]$, $r_2[D]$ with outward normals $-n_1$ and $+n_2$ respectively, where $n_i$ is the normal induced by $r_i$ (that is, $V$ is the region enclosed between the two patches; this is a hypothesis on the pair, part of its data). Then, for every $t\in I$:

(a) [Total current flux is surface independent]

${\int }_{S_1}(J+{\epsilon }_0{\mathrm{\partial }}_{t}E)(\cdot ,t)\cdot n_{1}dS={\int }_{S_2}(J+{\epsilon }_0{\mathrm{\partial }}_{t}E)(\cdot ,t)\cdot n_{2}dS=\frac{1}{{\mu }_0}{\oint }_{\gamma }B(\cdot ,t)\cdot d\mathbf{r},$

where $S_i=r_i[D]$ and the flux is that of Unit normal fields, orientations, and flux through a regular surface patch.

(b) [Conduction-only flux need not be] For every $t\in I$ the difference of the conduction fluxes equals the negative of the difference of the displacement fluxes:

${\int }_{S_1}J(\cdot ,t)\cdot n_{1}dS-{\int }_{S_2}J(\cdot ,t)\cdot n_{2}dS=-{\epsilon }_0({\int }_{S_1}{\mathrm{\partial }}_{t}E(\cdot ,t)\cdot n_{1}dS-{\int }_{S_2}{\mathrm{\partial }}_{t}E(\cdot ,t)\cdot n_{2}dS)\mathrm{.}$

Hence the conduction-only fluxes are equal exactly when the displacement fluxes are equal, and neither conclusion is forced by the Maxwell equations alone.

(c) [Explicit realisation] In this clause all displayed variables and constants are numerical values in the SI units of Framework-local SI units, the vacuum constants and the exact/measured status of their values: spatial coordinates in metres, $t$ in seconds, $E$ in V m${}^{-1}$, $B$ in T, $\rho$ in C m${}^{-3}$ and $J$ in A m${}^{-2}$. Thus the profile argument $z$ means physical height divided by $1\mathrm{m}$, the implicit current-density scale is $1\mathrm{A}{\mathrm{m}}^{-2}$, and the geometric length scale is $1\mathrm{m}$. The following declared smooth data satisfies all the hypotheses above and has different conduction fluxes through the two surfaces. On $\mathrm{\Omega }={\mathbb{R}}^3$, $I=\mathbb{R}$, fix $\chi \in C^{\mathrm{\infty }}(\mathbb{R})$ with $0\le \chi \le 1$, $\chi (z)=1$ for $z\le -1$ and $\chi (z)=0$ for $z\ge 0$, and set

$E(x,t):=\frac{t}{{\epsilon }_0}\chi (z)e_z,B:=0,\rho (x,t):=t{\chi }^{\mathrm{\prime }}(z),J(x,t):=-\chi (z)e_z\mathrm{.}$

Then $\mathrm{div}E=\rho \mathrm{/}{\epsilon }_0$, $\mathrm{div}B=0$, $\mathrm{curl}E=0=-{\mathrm{\partial }}_{t}B$, $\mathrm{curl}B=0={\mu }_{0}J+c^{-2}{\mathrm{\partial }}_{t}E$ (the last identity because $c^{-2}{\epsilon }_0^{-1}={\mu }_0$), and ${\mathrm{\partial }}_t\rho +\mathrm{div}J=0$. Take $D=[-1,1{]}^2$ and its interior $D^{\circ }=(-1,1{)}^2$. Let $\sigma$ be the standard smooth step of The standard smooth step function and define $h(s):=\sigma (2(1-s^2))$, $\eta (u,v):=h(u)h(v)$. Then $\eta \in C_c^{\mathrm{\infty }}({\mathbb{R}}^2)$, $0\le \eta \le 1$, $\eta =1$ in a neighbourhood of the origin, $\eta >0$ at every point of $D^{\circ }$, and $\eta =0$ on $\mathrm{\partial }D$ and outside $D$. Take the graphs

$r_1(u,v):=(u,v,-2\eta (u,v)),r_2(u,v):=(u,v,0)\mathrm{.}$

Both induced boundary chains are the square $\{(u,v,0):(u,v)\in \mathrm{\partial }D\}$, the region $V=\{(u,v,w):(u,v)\in D^{\circ },-2\eta (u,v)<w<0\}$ is enclosed between the patches, and

${\int }_{S_1}J\cdot n_{1}dS=-{\int }_D\chi (-2\eta (u,v))dudv<0,{\int }_{S_2}J\cdot n_{2}dS=-{\int }_D\chi (0)dudv=0,$

the first inequality because $\eta =1$ in a neighbourhood of the origin, so $\chi (-2\eta )=1$ on a nonempty open subset of $D$, whose area is positive, while both total current fluxes in (a) vanish, consistently with the loop integral ${\oint }_{\gamma }B\cdot d\mathbf{r}=0$. In the standard charging-capacitor discussion the two spanning surfaces cut the conduction current in different ways and the difference is carried by the displacement term; the realisation above exhibits exactly this phenomenon in a self-contained smooth model. An idealised thin-wire current is not a source type of this framework and the capacitor interpretation is explanatory, not an additional empirical claim.

The surface-choice phenomenon in the integral Ampere law is the standard motivation of the displacement current in the cited treatments; the realisation is the framework's own smooth slab model, chosen because idealised thin-wire currents are not among the declared source types. The mathematical content uses the imported divergence theorem, patch flux and step-function interfaces, not textbook authority.