= Arithmetic-harmonic bounds for dielectric capacitance
{title2=$\langle\varepsilon^{-1}\rangle^{-1}A/d\leq C\leq\langle\varepsilon\rangle A/d$}
For a <parallel-plate capacitor> of area $A$ and separation $d$, filled by a spatially varying positive scalar <permittivity>, neglect edge fields. At fixed terminal voltage, the physical <electric potential> minimizes $\int\varepsilon|\nabla V|^2\,d\mathcal V/2$. A linear trial potential gives the arithmetic upper bound. At fixed plate charge, minimizing $\int|\mathbf D|^2/(2\varepsilon)\,d\mathcal V$ over divergence-free <electric displacement fields> gives the harmonic lower bound by using a uniform trial displacement. The averages are volume averages. Layers parallel to the field attain the arithmetic bound; layers perpendicular to the field attain the harmonic bound. Equal volumes of vacuum and relative <permittivity> $\epsilon$ therefore give $2\epsilon C_0/(1+\epsilon)\leq C\leq(1+\epsilon)C_0/2$, where $C_0=\varepsilon_0A/d$.
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