The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models

For the Lebesgue/surface/distributional source and admissibility branches (c)-(d), assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). The smooth formulation in (a) and the smooth conserved-current predicate in (b) do not themselves use Choice. Let the laboratory region and its chart, the field and source objects and the units be those of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, The electric and magnetic fields as fixed-time spatial vector fields, Charge parameter, frame charge and current densities, volume charge and flux, and the four-current and Framework-local SI units, the vacuum constants and the exact/measured status of their values, and fix ${\mu }_0>0$, $c>0$, ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$.

(a) [Steady-current regime]. A steady-current (magnetostatic) configuration on $\mathrm{\Omega }\times I$ consists of a field pair $E,B:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ and source fields $\rho :\mathrm{\Omega }\times I\to \mathbb{R}$, $J:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ that are independent of $t$ and satisfy the smooth branch of The microscopic vacuum Maxwell equations: smooth and weak branches with the operators of Divergence and curl of a $C^1$ vector field and Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$. For such a configuration the formal distributional branch of Charge continuity as a compatibility condition of the vacuum Maxwell equations gives $\mathrm{div}J=0$ distributionally; this is pointwise when $J$ is $C^1$. In the jointly $C^2$ field/$C^1$ source branch its classical proof applies directly. The locally integrable distributional reading here inherits Countable Choice, whereas the stated classical $C^2\mathrm{/}{C}^1$ compatibility calculation is choice-free, and the four adopted equations are equivalent to the electrostatic pair and the magnetostatic pair $\mathrm{div}E=\rho \mathrm{/}{\epsilon }_0,\mathrm{curl}E=0,\mathrm{div}B=0,\mathrm{curl}B={\mu }_{0}J\mathrm{.}$

(b) [Compact smooth current class]. The compact smooth current class of this framework consists of the currents $J\in C_c^{\mathrm{\infty }}(\mathrm{\Omega };{\mathbb{R}}^3)$ with $\mathrm{div}J=0$, extended by zero to ${\mathbb{R}}^3$. A magnetostatic field of such a current is a field $B$ of class $C^1$ on ${\mathbb{R}}^3$ satisfying $\mathrm{div}B=0$ and $\mathrm{curl}B={\mu }_{0}J$ on ${\mathbb{R}}^3$ with the extended $J$. This is the hypothesis class of the vector-potential, Biot-Savart and dipole theorems of this page; this definition asserts neither existence nor uniqueness of $B$ in the class.

(c) [Ideal source objects]. Define three ideal current source objects outside the compact class, together with the data selecting their fields. Curve parameters $s$ and $L$ and the radius $a$ carry metres; the signed line-current amplitude $I$ carries amperes and the signed sheet-current amplitude $K$ carries A m${}^{-1}$. The field $B$ carries tesla, as in Framework-local SI units, the vacuum constants and the exact/measured status of their values.

(c1) [Closed thin loop]. For a closed embedded $C^2$ curve $\gamma :\mathbb{R}\mathrm{/}L\mathbb{Z}\to {\mathbb{R}}^3$ with $\mathrm{\mid }{\gamma }^{\mathrm{\prime }}\mathrm{\mid }\equiv 1$ ($L>0$) and $I\in \mathbb{R}$, the ideal loop current is the distribution $⟨J_{\gamma },\psi ⟩:=I{\int }_0^L{\gamma }^{\mathrm{\prime }}(s)\cdot \psi (\gamma (s))ds,\psi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^3;{\mathbb{R}}^3),$ with ${\rho }_{\gamma }:=0$.

(c2) [Infinite straight wire]. For $\gamma (s)=(0,0,s)$, $s\in \mathbb{R}$, and $I\in \mathbb{R}$, the ideal wire current is $⟨J_{\mathrm{w}},\psi ⟩:=I{\int }_{\mathbb{R}}{\psi }_3(0,0,s)ds,$ with ${\rho }_{\mathrm{w}}:=0$.

(c3) [Infinite cylindrical sheet]. For $a>0$, $S=\{\rho =a\}$, $\rho =\sqrt{x_1^2+x_2^2}$, $e_{\rho }:={\rho }^{-1}(x_1,x_2,0)$ and $e_{\phi }:={\rho }^{-1}(-x_2,x_1,0)$ on $\{\rho >0\}$, and $K\in \mathbb{R}$, the ideal solenoid sheet current is $⟨J_S,\psi ⟩:=K{\int }_S{e}_{\phi }\cdot \psi dS$ (Surface integration on compact C1 hypersurfaces, read on compact cylinder segments), with ${\rho }_S:=0$. All three pairings are finite because $\psi$ has compact support.

