电磁场数值分析与计算课程笔记
无PPT，按照板书整理，若有错误敬请指正。
目录
1.电磁场数值分析与计算01-场论
2.电磁场数值分析与计算02-Maxwell方程组
3.电磁场数值分析与计算03-电磁场数值分析的定解问题
4.电磁场数值分析与计算04-边界条件
5.电磁场数值分析与计算05-有限元方法介绍
6.电磁场数值分析与计算06-2D有限元分析
上一课的公式总结
\[Maxwell方程组\left\{ {\begin{array}{*{20}{c}} { {\bf{\nabla}} \cdot {\bf{D}} = \rho} \\ { {\bf{\nabla}} \cdot {\bf{B}} = 0} \\ { {\bf{\nabla}} \times {\bf{E}} = - \frac{\partial {\bf{B}}}{\partial t}} \\ {\bf{\nabla}} \times {\bf{H}} = {\bf{J}} + \frac{\partial {\bf{D}}}{\partial t} \end{array}} \right.\] 电磁性能关系式：\({\bf{D}}=\varepsilon{\bf{E}}\)，\({\bf{B}}=\mu{\bf{H}}\)，\({\bf{J}}=\sigma{\bf{E}}\)
电磁场控制方程的描述
电磁场数值分析的任务：求解一个与特定问题相联系的定解问题。
定解问题：在某一个确定区域内成立的微分方程+定解条件。
定解条件：
对于静态场而言，定解条件指的是微分方程中的未知函数在求解区域边界上满足的条件，又被称作边界条件。
对于时变场而言，定解条件指的是边界条件+初始条件。初始条件指的是整个区域未知函数在初始时刻的值。
工程电磁场问题（写出定解问题表达式）：
①选择什么物理量作为控制方程的未知函数，建立什么样的微分方程。
②四个方程，五个未知数，直接求解困难⇒引入位函数（电位、磁位）
电磁场中的位函数
矢量磁位
由描述磁的方程\({ {\bf{\nabla}} \cdot
{\bf{B}} =
0}\)与矢量微分算子的“旋度场无源”这一性质，可以将矢量\(\bf{B}\)替换为一个矢量的旋度。
定义矢量磁位\({\bf{B}} = {\bf{\nabla}} \times
{\bf{A}}\)。
标量电位
由磁生电的方程\({\bf{\nabla}} \times
{\bf{E}} = - \frac{\partial {\bf{B}}}{\partial t}\)
移项得\({\bf{\nabla}} \times {\bf{E}} +
\frac{\partial {\bf{B}}}{\partial t} = 0\)
将\(\bf{B}\)表示为矢量磁位得\({\bf{\nabla}} \times {\bf{E}} +
\frac{\partial}{\partial t} ({\bf{\nabla}} \times {\bf{A}}) =
0\)
交换偏微分与矢量微分运算次序，得\({\bf{\nabla}} \times {\bf{E}} + {\bf{\nabla}}
\times \frac{\partial {\bf{A}}}{\partial t} = 0\)
合并同类项得\({\bf{\nabla}} \times ({\bf{E}} +
\frac{\partial {\bf{A}}}{\partial t}) = 0\)
由矢量微分算子的性质“梯度场无旋”，可以将\({\bf{E}} + \frac{\partial {\bf{A}}}{\partial
t}\)替换为一个标量的梯度\({\bf{E}} +
\frac{\partial {\bf{A}}}{\partial t} =
-{\bf{\nabla}}\Phi\)。
整理得标量电位\({\bf{E}} = -{\bf{\nabla}}\Phi
-\frac{\partial {\bf{A}}}{\partial t}\)
注：所替换的标量的梯度前有负号，是因为沿电场线方向，电势逐渐降低。电场增加的方向与电势增加的方向相反，所以电场强度为电势梯度的负数。
场矢量的微分方程
三个假设
①所研究区域内媒质是线性、各向同性的（\(\varepsilon\)、\(\mu\)、\(\sigma\)为常量）
②不包含电源区（\({\bf{J_s}}=0\)，电流密度\(\bf{J}\)包括源电流密度\(\bf{J_s}\)与涡电流密度\(\bf{J_e}\)）
③自由电荷体密度为0（\(\rho=0\)，\({\bf{\nabla}} \cdot {\bf{D}} = 0\)）
\(\bf{B}\)的方程推导
从电生磁的方程出发\({\bf{\nabla}} \times
{\bf{H}} = {\bf{J}} + \frac{\partial {\bf{D}}}{\partial
t}\)①
两端同时取旋度\({\bf{\nabla}} \times
