[物理学与PDEs]第1章第8节 静电场和静磁场 8.3 静磁场

1. 静磁场: 由稳定电流形成的磁场.

2. 此时, Maxwell 方程组为 $$\beex \bea \Div{\bf D}&=\rho_f,\\ \rot {\bf E}&={\bf 0},\\ \Div{\bf B}&=0,\\ \rot{\bf H}&={\bf j}_f. \eea \eeex$$ 电荷守恒律方程: $$\bex \Div{\bf j}_f=0. \eex$$

3. 矢势法处理静磁场

(1) $$\bex \Div{\bf B}=0\ra \exists\ {\bf A},\st \Div{\bf A}=0,\ \rot{\bf A}={\bf B}. \eex$$ 称 ${\bf A}$ 为静磁场的矢势.

(2) $$\beex \bea {\bf j}_f&=\rot{\bf H}=\rot\sex{\cfrac{1}{\mu}\rot{\bf A}}\\ &=\n\cfrac{1}{\mu}\times\rot{\bf A}+\cfrac{1}{\mu}\rot\rot{\bf A}\\ &=\n\cfrac{1}{\mu}\times\rot{\bf A}-\cfrac{1}{\mu}\lap{\bf A}. \eea\eeex$$ 故 ${\bf A}$ 满足主部分离的二阶椭圆型方程组.

(3) 当 $\mu$ 为常数时, $$\bex -\lap{\bf A}=\mu{\bf j}_f\ra {\bf A}=\cfrac{\mu}{4\pi}\int_\Omega \cfrac{{\bf j}_f(P')\rd V_{P'}}{r_{P'P}}. \eex$$ 电磁场能量密度 $$\beex \bea \cfrac{1}{2}{\bf B}\cdot{\bf H} &=\cfrac{1}{2}\rot{\bf A}\cdot {\bf H}\\ &=\cfrac{1}{2}\Div({\bf A}\cdot{\bf H})+\cfrac{1}{2}{\bf A}\cdot\rot{\bf H}\\ &=\cfrac{1}{2}\Div({\bf A}\cdot{\bf H})+\cfrac{1}{2}{\bf A}\cdot {\bf j}_f; \eea \eeex$$ 电磁场能量 $$\bex U_{e,m}=\cfrac{1}{2}\int_\Omega {\bf A}\cdot{\bf j}_f\rd V. \eex$$

4. 标势法处理静磁场

(1) $$\beex \bea \Div{\bf j}_f=0&\ra \exists\ {\bf F},\st \Div{\bf F}=0,\ \rot{\bf F}={\bf j}_f\\ &\ra \rot ({\bf H}-{\bf F})={\bf 0}\\ &\ra \exists\ \phi,\st {\bf H}-{\bf F}=-\n\phi. \eea \eeex$$ 称 $\phi$ 为静磁场的标势. 而有 $$\beex \bea {\bf B}&=\mu {\bf H}=\mu({\bf F}-\n \phi),\\ -\Div(\mu \n\phi)&=-\Div(\mu {\bf F})=-\n \mu\cdot {\bf F}\\ &=0\quad\sex{\mu\mbox{ 为常数时}}. \eea \eeex$$

(2) 交界面条件 $$\beex \bea &\quad 0=[{\bf H}]\times{\bf n}=[{\bf F}-\n\phi]\times {\bf n}\\ &\ra [\phi]=0\quad\sex{[{\bf B}]\cdot{\bf n}=0\ra F\mbox{ 连续}};\\ &\quad 0=[{\bf B}]\cdot{\bf n}=[\mu {\bf F}-\mu \n\phi]\cdot{\bf n}\\ &\ra \sez{\mu\cfrac{\p \phi}{\p n}}=[\mu] {\bf F}\cdot {\bf n}. \eea \eeex$$

(3) 电磁能量密度 $$\beex \bea \cfrac{1}{2}{\bf B}\cdot{\bf H} &=\cfrac{1}{2}\mu\sex{{\bf F}-\n\phi}\cdot \sex{{\bf F}-\n\phi}\\ &=\cfrac{1}{2}\mu |{\bf F}|^2-\mu {\bf F}\cdot\n\phi+\cfrac{1}{2}\mu |\n\phi|^2\\ &=\cfrac{1}{2}\mu\sex{|{\bf F}|^2+|\n\phi|^2} -\Div(\mu{\bf F}\phi)+\phi\Div(\mu {\bf F})\\ &=\cfrac{1}{2}\mu\sex{|{\bf F}|^2+|\n\phi|^2} -\Div(\mu{\bf F}\phi)+\phi {\bf F}\cdot\n\nu; \eea \eeex$$ 电磁场能量 $$\beex \bea U_{e,m}&=\cfrac{1}{2}\int_\Omega {\bf B}\cdot{\bf H}\rd V\\ &=\cfrac{1}{2}\int_\Omega \mu|{\bf F}|^2+2\phi {\bf F}\cdot\n\mu+\mu|\n\phi|^2\rd V. \eea \eeex$$

5. 静电场与静磁场的比较 $$\bex \ba{rl} \mbox{静电场 }{\bf E}&\quad\quad\mbox{静磁场 }{\bf B}\\ ----------&\quad\quad----------\\ \mbox{无旋场 (纵场)}&\quad\quad\mbox{无源场 (横场)}\\ {\bf E}=-\n \phi&\quad\quad{\bf B}=\rot{\bf A}\quad(\Div{\bf A}=0)\\ -\Div(\ve \n\phi)=\rho_f &\quad\quad\rot\sex{\cfrac{1}{\mu}\rot{\bf A}}={\bf j}_f\\ -\lap\phi= \cfrac{1}{\ve}\rho_f\ \sex{\ve\mbox{ 为常数时 }}&\quad\quad-\lap{\bf A}=\mu{\bf j}_f\ \sex{\mu\mbox{ 为常数时}}\\ \mbox{能量 }U=\cfrac{1}{2}\int_\Omega \rho_f\phi\rd V&\quad\quad\mbox{能量 }U=\cfrac{1}{2}\int_\Omega {\bf j}_f\cdot{\bf A}\rd V\\ ----------&\quad\quad---------- \ea \eex$$