= Dielectric capacitance measurement
The motivating observation is that inserting an insulating material between fixed <capacitor> plates increases its <capacitance>. https://www.feynmanlectures.caltech.edu/II_10.html#Ch10-S1[Feynman, Leighton and Sands, II.10–1, opening paragraphs: observed increase in capacitance with an inserted insulator]{ref}. To investigate this response, use two flat parallel metal electrodes with measured overlapping <area> $A$ and separation $d$, a uniform specimen filling the gap, and an instrument that measures <electrical impedance>. Record the <capacitance> with and without the specimen at the same small applied <electric potential difference>, <frequency> and <temperature>, after <capacitance measurement calibration>. An increase in corrected <capacitance> indicates a larger <permittivity>, provided the geometry is unchanged. A guard electrode reduces the contribution of the <fringing field of a capacitor>. <Materials Measurement: Dielectric Materials>{ref}.
For negligible <dielectric loss> in the <parallel-plate capacitor> approximation,
$$
C=\frac{\epsilon A}{d},\qquad C_0=\frac{\epsilon_0A}{d},\qquad
\epsilon_r=\frac{C}{C_0}=\frac{Cd}{\epsilon_0A}.
$$
An air reference approximates $C_0$; it is not an exact <vacuum> reference. For the following synthetic example, stipulate $A=0.010\,\mathrm{m^2}$, $d=1.00\,\mathrm{mm}$ and the rounded <vacuum permittivity> $\epsilon_0\approx8.854\times10^{-12}\,\mathrm{F/m}$. Then $C_0=88.54\,\mathrm{pF}$ in this model, and <relative permittivity> four gives $C=354.16\,\mathrm{pF}$. A residual parallel <capacitance> of $20\,\mathrm{pF}$ would make an uncorrected sample/reference ratio $374.16/108.54\approx3.447$, rather than four. This additive error does not cancel in a ratio.
\Image[/synthetic-dielectric-capacitance-frequency-response.png]
{title=Synthetic <capacitance> response to <relative permittivity> and <frequency>}
{description=Left: the ideal scalar <capacitance> and a hypothetical additive stray <capacitance> of $20\,\mathrm{pF}$. Right: a hypothetical <dielectric relaxation> model, $\epsilon_r^*=2+2/(1+if/f_0)$ with $f_0=1\,\mathrm{kHz}$, for the same geometry. The solid curve is the parallel <capacitance> $C_p=C_0\epsilon_r'$; the dashed curve is the loss component $C_0\epsilon_r''$, not an independently measured storage <capacitance>. Every plotted value is synthetic.}
At nonzero <frequency> $f$, measure both magnitude and <phase> of the <electric current> relative to the applied <electric potential difference>. Define $Y=I/V=1/Z$, where $Z$ is <electrical impedance>. For the stated <complex permittivity> convention, the ideal uniform specimen gives
$$
Y=i\omega C_0\epsilon_r^*=\omega C_0\epsilon_r''+i\omega C_0\epsilon_r',\qquad
C_p=\frac{\operatorname{Im}Y}{\omega},\qquad \omega=2\pi f.
$$
Thus a parallel <capacitor> and <electrical conductance> describe the readout at that <frequency>; a single scalar <capacitance> cannot describe both components. The diagram's assumed <dielectric relaxation> gives $C_p=265.62\,\mathrm{pF}$ and $C_0\epsilon_r''=88.54\,\mathrm{pF}$ at $1\,\mathrm{kHz}$. These are model predictions, not observations.
Repeat the reference measurement after removing the specimen and vary the small applied <electric potential difference> to test drift and <linearity>. Record specimen thickness, electrode <area>, contact conditions, <frequency> and <temperature>. Air gaps act as additional layers in series and can lower inferred <relative permittivity>; roughness, uncertain thickness and the <fringing field of a capacitor> limit the geometric formula. At sufficiently high <frequency>, the fixture itself changes the <electromagnetic field>, so low-<frequency> guarding and a lumped <capacitor> model require reassessment. <Materials Measurement: Dielectric Materials>{ref}.
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