38 lines
1.4 KiB
TeX
38 lines
1.4 KiB
TeX
|
||
\chapter{Maxwell Equations}
|
||
|
||
|
||
因为在柱坐标系下，$\overline{\overline\mu}$是对角的，所以Maxwell方程组中电场$\bf
|
||
E$的旋度
|
||
|
||
所以$\bf H$的各个分量可以写为：
|
||
\begin{subequations}
|
||
\begin{eqnarray}
|
||
H_r=\frac{1}{\mathbf{i}\omega\mu_r}\frac{1}{r}\frac{\partial
|
||
E_z}{\partial\theta } \\
|
||
H_\theta=-\frac{1}{\mathbf{i}\omega\mu_\theta}\frac{\partial E_z}{\partial r}
|
||
\end{eqnarray}
|
||
\end{subequations}
|
||
同样地，在柱坐标系下，$\overline{\overline\epsilon}$是对角的，所以Maxwell方程组中磁场$\bf
|
||
H$的旋度
|
||
\begin{subequations}
|
||
\begin{eqnarray}
|
||
&&\nabla\times{\bf H}=-\mathbf{i}\omega{\bf D}\\
|
||
&&\left[\frac{1}{r}\frac{\partial}{\partial
|
||
r}(rH_\theta)-\frac{1}{r}\frac{\partial
|
||
H_r}{\partial\theta}\right]{\hat{\bf
|
||
z}}=-\mathbf{i}\omega{\overline{\overline\epsilon}}{\bf
|
||
E}=-\mathbf{i}\omega\epsilon_zE_z{\hat{\bf z}} \\
|
||
&&\frac{1}{r}\frac{\partial}{\partial
|
||
r}(rH_\theta)-\frac{1}{r}\frac{\partial
|
||
H_r}{\partial\theta}=-\mathbf{i}\omega\epsilon_zE_z
|
||
\end{eqnarray}
|
||
\end{subequations}
|
||
由此我们可以得到关于$E_z$的波函数方程：
|
||
\begin{eqnarray}
|
||
\frac{1}{\mu_\theta\epsilon_z}\frac{1}{r}\frac{\partial}{\partial r}
|
||
\left(r\frac{\partial E_z}{\partial r}\right)+
|
||
\frac{1}{\mu_r\epsilon_z}\frac{1}{r^2}\frac{\partial^2E_z}{\partial\theta^2}
|
||
+\omega^2 E_z=0
|
||
\end{eqnarray}
|