(c4) [Admissibility]. A field $B$ is admitted for one of these declared models when: $B$ is locally integrable on ${\mathbb{R}}^3$ (hence a distribution by Regular distribution from a locally integrable function, Locally integrable functions embed in distributions) and $C^1$ off the support of the current; $\mathrm{div}B=0$ and $\mathrm{curl}B=0$ hold pointwise off that support; the distributional identities $\mathrm{div}B=0$ and $\mathrm{curl}B={\mu }_{0}J$ hold against every scalar divergence test $\phi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^3)$ and vector curl test $\psi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^3;{\mathbb{R}}^3)$ in the sense of Distributional derivative; $B$ has the model's declared symmetry: in (c2) and (c3), $B(x+h{e}_3)=B(x)$ for every $h\in \mathbb{R}$, and $B(R_{\theta }x)=R_{\theta }B(x)$ for every $\theta \in \mathbb{R}$, at off-source points, where $R_{\theta }(x_1,x_2,x_3)=(x_1\cos \theta -x_2\sin \theta ,x_1\sin \theta +x_2\cos \theta ,x_3)$; no symmetry is required in (c1); and $B$ obeys the decay requirement: $\mathrm{\mid }B(x)\mathrm{\mid }\to 0$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$ in (c1), and in (c2), (c3), ${\sup }_{(\theta ,z)\in C}\mathrm{\mid }B(r\cos \theta ,r\sin \theta ,z)\mathrm{\mid }\to 0$ as $r\to \mathrm{\infty }$ for every nonempty compact $C\subset {\mathbb{R}}^2$. The companion examples exhibit and verify the fields of the two axisymmetric models in this class; the definition itself asserts no existence, uniqueness, finite energy or limiting relation for the declared models, and no compact-class theorem of this page applies to them.

(d) [Scope]. This is a model restriction: the steady-current regime for compact smooth currents, and three explicitly declared ideal noncompact or singular models. It excludes point charges, weaker-regularity volume sources, moving sheets, normal sheet currents and any claim of empirical exactness; the equations in (c4) define a mathematical admissibility predicate; its physical adoption for these source objects is a separate postulate, and no product of distributions is formed.

Well-definedness of the clauses. In (c1)-(c3) the pairings are order-zero continuous linear functionals on the test spaces of Test function space d of an open set: on a fixed compact test support their absolute values are bounded by $\mathrm{\mid }I\mathrm{\mid }L\mathrm{\parallel }\psi {\mathrm{\parallel }}_{\mathrm{\infty }}$, by $\mathrm{\mid }I\mathrm{\mid }$ times the finite length of the axis projection times $\mathrm{\parallel }\psi {\mathrm{\parallel }}_{\mathrm{\infty }}$, and by $\mathrm{\mid }K\mathrm{\mid }$ times the finite area of a containing compact cylindrical face times $\mathrm{\parallel }\psi {\mathrm{\parallel }}_{\mathrm{\infty }}$, respectively. The cylinder chart $X(\theta ,z)=(a\cos \theta ,a\sin \theta ,z)$ has Gram density $a$, and finitely many angular charts cover each compact face, so the surface pairing is the chart-defined integral of Surface integration on compact C1 hypersurfaces; the scalar triple product and the direction $e_{\phi }$ are those of The cross product in ${\mathbb{R}}^3$. The divergence pairing uses scalar tests and the curl pairing vector tests under Distributional derivative, a vector distribution being a triple of scalar distributions in ${\mathcal{D}}^{\mathrm{\prime }}({\mathbb{R}}^3)$ of Distribution; a locally integrable field is a distribution in the sense of the regular-distribution embedding Regular distribution from a locally integrable function, Locally integrable functions embed in distributions. Smoothness statements use $C^k$ Euclidean maps and diffeomorphisms. The finite area of a compact angular face is the Riemann integral of the constant function one over its rectangular chart domain, with the properties of Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in ${\mathbb{R}}^m$. The declared ideal objects lie outside the source classes of Distributional Maxwell equations with point and stationary-sheet sources: that definition excludes line and noncompact sheet currents, so no equation for these singular objects is inherited from it, and Countable Choice is inherited only through the compact-face surface and regular-distribution interfaces.

Griffiths, Section 5.2.1, printed pp. 215-216, and the opening of Tong, Chapter 3, printed pp. 41-42, describe steady currents and the reduction to $\mathrm{\nabla }\cdot J=0$ and the static field equations; Griffiths, Section 5.1.3, printed pp. 208-214, supplies the line, surface and volume current-density conventions reproduced in (c). Landau-Lifshitz, Section 43, printed pp. 101-103, and Chew, Lecture 4 Section 4.1, printed pp. 39-41, give the constant-field equations and the vector-potential formulation in the same regime. The ideal noncompact models are declared modelling hypotheses of this framework, not measurements or exact descriptions of any apparatus.