{\bf{\nabla}} \times {\bf{H}} = {\bf{\nabla}} \times {\bf{J}} +
{\bf{\nabla}} \times \frac{\partial {\bf{D}}}{\partial
t}\)②
交换运算次序\({\bf{\nabla}} \times
{\bf{\nabla}} \times {\bf{H}} = {\bf{\nabla}} \times {\bf{J}} +
\frac{\partial}{\partial t}({\bf{\nabla}} \times
{\bf{D}})\)③
代入关系式\({\bf{B}}=\mu{\bf{H}}\)，\({\bf{J}}=\sigma{\bf{E}}\)，\({\bf{D}}=\varepsilon{\bf{E}}\)，得\({\bf{\nabla}} \times \frac{1}{\mu}{\bf{\nabla}}
\times {\bf{B}} = {\bf{\nabla}} \times \sigma{\bf{E}} +
\frac{\partial}{\partial t}({\bf{\nabla}} \times
\varepsilon{\bf{E}})\)④
\(\varepsilon\)、\(\mu\)、\(\sigma\)为常量，可以整理为\({\bf{\nabla}} \times {\bf{\nabla}} \times {\bf{B}}
= \mu\sigma{\bf{\nabla}} \times {\bf{E}} +
\mu\varepsilon\frac{\partial}{\partial t}({\bf{\nabla}} \times
{\bf{E}})\)⑤
代入磁生电的方程\({\bf{\nabla}} \times
{\bf{E}} = - \frac{\partial {\bf{B}}}{\partial t}\)，得\({\bf{\nabla}} \times {\bf{\nabla}} \times {\bf{B}}
= -\mu\sigma\frac{\partial {\bf{B}}}{\partial t} -
\mu\varepsilon\frac{\partial^2 {\bf{B}}}{\partial t^2}\)⑥
由矢量恒等式\({\bf{\nabla}} \times
{\bf{\nabla}} \times {\bf{B}} = {\bf{\nabla}}({\bf{\nabla}} \cdot
{\bf{B}}) - {\bf{\nabla}}^2 {\bf{B}}\)与描述磁的方程\({\bf{\nabla}} \cdot {\bf{B}} = 0\)
得\(\bf{B}\)的方程式\({\bf{\nabla}}^2 {\bf{B}} -\mu\sigma\frac{\partial
{\bf{B}}}{\partial t} - \mu\varepsilon\frac{\partial^2
{\bf{B}}}{\partial t^2} =0\)⑦
\(\bf{E}\)的方程推导
从磁生电的方程出发\({\bf{\nabla}} \times
{\bf{E}} = - \frac{\partial {\bf{B}}}{\partial t}\)①
两端同时取旋度\({\bf{\nabla}} \times
{\bf{\nabla}} \times {\bf{E}} = - {\bf{\nabla}} \times \frac{\partial
{\bf{B}}}{\partial t}\)②
交换运算次序\({\bf{\nabla}} \times
{\bf{\nabla}} \times {\bf{E}} = - \frac{\partial}{\partial
t}({\bf{\nabla}} \times {\bf{B}})\)③
代入关系式\({\bf{B}}=\mu{\bf{H}}\)与电生磁的方程\({\bf{\nabla}} \times {\bf{H}} = {\bf{J}} +
\frac{\partial {\bf{D}}}{\partial t}\)
得\({\bf{\nabla}} \times {\bf{\nabla}} \times
{\bf{E}} = - \frac{\partial}{\partial t}[\mu({\bf{J}} + \frac{\partial
{\bf{D}}}{\partial t})] = -\mu(\frac{\partial {\bf{J}}}{\partial t} +
\frac{\partial^2 {\bf{D}}}{\partial t^2})\)④
代入关系式\({\bf{J}}=\sigma{\bf{E}}\)与\({\bf{D}}=\varepsilon{\bf{E}}\)
