导电媒质中的平面波

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

在导电媒质中,有:

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

只要令

\[\varepsilon_c=\varepsilon-j\frac{\sigma}{\varepsilon}\]

就能得到与上一节相同形式的公式:

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

波阻抗变为复数:

\[\eta_c=\sqrt{\frac{\mu}{\varepsilon_c}}\]

低损耗媒质与良导电媒质

低损耗媒质

低损耗媒质满足:$\sigma \ll \omega \varepsilon$,此时可认为:

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

则 $\alpha,\beta,\eta_c$ 可认为:

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

良导体媒质

良导体媒质满足:$\sigma \gg \omega \varepsilon$,此时可认为:

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

则 $\alpha,\beta,\eta_c$ 可认为:

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

定义电磁波的振幅衰减到表面值的 $1/e$ 的深度称为 趋肤深度 $\delta$

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