= Electric susceptibility
{title2=$\chi_e$}
{wiki}
The <dimensionless quantity> relating induced <electric polarization> to the internal <electric field> through $\mathbf P=\epsilon_0\chi_e\mathbf E$ in the static scalar model. It describes the material contribution; <relative permittivity> includes the <vacuum> term as well, $\epsilon_r=1+\chi_e$. Zero <electric susceptibility> gives $\epsilon_r=1$, not zero <permittivity>.
The scalar $\chi_e$ assumes a local response with <linearity> and <isotropy>, settled <electric polarization>, fixed material conditions, and no residual <electric polarization> at zero <electric field>. It relates $\mathbf P$ to the averaged total internal <electric field>, which already includes the field of the bound <electric charges>. It does not relate $\mathbf P$ to the field the external <electrodes> would produce in <vacuum>. The distinction is essential for a self-consistent calculation.<Classical Electromagnetism: An intermediate level course by Richard Fitzpatrick/Polarization>[equations (807)–(812): total field, displacement, linear susceptibility and relative dielectric constant]{ref}
Substitute the scalar relation into the definition of <electric displacement field>:
$$
\mathbf D=\epsilon_0\mathbf E+\mathbf P
=\epsilon_0(1+\chi_e)\mathbf E.
$$
Comparison with $\mathbf D=\epsilon\mathbf E$ proves $\epsilon=\epsilon_0(1+\chi_e)$ and $\epsilon_r=1+\chi_e$. Both $\mathbf D$ and $\mathbf P$ have units $\mathrm{C/m^2}$, so $\chi_e$ is a <dimensionless quantity>. In a stipulated static model with $\epsilon_r=3$, $\chi_e=2$: at a prescribed internal <electric field>, the material contribution $P$ is twice $\epsilon_0E$. This is a hypothetical response, not a measured material constant. The <electric susceptibility> is a convenient bridge from <microscopic dielectric polarization> to a macroscopic <capacitance> measurement, but it is not a measure of <electrical conductivity>.
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