The ideal infinite parallel-plate capacitor

Local integrability, classical relations and continuity: $E$ is bounded by $\sigma \mathrm{/}{\epsilon }_0$ and continuous off the two planes, so for every compactly supported smooth test the product $E\cdot \mathrm{\nabla }\psi$ is compactly supported and the integral is finite; $\phi$ is bounded by $\sigma h\mathrm{/}{\epsilon }_0$, hence locally integrable, and continuous because the slab formula, the two constants and the plane values match at $x_3=\pm h$. On each of the three open regions $E=-\mathrm{\nabla }\phi$ classically, with $\mathrm{\mid }E\mathrm{\mid }=\sigma \mathrm{/}{\epsilon }_0$ and normal direction inside the slab and $E=0$ outside; this is the regularity asserted in (a) and the uniform-field statement of (b).

Distributional divergence: for $\psi \in \mathcal{D}({\mathbb{R}}^3)$, the definition of the distributional derivative gives $⟨\mathrm{div}E,\psi ⟩=-{\int }_{{\mathbb{R}}^3}E\cdot \mathrm{\nabla }\psi d{\lambda }_3$, and since $E$ vanishes off the slab and equals $-(\sigma \mathrm{/}{\epsilon }_0)e_3$ on it, this is $\frac{\sigma }{{\epsilon }_0}{\int }_{{\mathbb{R}}^3}{\mathbf{1}}_{\{\mathrm{\mid }{x}_3\mathrm{\mid }<h\}}{\mathrm{\partial }}_3\psi d{\lambda }_3$. Fubini writes the integral as the iterated integral ${\int }_{{\mathbb{R}}^2}({\int }_{-h}^h{\mathrm{\partial }}_3\psi (x_1,x_2,s)ds)d{x}_{1}d{x}_2$, and the inner integral equals $\psi (x_1,x_2,h)-\psi (x_1,x_2,-h)$ by the fundamental theorem; hence $⟨\mathrm{div}E,\psi ⟩=\frac{\sigma }{{\epsilon }_0}({\delta }_{{\mathrm{\Pi }}_{+}}-{\delta }_{{\mathrm{\Pi }}_{-}})(\psi )$ for every test $\psi$, i.e. $\mathrm{div}E=\frac{\sigma }{{\epsilon }_0}({\delta }_{{\mathrm{\Pi }}_{+}}-{\delta }_{{\mathrm{\Pi }}_{-}})$ in ${\mathcal{D}}^{\mathrm{\prime }}({\mathbb{R}}^3)$, with no other source.

Distributional curl: $E_1=E_2=0$ are the zero distribution; $E_3=-(\sigma \mathrm{/}{\epsilon }_0){\mathbf{1}}_{\{\mathrm{\mid }{x}_3\mathrm{\mid }<h\}}$ is a function of $x_3$ alone, so pairing its distributional ${\mathrm{\partial }}_1$- and ${\mathrm{\partial }}_2$-derivatives against a test gives integrals of ${\mathrm{\partial }}_1\psi$ and ${\mathrm{\partial }}_2\psi$ over the slab, and each inner one-dimensional integral over $\mathbb{R}$ of a compactly supported derivative vanishes by the fundamental theorem; hence ${\mathrm{\partial }}_1{E}_3={\mathrm{\partial }}_2{E}_3=0$ in ${\mathcal{D}}^{\mathrm{\prime }}({\mathbb{R}}^3)$, while ${\mathrm{\partial }}_3{E}_1={\mathrm{\partial }}_3{E}_2=0$ because $E_1,E_2$ vanish identically. All three components of $\mathrm{curl}E=({\mathrm{\partial }}_2{E}_3-{\mathrm{\partial }}_3{E}_2,{\mathrm{\partial }}_3{E}_1-{\mathrm{\partial }}_1{E}_3,{\mathrm{\partial }}_1{E}_2-{\mathrm{\partial }}_2{E}_1)$ therefore vanish: $\mathrm{curl}E=0$. Together with the divergence identity above this is the asserted static electrostatic realization of (a) with the declared plane sources, and no other source.

