Declared ideal infinite wire and cylindrical current sheet: explicit fields and their distributional equations

The field of (a) is $B=\frac{{\mu }_{0}I}{2\pi }f$ with $f={\rho }^{-1}{e}_{\phi }$; by [F1] it is locally integrable on ${\mathbb{R}}^3$ and $C^{\mathrm{\infty }}$ off $\gamma$, with $\mathrm{div}B=\mathrm{curl}B=0$ pointwise off $\gamma$, and all these properties are stable under multiplication by the constant $\frac{{\mu }_{0}I}{2\pi }$.

The field of (b) is $B={\mu }_{0}Kg$ with $g$ the piecewise constant cylinder field: it is bounded, hence locally integrable, constantly ${\mu }_{0}K{e}_3$ for $\rho <a$ and $0$ for $\rho >a$, so it is $C^{\mathrm{\infty }}$ off $S$ with $\mathrm{div}B=\mathrm{curl}B=0$ there; it is invariant under translations along and rotations about the third axis, and its supremum on any compact set of cross-sectional parameters is $0$ for every $\rho >a$, hence its transverse far-zone supremum is $0$.

For the wire, $⟨\mathrm{curl}{T}_B,\psi ⟩=\frac{{\mu }_{0}I}{2\pi }⟨\mathrm{curl}{T}_f,\psi ⟩={\mu }_{0}IA(\psi )={\mu }_0⟨J_{\mathrm{w}},\psi ⟩$ and $⟨\mathrm{div}{T}_B,\phi ⟩=0$ for all tests by [F1] and [F3]; moreover $B$ depends only on $\rho$ through the scalar factor $\frac{{\mu }_{0}I}{2\pi \rho }$ multiplying $e_{\phi }$, so it is invariant under translations along $\gamma$ and rotations about it, and its transverse far-zone supremum is $\frac{{\mu }_0\mathrm{\mid }I\mathrm{\mid }}{2\pi \rho }\to 0$. The circulation of $B$ along the increasing-$t$ traversal ${\gamma }_{r,h}$ of (a) is $\frac{{\mu }_{0}I}{2\pi }$ times the circulation $2\pi$ of $f$, that is ${\mu }_{0}I$, by [F1] and the linearity of the line integral [F4].

For the sheet, $⟨\mathrm{curl}{T}_B,\psi ⟩={\mu }_{0}K⟨\mathrm{curl}{T}_g,\psi ⟩={\mu }_{0}K{\int }_S{e}_{\phi }\cdot \psi dS={\mu }_0⟨J_S,\psi ⟩$ and $⟨\mathrm{div}{T}_B,\phi ⟩=0$ for all tests by [F2] and [F3]. The one-sided values across $S$ are $B_{+}=0$ and $B_{-}={\mu }_{0}K{e}_3$ in the direction from the interior to the exterior, so $[B]=B_{+}-B_{-}=-{\mu }_{0}K{e}_3$ and, by [F4], $n\times [B]=e_{\rho }\times (-{\mu }_{0}K{e}_3)={\mu }_{0}K{e}_{\phi }$ and $n\cdot [B]=0$; this is the jump displayed in (b), computed directly from the one-sided values of the declared field.

For the wire, every clause of the admissibility predicate F3 holds: local integrability and off-source $C^{\mathrm{\infty }}$ regularity from step 1.1, the two pointwise vanishing identities off $\gamma$ from step 1.1, the two distributional identities and the line-measure value from step 2.1, the translation and rotation symmetry from step 2.1, and the transverse decay from step 2.1. Hence the field of (a) is admitted for the declared wire model; the decay requirement is what rules out adding a constant longitudinal field, and nothing else is claimed.

For the sheet, every clause of F3 holds: local integrability, off-source $C^{\mathrm{\infty }}$ regularity, the two pointwise vanishing identities and the symmetry and the vanishing transverse far-zone supremum from step 1.2, and the two distributional identities from step 2.2, with the one-sided jump of step 2.2 as the interface content. Hence the field of (b) is admitted for the declared solenoid model.

Steps 3.1 and 3.2 prove that both explicit fields are admitted under the predicate (c4) of [F3]; the physical reading of the declared models is the separate adopted assumption [F5], which adds no existence, uniqueness, self-energy, inductance, force or apparatus claim. Both models are declared noncompact idealisations: no finite-energy or localized-current conclusion is drawn, nothing is asserted on the sources, and the fields are selected within the declared symmetry and decay class rather than shown to be the only solutions off the sources. Countable Choice is inherited only through the cited measure, surface, regular-distribution and Newtonian interfaces [F6]; no stronger choice principle occurs.