设 $\Omega$ 为一有界区域, 外部为理想导体 $(\sigma=+\infty)$, 则 $\Omega$ 中电磁场满足 Maxwell 方程组 $$\beex \bea \ve\cfrac{\p{\bf E}}{\p t}-\cfrac{1}{\mu}\rot{\bf B}&=-{\bf j},\\ \cfrac{\p{\bf B}}{\p t}+\rot{\bf E}&={\bf 0},\\ \Div{\bf E}&=\cfrac{\rho}{\ve},\\ \Div{\bf B}&=0. \eea \eeex$$ 电荷守恒律方程为 $$\bex \cfrac{\p\rho}{\p t}+\Div{\bf j}=0. \eex$$ 而边界上条件为 $$\bex {\bf E}\times{\bf n}=0,\quad \cfrac{\p}{\p t}({\bf B}\cdot {\bf n})=0,\quad\mbox{on }\p \Omega. \eex$$ 初始条件为 $$\bex {\bf E}={\bf E}_0,\quad {\bf B}={\bf B}_0,\quad\mbox{on }\sed{t=0}\times\Omega \eex$$ 须满足相容性条件: $$\beex \bea \Div{\bf E}_0=\cfrac{\rho_0}{\ve_0},&\quad\rho_0=\rho(0,x,y,z),\\ \Div{\bf B}_0=0,&\\ {\bf E}_0\times{\bf n}={\bf 0},&\quad\mbox{on }\p\Omega. \eea \eeex$$