Potential difference and voltage (b): the trace values of [F1] give $\phi {\mathrm{\mid }}_{{\mathrm{\Pi }}_{+}}-\phi {\mathrm{\mid }}_{{\mathrm{\Pi }}_{-}}=\sigma h\mathrm{/}{\epsilon }_0-(-\sigma h\mathrm{/}{\epsilon }_0)=2\sigma h\mathrm{/}{\epsilon }_0=\sigma d\mathrm{/}{\epsilon }_0$ because $d=2h$; combined with the constant normal field of magnitude $\sigma \mathrm{/}{\epsilon }_0$ between the planes and the zero field outside, this is (b), and it also exhibits the potential difference of (a).

Charge, capacitance and energy (c): fix a Lebesgue-measurable $P\subseteq {\mathbb{R}}^2$ with finite positive area $A={\lambda }_2(P)$. The positive plate carries the constant density $+\sigma$, so the charge on the portion over $P$ is $Q=\sigma {\lambda }_2(P)=\sigma A$. The slab of area $A$ between the plates is $P\times (-h,h)$, and on it $\mathrm{\mid }E{\mathrm{\mid }}^2={\sigma }^2\mathrm{/}{\epsilon }_0^2$ is constant; by [F3] the slab has ${\lambda }_3$-measure $A\cdot 2h=Ad$, so Tonelli gives ${\int }_{P\times (-h,h)}\mathrm{\mid }E{\mathrm{\mid }}^{2}d{\lambda }_3=({\sigma }^2\mathrm{/}{\epsilon }_0^2)Ad$ and $U=\frac{{\epsilon }_0}{2}\cdot \frac{{\sigma }^2}{{\epsilon }_0^2}Ad=\frac{{\sigma }^{2}Ad}{2{\epsilon }_0}$. With $V=\sigma d\mathrm{/}{\epsilon }_0$ from [step 1.4], the ratio $C:=Q\mathrm{/}V=(\sigma A)\mathrm{/}(\sigma d\mathrm{/}{\epsilon }_0)={\epsilon }_{0}A\mathrm{/}d$ is positive and independent of $\sigma$ and $V$; and the four displayed energies agree: $\frac{1}{2}QV=\frac{1}{2}\sigma A\cdot \frac{\sigma d}{{\epsilon }_0}=\frac{{\sigma }^{2}Ad}{2{\epsilon }_0}$, $Q^2\mathrm{/}(2C)=(\sigma A{)}^2\cdot \frac{d}{2{\epsilon }_{0}A}=\frac{{\sigma }^{2}Ad}{2{\epsilon }_0}$ and $\frac{1}{2}C{V}^2=\frac{1}{2}\cdot \frac{{\epsilon }_{0}A}{d}\cdot \frac{{\sigma }^2{d}^2}{{\epsilon }_0^2}=\frac{{\sigma }^{2}Ad}{2{\epsilon }_0}$. This is (c).

Relation to the bounded-conductor theorem and scope (d): the relations $Q=CV$, $C>0$ and $U=\frac{1}{2}QV=\frac{Q^2}{2C}=\frac{1}{2}C{V}^2$ verified in [step 2.1] have the same scalar form as Capacitance and energy for specified linear conductor geometry(d), under that theorem's hypotheses with only conductor $1$ driven and every other conductor and the outer boundary grounded, taking $C=C_{11}$, $Q=Q_1$ and $V=V_1$ [F4]. For general conductor-potential data its conclusions instead use the matrix relation $\mathbf{q}=\mathsf{C}\mathbf{v}$ and the energy $U=\frac{1}{2}{\sum }_j{v}_j{q}_j$; the present verification is direct, from the explicitly declared piecewise model, because the infinite planes, the unbounded field and the plane densities are outside that theorem's compact-conductor and declared sheet-source hypotheses, and no approximation theorem from finite plates to the infinite model is asserted. The uniform-field idealization neglects the fringing field near finite edges, which is not computed, and omits finite thickness, material, transient and measurement content; the direct argument retains only ${\mathrm{A}\mathrm{C}}_{\omega }$ as declared, and no empirical claim is made. This is (d).