得\({\bf{\nabla}} \times {\bf{\nabla}} \times
{\bf{E}} = -\mu\sigma\frac{\partial {\bf{E}}}{\partial t} -
\mu\varepsilon\frac{\partial^2 {\bf{E}}}{\partial t^2}\)⑤
由描述电的方程与假设\(\rho=0\)知\({\bf{\nabla}} \cdot {\bf{D}} =
0\)，又由关系式\({\bf{D}}=\varepsilon{\bf{E}}\)知\({\bf{\nabla}} \cdot {\bf{E}} = 0\)⑥
由矢量恒等式\({\bf{\nabla}} \times
{\bf{\nabla}} \times {\bf{E}} = {\bf{\nabla}}({\bf{\nabla}} \cdot
{\bf{E}}) - {\bf{\nabla}}^2 {\bf{E}}\)与\({\bf{\nabla}} \cdot {\bf{E}} = 0\)
得\(\bf{E}\)的方程式\({\bf{\nabla}}^2 {\bf{E}} -\mu\sigma\frac{\partial
{\bf{E}}}{\partial t} - \mu\varepsilon\frac{\partial^2
{\bf{E}}}{\partial t^2} =0\)⑦
在\(\bf{B}\)的方程基础下代入关系式\({\bf{B}}=\mu{\bf{H}}\)，可以得到\(\bf{H}\)的方程：
\({\bf{\nabla}}^2 {\bf{H}}
-\mu\sigma\frac{\partial {\bf{H}}}{\partial t} -
\mu\varepsilon\frac{\partial^2 {\bf{H}}}{\partial t^2} =0\)
在\(\bf{E}\)的方程基础下代入关系式\({\bf{J}}=\sigma{\bf{E}}\)，可以得到\(\bf{J}\)的方程：
\({\bf{\nabla}}^2 {\bf{D}}
-\mu\sigma\frac{\partial {\bf{D}}}{\partial t} -
\mu\varepsilon\frac{\partial^2 {\bf{D}}}{\partial t^2} =0\)
在\(\bf{E}\)的方程基础下代入关系式\({\bf{D}}=\varepsilon{\bf{E}}\)，可以得到\(\bf{D}\)的方程：
\({\bf{\nabla}}^2 {\bf{J}}
-\mu\sigma\frac{\partial {\bf{J}}}{\partial t} -
\mu\varepsilon\frac{\partial^2 {\bf{J}}}{\partial t^2} =0\)
可以观察到五个方程在形式上完全相同，是齐次波动方程、单一矢量满足的微分方程（矢量方程）。
方程中电场和磁场解耦，但是求解困难，需要引入电磁位以便求解。
在没有电流与电荷的空间，\(\bf{B}\)方程可简化为\({\bf{\nabla}}^2 {\bf{B}} =
\mu_0\varepsilon_0\frac{\partial^2 {\bf{B}}}{\partial
t^2}\)
\(\bf{E}\)、\(\bf{H}\)、\(\bf{D}\)的方程同理，替换调方程中的\(\bf{B}\)即可。
若取真空磁导率\(\mu_0=4\pi\times
10^{-7}H/m\)，真空电容率\(\varepsilon_0=8.85\times
10^{-12}F/m\)则可计算出\(v=\frac{1}{\sqrt{\mu_0\varepsilon_0}}=3 \times
10^8
m/s\)，为真空中的光速，Maxwell方程组预言了光是一种电磁波。
静态场中电磁位的引入及对应的微分方程
静电场
静态场不随时间变化，静电场中有\({\bf{\nabla}} \times {\bf{E}} = 0\)
令\({\bf{E}} =
-{\bf{\nabla}}\Phi\)（标量电位）
由\({\bf{\nabla}} \cdot {\bf{D}} =
\rho\)与\({\bf{D}}=\varepsilon{\bf{E}}\)得\({\bf{\nabla}} \cdot \varepsilon{\bf{E}} =
{\bf{\nabla}} \cdot \varepsilon(-{\bf{\nabla}}\Phi) =
\rho\)
进而得\({\bf{\nabla}}^2 \Phi =
-\frac{\rho}{\varepsilon}\)，该式被称作静电场电位的泊松方程。
在体电荷为零的区域，有\({\bf{\nabla}}^2 \Phi = 0\)，该式被称作静电场电位的拉普拉斯方程。
静磁场
在电流密度为0的区域（\({\bf{J_s}}=0\)），有\({\bf{\nabla}} \times {\bf{H}} = 0\)
