# 電極化率

${\displaystyle \mathbf {P} \ {\stackrel {def}{=}}\ \varepsilon _{0}\chi _{e}\mathbf {E} }$

${\displaystyle \mathbf {D} \ {\stackrel {\mathrm {def} }{=}}\ \varepsilon _{0}\mathbf {E} +\mathbf {P} }$

${\displaystyle \mathbf {D} =\varepsilon _{0}(1+\chi _{e})\mathbf {E} =\varepsilon \mathbf {E} }$

${\displaystyle \varepsilon _{r}\ {\stackrel {\mathrm {def} }{=}}\ \varepsilon /\varepsilon _{0}}$

${\displaystyle \chi _{e}\ =\varepsilon _{r}-1}$

${\displaystyle \chi _{e}\ =0}$

## 色散性質和因果關係

${\displaystyle \mathbf {P} (t)={\frac {\varepsilon _{0}}{\sqrt {2\pi }}}\int _{-\infty }^{t}\chi _{e}(t-t')\mathbf {E} (t')\,dt'}$

${\displaystyle \mathbf {P} (t)={\frac {\varepsilon _{0}}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\chi _{e}(t-t')\mathbf {E} (t')\,dt'}$

{\displaystyle {\begin{aligned}\mathbf {P} (\omega )&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\mathbf {P} (t)e^{i\omega t}\,dt\\&={\frac {\varepsilon _{0}}{2\pi }}\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }\chi _{e}(t-t')\mathbf {E} (t')\,dt'\right]e^{i\omega t}\,dt\\&={\frac {\varepsilon _{0}}{2\pi }}\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }\chi _{e}(t-t')e^{i\omega (t-t')}\,dt\right]\mathbf {E} (t')e^{i\omega t'}\,dt'\\&={\frac {\varepsilon _{0}}{2\pi }}\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }\chi _{e}(t'')e^{i\omega (t'')}\,dt''\right]\mathbf {E} (t')e^{i\omega t'}\,dt'\\&={\frac {\varepsilon _{0}}{2\pi }}\left[\int _{-\infty }^{\infty }\chi _{e}(t'')e^{i\omega (t'')}\,dt''\right]\left[\int _{-\infty }^{\infty }\mathbf {E} (t')e^{i\omega t'}\,dt'\right]\\&=\varepsilon _{0}\chi _{e}(\omega )\mathbf {E} (\omega )\\\end{aligned}}}