连续介质与宏观Maxwell方程
微观Maxwell方程$\quad$当系统尺度远大于$10^{-14}$m，电子与原子核可当成质点处理
$$\nabla\cdot\boldsymbol{e}=\eta/\epsilon_0\qquad\nabla\times\boldsymbol{e}=-\frac{\partial \boldsymbol{b}}{\partial t}$$
$$\nabla\cdot\boldsymbol{b}=0\qquad\nabla\times\boldsymbol{b}=\frac{1}{c^2}\frac{\partial \boldsymbol{e}}{\partial t}+\mu_0\boldsymbol{\vartheta}$$
$$\eta(\boldsymbol{r},t)=\sum_j q_j\;\delta(\boldsymbol{r}-\boldsymbol{r}_j(t))\qquad \boldsymbol{\vartheta}=\sum_j q_j\boldsymbol{v}_j\;\delta(\boldsymbol{r}-\boldsymbol{r}_j(t))$$
平均化$\quad$取试探函数$f(\boldsymbol{r})$，一般为中心平坦，边缘快速衰减的函数，可定义空间期望
$$\langle F(\boldsymbol{r},t)\rangle=\int\text{d}^3 r’f(\boldsymbol{r}’)F(\boldsymbol{r}-\boldsymbol{r}’,t)\qquad \int\text{d}^3 r’f(\boldsymbol{r}’)=1$$
平均化算符与对空间和时间求导对易
$$\partial_\mu \langle F(\boldsymbol{r},t)\rangle=\langle \partial_\mu F(\boldsymbol{r},t)\rangle$$
宏观Maxwell方程$\quad$将微观Maxwell方程平均化，得到
$$\epsilon_0\nabla\cdot\boldsymbol{E}=\langle\eta(\boldsymbol{r},t)\rangle\qquad\nabla\times\boldsymbol{E}=-\frac{\partial \boldsymbol{B}}{\partial t}$$
$$\nabla\cdot\boldsymbol{B}=0\qquad\frac{1}{\mu_0}\nabla\times\boldsymbol{B}=\epsilon_0\frac{\partial \boldsymbol{E}}{\partial t}+\langle\boldsymbol{\vartheta}(\boldsymbol{r},t)\rangle$$
$$\langle\boldsymbol{e}(\boldsymbol{r},t)\rangle=\boldsymbol{E}\qquad \langle\boldsymbol{b}(\boldsymbol{r},t)\rangle=\boldsymbol{B}$$
微观电荷与电流密度平均化中的束缚电荷与电流部分贡献了电极化和磁极化矢量
$$\langle\eta(\boldsymbol{r},t)\rangle=\rho(\boldsymbol{r},t)-\nabla\cdot\boldsymbol{P}(\boldsymbol{r},t)+\cdots$$
$$\rho(\boldsymbol{r},t)=\left\langle\sum_{j\in\text{free}} q_j\;\delta(\boldsymbol{r}-\boldsymbol{r}_j)\right\rangle+\left\langle\sum_{n\in\text{molecules}} q_n\;\delta(\boldsymbol{r}-\boldsymbol{r}_n)\right\rangle$$
$$\boldsymbol{P}=\left\langle\sum_{n\in\text{molecules}} \boldsymbol{p}_n\;\delta(\boldsymbol{r}-\boldsymbol{r}_n)\right\rangle\qquad q_n=\sum_{j\in n}q_j\quad \boldsymbol{p}_n=\sum_{j\in n}q_j\boldsymbol{r}_{jn}$$
$$\langle\boldsymbol{\vartheta}(\boldsymbol{r},t)\rangle=\boldsymbol{j}(\boldsymbol{r},t)+\frac{\partial \boldsymbol{P}(\boldsymbol{r},t)}{\partial t}+\nabla\times\boldsymbol{M}+\cdots$$
$$\boldsymbol{j}(\boldsymbol{r},t)(\boldsymbol{r},t)=\left\langle\sum_{j\in\text{free}} q_j\boldsymbol{v}_j\;\delta(\boldsymbol{r}-\boldsymbol{r}_j)\right\rangle+\left\langle\sum_{n\in\text{molecules}} q_n\boldsymbol{v}_n\;\delta(\boldsymbol{r}-\boldsymbol{r}_n)\right\rangle$$
$$\boldsymbol{M}=\left\langle\sum_{n\in\text{molecules}} \boldsymbol{m}_n\;\delta(\boldsymbol{r}-\boldsymbol{r}_n)\right\rangle\qquad \boldsymbol{m}_n=\sum_{j\in n}\frac{q_j}{2}(\boldsymbol{r}_{jn}\times \boldsymbol{v}_{jn})$$
由上即得宏观Maxwell方程
$$\nabla\cdot\boldsymbol{D}=\rho\qquad\nabla\times\boldsymbol{E}=-\frac{\partial \boldsymbol{B}}{\partial t}$$
$$\nabla\cdot\boldsymbol{B}=0\qquad\nabla\times\boldsymbol{H}=\frac{\partial \boldsymbol{D}}{\partial t}+\boldsymbol{j}$$
$$\boldsymbol{D}=\epsilon_0\boldsymbol{E}+\boldsymbol{P}\qquad\boldsymbol{H}=\displaystyle\frac{\boldsymbol{B}}{\mu_0}-\boldsymbol{M}$$