电磁场的守恒与对称性
$$\dd W_{机械}=\rho\dd V\boldsymbol{E}\cdot\dd\boldsymbol{l}=\boldsymbol{E}\cdot\boldsymbol{j}\dd V\dd t\\
\frac{\dd W_{机械}}{\dd t}=\int_V\boldsymbol{E}\cdot\boldsymbol{j}\dd V\\
\boldsymbol{E}\cdot\boldsymbol{j}=\boldsymbol{E}\cdot(\frac{1}{\mu_0}\nabla\times\boldsymbol{B}-\varepsilon_0\frac{\partial \boldsymbol{E}}{\partial t})-\boldsymbol{B}\cdot(\frac{1}{\mu_0}\nabla\times\boldsymbol{E}+\frac{1}{\mu_0}\frac{\partial \boldsymbol{B}}{\partial t})\\
=-\nabla\cdot(\boldsymbol{E}\times\boldsymbol{H})-\frac{\partial}{\partial t}\left[\frac{1}{2}(\varepsilon_0E^2+\frac{1}{\mu_0}B^2)\right]\\
\overset{定义}{=}-\nabla\cdot\boldsymbol{S}_P-\frac{\partial w}{\partial t}\\
\dd W_{机械}=-\left[\oint_S\boldsymbol{S}_P\cdot\dd\boldsymbol{S}\right]\dd t-\dd\left[\int_Vw\dd V\right]\\
\frac{\dd \boldsymbol{G}_{机械}}{\dd t}=\int_V\boldsymbol{f}\dd V=\int_V(\rho\boldsymbol{E}+\boldsymbol{j}\times\boldsymbol{B})\dd V\\
\boldsymbol{f}=\rho\boldsymbol{E}+\boldsymbol{j}\times\boldsymbol{B}=\varepsilon_0(\nabla\cdot\boldsymbol{E})\boldsymbol{E}+(\frac{1}{\mu_0}\nabla\times\boldsymbol{B}-\varepsilon_0\frac{\partial \boldsymbol{E}}{\partial t})\times\boldsymbol{B}\\
+\frac{1}{\mu_0}(\nabla\cdot\boldsymbol{B})\boldsymbol{B}+(\varepsilon_0\nabla\times\boldsymbol{E}+\varepsilon_0\frac{\partial \boldsymbol{B}}{\partial t})\times\boldsymbol{E}\\
=\nabla\cdot(\varepsilon_0\boldsymbol{EE}+\frac{1}{\mu_0}\boldsymbol{BB})-\nabla\cdot\left[\frac{1}{2}(\varepsilon_0 E^2+\frac{1}{\mu_0}B^2)\mathbb{I}\right]-\frac{\partial}{\partial t}\frac{\boldsymbol{S}_P}{c^2}\\
\overset{定义}{=}-\nabla\cdot\mathbb{T}-\frac{\partial\boldsymbol{g}}{\partial t}\\
\dd \boldsymbol{G}_{机械}=-\left[\oint_S\mathbb{T}\cdot\dd\boldsymbol{S}\right]\dd t-\dd\left[\int_V\boldsymbol{g}\dd V\right]\\
\boldsymbol{G}_{电磁}\overset{单粒子}{=}\int_\infty\boldsymbol{g}\dd V’=\varepsilon_0\int_\infty\boldsymbol{E}\times(\nabla\times\boldsymbol{A})\dd V’\\
=\varepsilon_0\int_\infty[\nabla(\boldsymbol{E}\cdot\boldsymbol{A})-\boldsymbol{A}\times(\nabla\times\boldsymbol{E})-\boldsymbol{E}\cdot(\nabla\boldsymbol{A})-\boldsymbol{A}\cdot(\nabla\boldsymbol{E})]\dd V’\\
=\varepsilon_0\int_\infty[\nabla(\boldsymbol{E}\cdot\boldsymbol{A})-\nabla\cdot(\boldsymbol{EA}+\boldsymbol{AE})+(\nabla\cdot\boldsymbol{E})\boldsymbol{A}+(\nabla\cdot\boldsymbol{A})\boldsymbol{E}]\dd V’\\
=\varepsilon_0[\int_\infty\boldsymbol{E}\cdot\boldsymbol{A}\dd\boldsymbol{S’}-\int_\infty(\boldsymbol{EA}+\boldsymbol{AE})\cdot\dd\boldsymbol{S’}+\int_\infty\frac{q\delta(\boldsymbol{r}-\boldsymbol{r’})}{\varepsilon_0}\boldsymbol{A}\dd V’]\\
=q\boldsymbol{A}\\
\boldsymbol{G}\overset{单粒子}{=}m\boldsymbol{v}+q\boldsymbol{A}\\
