<center><img src="https://suncos-01-1254144885.cos.ap-shanghai.myqcloud.com/Hexo/%E5%85%89%E9%80%9F%E5%8F%8A%E6%8A%98%E5%B0%84%E7%8E%87%E8%A1%A8%E8%BE%BE%E5%BC%8F%E6%8E%A8%E5%AF%BC.jpg" style="width: 90%"></center>
在没有自由电荷和传导电流的情况下,麦克斯韦方程组表达式如下:
$$
\nabla \bullet \mathbf{D} = 0
$$
$$
\nabla \bullet \mathbf{B} = 0
$$
$$
\nabla \times \mathbf{E} = - \frac{\partial\mathbf{B}}{\partial t}
$$
$$
\nabla \times \mathbf{H} = - \frac{\partial\mathbf{D}}{\partial t}
$$
其中,
$$
\mathbf{D} = \varepsilon\mathbf{E} = \varepsilon_{0}\varepsilon_{r}\mathbf{E}
$$
$$
\mathbf{B} = \mu\mathbf{H} = \mu_{0}\mu_{r}\mathbf{H}
$$
对$$\nabla \times \mathbf{E} = - \frac{\partial\mathbf{B}}{\partial
t}$$两边取旋度,可得:
$$
\nabla \times \left( \nabla \times \mathbf{E} \right) = - \nabla \times \frac{\partial\mathbf{B}}{\partial t} = - \frac{\partial}{\partial t}\left( \nabla \times \mathbf{B} \right) = - \text{εμ}\frac{\partial^{2}\mathbf{E}}{\partial t^{2}}
$$
利用矢量分析公式和
$$\nabla \bullet
\mathbf{E}\mathbf{=}\frac{\mathbf{1}}{\varepsilon}\nabla \bullet \mathbf{D} =0$$
两遍取旋度,可得:
$$
\nabla \times \left( \nabla \times \mathbf{E} \right)\mathbf{=}\nabla\left( \nabla \bullet \mathbf{E} \right)\mathbf{-}\nabla^{2}\mathbf{E} = \nabla^{2}\mathbf{E}
$$
由此可得:
$$
\nabla^{2}\mathbf{E}\mathbf{-}\text{εμ}\frac{\partial^{2}\mathbf{E}}{\partial t^{2}} = 0
$$
或
$$
\nabla^{2}\mathbf{E}\mathbf{-}v^{2}\frac{\partial^{2}\mathbf{E}}{\partial t^{2}} = 0
$$
其中,v表示电磁波传播速度,电磁波真空中的传播速度为
$$c =
\frac{1}{\sqrt{\varepsilon_{0}\mu_{0}}} = 2.997924 \times 10^{8}\ m/s$$
折射率表达式为:
$$n =
\frac{c}{v} = {\sqrt{\varepsilon_{r}\mu_{r}}}
$$

光速及折射率表达式推导