= Permittivity
{title2=$\epsilon$}
{wiki}
<Permittivity> describes how an <electric field> and a material's <electric polarization> combine into the <electric displacement field>. In a static, local response with <linearity> and <isotropy>, and no <electric polarization> at zero <electric field>, the <constitutive equation> is
$$
\mathbf D=\epsilon\mathbf E,\qquad \epsilon=\epsilon_0\epsilon_r.
$$
Here $\epsilon_0$ is <vacuum permittivity>, while $\epsilon_r$ is the <relative permittivity>, a <dimensionless quantity>. The <SI> unit of <permittivity> is <farad> per metre, $\mathrm{F/m}$: multiplying it by an <electric field> in $\mathrm{V/m}$ gives an <electric displacement field> in $\mathrm{C/m^2}$. A larger positive <permittivity> lets the same <parallel-plate capacitor> hold more plate <electric charge> at the same <electric potential difference>.
The material contribution is <electric polarization> $\mathbf P$, the <electric dipole moment> per unit <volume>. The <electric displacement field> and <electric susceptibility> $\chi_e$ satisfy
$$
\mathbf D=\epsilon_0\mathbf E+\mathbf P,\qquad
\mathbf P=\epsilon_0\chi_e\mathbf E,\qquad
\epsilon_r=1+\chi_e.
$$
Substitution gives the first <constitutive equation>. The <electric field> here is the averaged, total field inside the material, not just the field of the external plate <electric charges>. The <continuum approximation> averages over many <chemical atoms> or <molecules>; locality means that the <electric polarization> at a position depends on the <electric field> there. Static equilibrium, fixed material conditions and sufficiently small <electric fields> are needed for this scalar proportionality. <Classical Electromagnetism: An intermediate level course by Richard Fitzpatrick/Polarization>{ref}.
For an ideal <parallel-plate capacitor> completely filled by a uniform <dielectric>, hold the free plate <electric charge> magnitude $Q$ fixed and neglect the <fringing field of a capacitor>. Writing $A$ for the plate <area>, <Gauss's law> gives $D=Q/A$. Thus
$$
E=\frac{Q}{\epsilon_0\epsilon_r A},\qquad
P=\frac{Q}{A}\left(1-\frac1{\epsilon_r}\right).
$$
The <electric dipoles> point along the <electric field>. Their surface <electric charges>, described by <polarization density>, oppose the field from the free plate <electric charges>. For a hypothetical <dielectric> with $\epsilon_r=4$, the internal <electric field> is one quarter of its <vacuum> value, while $P=3D/4$. At fixed <electric potential difference> instead, the internal <electric field> stays approximately $V/d$ and the plate <electric charge> grows fourfold. These are different experimental constraints. https://www.feynmanlectures.caltech.edu/II_10.html#Ch10-S3[Feynman, Leighton and Sands, II.10–3, equations (10.5)–(10.11): polarization surface charge, field reduction and capacitance]{ref}.
\Image[/dielectric-polarization-fixed-charge-fields.png]
{title=<Electric polarization> and <electric field> reduction at fixed plate <electric charge>}
{description=An original schematic and normalized model comparison for a uniform <dielectric> with <relative permittivity> four. The arrows for <electric dipole moments> run from negative to positive <electric charge>; the two field contributions oppose each other. The bar values compare <electric displacement field> $D$, $\epsilon_0E$, and <electric polarization> $P$, which all have units $\mathrm{C/m^2}$. The schematic is not to microscopic scale.}
<Permittivity> is a useful material summary, not a universal constant of the material under every condition. <Temperature>, <frequency> and <electric field> strength can change the response. In a <linear anisotropic dielectric>, <permittivity> is a <tensor>, $D_i=\sum_j\epsilon_{ij}E_j$, so the <electric displacement field> need not be parallel to the <electric field>. At nonzero <frequency>, delayed <electric polarization> requires <complex permittivity> and may produce <dielectric loss>. https://www.feynmanlectures.caltech.edu/II_32.html#Ch32-S2[Feynman, Leighton and Sands, II.32–2, equations (32.5)–(32.8) and following paragraph: frequency dependence and phase lag]{ref}.
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