\frac{\dd \boldsymbol{L}_{机械}}{\dd t}=\int_V\boldsymbol{r}\times(\rho\boldsymbol{E}+\boldsymbol{j}\times\boldsymbol{B})\dd V\\
=-\int_V\boldsymbol{r}\times(\nabla\cdot\mathbb{T}+\frac{\partial \boldsymbol{g}}{\partial t})\dd V\\
=\int_V\left[\nabla\cdot(\mathbb{T}\times\boldsymbol{r})-\frac{\partial (\boldsymbol{r}\times\boldsymbol{g})}{\partial t}\right]\dd V\\
\dd \boldsymbol{L}_{机械}=\left[\oint_S\mathbb{T}\times\boldsymbol{r}\cdot\dd\boldsymbol{S}\right]\dd t-\dd\left[\int_V\boldsymbol{r}\times\boldsymbol{g}\dd V\right]\\
\overset{定义}{=}-\left[\oint_S\mathbb{M}\cdot\dd\boldsymbol{S}\right]\dd t-\dd\left[\int_V\boldsymbol{l}\dd V\right]$$
$$\boldsymbol{E}\cdot\boldsymbol{j}_t=\boldsymbol{E}\cdot(\frac{1}{\mu_0}\nabla\times\boldsymbol{B}-\varepsilon_0\frac{\partial \boldsymbol{E}}{\partial t})-\boldsymbol{B}\cdot(\frac{1}{\mu_0}\nabla\times\boldsymbol{E}+\frac{1}{\mu_0}\frac{\partial \boldsymbol{B}}{\partial t})\\
=-\frac{1}{\mu_0}\nabla\cdot(\boldsymbol{E}\times\boldsymbol{B})-\frac{\partial}{\partial t}\left[\frac{1}{2}(\varepsilon_0E^2+\frac{1}{\mu_0}B^2)\right]\\
\boldsymbol{E}\cdot(\boldsymbol{j}_p+\boldsymbol{j}_m)=\boldsymbol{E}\cdot(\frac{\partial\boldsymbol{P}}{\partial t}+\nabla\times\boldsymbol{M})\\
=-\nabla\cdot(\boldsymbol{E}\times\boldsymbol{M})-\frac{\partial}{\partial t}\left[\frac{1}{2}(EP-BM)\right]\\
(\boldsymbol{E}\cdot\boldsymbol{j})_{机械}=-\nabla\cdot(\boldsymbol{E}\times\boldsymbol{H})-\frac{\partial}{\partial t}\left[\frac{1}{2}(ED+BH)\right]\\
w_{e,m}=\frac{1}{2}(DE+BH)\qquad\boldsymbol{S}_{e,m}=\boldsymbol{E}\times\boldsymbol{H}\\
\boldsymbol{f}_t=\nabla\cdot(\varepsilon_0\boldsymbol{EE}+\frac{1}{\mu_0}\boldsymbol{BB})-\nabla\cdot\left[\frac{1}{2}(\varepsilon_0 E^2+\frac{1}{\mu_0}B^2)\mathbb{I}\right]-\varepsilon_0\frac{\partial(\boldsymbol{E}\times\boldsymbol{B})}{\partial t}\\
\rho_p\boldsymbol{E}+(\boldsymbol{j}_p+\boldsymbol{j}_m)\times \boldsymbol{B}=-(\nabla\cdot\boldsymbol{P})\boldsymbol{E}+(\frac{\partial\boldsymbol{P}}{\partial t}+\nabla\times\boldsymbol{M})\times\boldsymbol{B}\\
+(\nabla\cdot\boldsymbol{B})\boldsymbol{M}-(\frac{\partial\boldsymbol{B}}{\partial t}+\nabla\times\boldsymbol{E})\times\boldsymbol{P}\\
=\nabla\cdot(\boldsymbol{BM}-\boldsymbol{PE})+\nabla\cdot\left[\frac{1}{2}(PE-BM)\mathbb{I}\right]-\frac{\partial(\boldsymbol{P}\times\boldsymbol{B})}{\partial t}\\
\boldsymbol{f}_{机械}=\nabla\cdot(\boldsymbol{DE}+\boldsymbol{BH})-\nabla\cdot\left[\frac{1}{2}(DE+BH)\mathbb{I}\right]+\frac{\partial(\boldsymbol{D}\times\boldsymbol{B})}{\partial t}\\
\\
\mathbb{T}_{e,m}=\frac{1}{2}\left(DE+BH\right)\mathbb{I}-\boldsymbol{DE}-\boldsymbol{BH}\qquad\boldsymbol{g}_{e,m}=\frac{\boldsymbol{S}_{e,m}}{c^2_{介质}}
$$