[物理学与PDEs]第1章第4节 电磁能量和电磁动量, 能量、动量守恒与转化定律 4.1 电磁能量, 能量守恒与转化定律

1. 设真空中运动的带电体的总机械能为 $U_m$, 则 $$\beex \bea \cfrac{\rd U_m}{\rd t} &=\int_\Omega {\bf f}\cdot {\bf v} \rd V\quad\sex{\mbox{功能关系}}\\ &=\int_\Omega (\rho {\bf e}+\rho {\bf v}\times {\bf B})\cdot{\bf v}\rd V\quad\sex{Lorentz\mbox{ 力}}\\ &=\int_\Omega {\bf j}\cdot{\bf E}\rd V\\ &=\int_\Omega \sex{ -\ve_0\cfrac{\p{\bf E}}{\p t}+\cfrac{1}{\mu_0}\rot{\bf B} }\cdot{\bf E}\rd V\quad\sex{Maxwell\mbox{ 方程组}}\\ &=-\cfrac{1}{2}\cfrac{\rd}{\rd t}\int_\Omega \ve_0E^2\rd V +\int_\Omega \cfrac{1}{\mu_0} \rot{\bf B}\cdot{\bf E}\rd V\\ &=-\cfrac{1}{2}\cfrac{\rd}{\rd t} \int_\Omega \ve_0E^2\rd V +\int_\Omega \cfrac{1}{\mu_0}\rot{\bf E}\cdot{\bf B}\rd V -\int_{\p\Omega} {\bf S}\cdot{\bf n}\rd S\\ &\quad\sex{ \Div({\bf E}\times {\bf B})=\rot{\bf E}\cdot{\bf B}-{\bf E}\cdot\rot{\bf B}\atop {\bf S}=\cfrac{1}{\mu_0}{\bf E}\times{\bf B}:\ Poynting\mbox{ 向量} }\\ &=-\cfrac{1}{2}\cfrac{\rd}{\rd t}\int_\Omega \sex{\ve_0E^2+\cfrac{1}{\mu_0}B^2}\rd V\quad\sex{Maxwell\mbox{ 方程组}}. \eea \eeex$$

2. 上式即代表电磁能量守恒与转化定律.