# 导电媒质中的平面波

\begin{align*} \newcommand{\dif}{\mathop{}\!\mathrm{d}}\\ \newcommand{\p}{\partial}\\ \newcommand{\bd}{\boldsymbol}\\ \newcommand{\E}{\mathscr{E}}\\ \newcommand{\db}[1]{\boldsymbol{\dot{#1}}} \end{align*}

$\nabla\times\db{H}=\dot{J}+j\omega \db{D}=\sigma\db{E}+j\omega \varepsilon \db{E}\\ =j\omega(\varepsilon-j\frac{\sigma}{\varepsilon})\db{E}$

$\varepsilon_c=\varepsilon-j\frac{\sigma}{\varepsilon}$

$\nabla^2 \db{E}(\bd{r})+\omega^2\mu\varepsilon_c \db{E}(\bd{r})=0\\ \nabla^2 \db{H}(\bd{r})+\omega^2\mu\varepsilon_c \db{H}(\bd{r})=0$

$\eta_c=\sqrt{\frac{\mu}{\varepsilon_c}}$

# 低损耗媒质与良导电媒质

## 低损耗媒质

$\sqrt{1+\left( \frac{\sigma}{\omega\varepsilon} \right)^2}\approx 1+\frac{1}{2}\left( \frac{\sigma}{\omega\varepsilon} \right)^2$

$\sigma \approx \frac{1}{2}\sigma \sqrt{\frac{\mu}{\varepsilon}}\\ \beta\approx \omega\sqrt{\mu\varepsilon}$

## 良导体媒质

$\sqrt{1+\left( \frac{\sigma}{\omega\varepsilon} \right)^2}\approx \frac{\sigma}{\omega\varepsilon}$

$\alpha\approx\beta\approx\sqrt{\varepsilon\mu\sigma/2}=\sqrt{\pi f \mu \sigma}\\ \eta_c\approx (1+j) \sqrt{\pi f \mu}$

$\delta = \frac{1}{\alpha}\approx \sqrt{\frac{2}{\omega\mu\sigma}}$