引入标量磁位\({\bf{H}} =
-{\bf{\nabla}}\Phi_m\)
由\({\bf{\nabla}} \cdot {\bf{B}} =
0\)与\({\bf{B}} =
\mu{\bf{H}}\)得\({\bf{\nabla}}^2 \Phi_m
= 0\)，该式被称作静磁场中标量磁位的拉普拉斯方程。
当有电流存在时，\({\bf{\nabla}} \times
{\bf{H}} = {\bf{J_s}}\)
由\({\bf{\nabla}} \cdot {\bf{B}} =
0\)引入矢量磁位\({\bf{B}} =
{\bf{\nabla}} \times {\bf{A}}\)
又由\({\bf{B}}=\mu{\bf{H}}\)得\({\bf{\nabla}} \times {\bf{\nabla}} \times {\bf{A}}
= \mu{\bf{J_s}}\)
由矢量恒等式\({\bf{\nabla}} \times
{\bf{\nabla}} \times {\bf{A}} = {\bf{\nabla}}({\bf{\nabla}} \cdot
{\bf{A}}) - {\bf{\nabla}}^2 {\bf{A}}\)与库伦规范\({\bf{\nabla}} \cdot {\bf{A}} = 0\)
得\({\bf{\nabla}}^2 {\bf{A}} =
-\mu{\bf{J_s}}\)，该式被称作静磁场中矢量磁位的泊松方程。
涡流场中电磁位的引入及对应的微分方程
时变电磁场中存在导电媒质⇒涡流
处理工程问题，时变场频率较低（\(f<10^{10}Hz\)），可以忽略电生磁方程中的\(\frac{\partial {\bf{D}}}{\partial
t}\)，即\({\bf{\nabla}} \times {\bf{H}}
= {\bf{J}}\)①
注：这里的电流密度\(\bf{J}=\bf{J_s}+\bf{J_e}\)，其中涡电流密度\(\bf{J_e}\)满足公式\({\bf{J_e}}=\sigma{\bf{E}}\)
由描述磁的公式\({\bf{\nabla}} \cdot {\bf{B}} =
0\)引入矢量磁位\({\bf{B}} =
{\bf{\nabla}} \times {\bf{A}}\)②
将②代入磁生电公式\({\bf{\nabla}} \times
{\bf{E}} = - \frac{\partial {\bf{B}}}{\partial t}\)，得\({\bf{\nabla}} \times {\bf{E}} = -
\frac{\partial}{\partial t}({\bf{\nabla}} \times
{\bf{A}})\)
交换微分运算次序并整理得到\({\bf{\nabla}}
\times ({\bf{E}} + \frac{\partial {\bf{A}}}{\partial t}) =
0\)③
引入标量电位\(\Phi\)，得\({\bf{E}} + \frac{\partial {\bf{A}}}{\partial t} =
-{\bf{\nabla}}\Phi\)，即\({\bf{E}} =
-\frac{\partial {\bf{A}}}{\partial t} -{\bf{\nabla}}\Phi\)④
则涡电流密度可表示为\({\bf{J_e}}=\sigma{\bf{E}}=-\sigma{\bf{\nabla}}\Phi-\sigma\frac{\partial
{\bf{A}}}{\partial t}\)⑤
由②与\({\bf{B}}=\mu{\bf{H}}\)得\({\bf{H}} = \frac{1}{\mu}{\bf{\nabla}} \times
{\bf{A}}\)⑥
将③⑤⑥代入①，得\({\bf{\nabla}} \times
{\bf{\nabla}} \times {\bf{A}} = \mu\bf{J_s} -\mu\sigma{\bf{\nabla}}\Phi
- \mu\sigma\frac{\partial {\bf{A}}}{\partial t}\)⑦
由矢量恒等式\({\bf{\nabla}} \times
{\bf{\nabla}} \times {\bf{A}} = {\bf{\nabla}}({\bf{\nabla}} \cdot
{\bf{A}}) - {\bf{\nabla}}^2 {\bf{A}}\)得\({\bf{\nabla}}({\bf{\nabla}} \cdot {\bf{A}}) -
{\bf{\nabla}}^2 {\bf{A}} + \mu\sigma{\bf{\nabla}}\Phi +
\mu\sigma\frac{\partial {\bf{A}}}{\partial t} =
\mu\bf{J_s}\)⑧
由描述电的公式\({\bf{\nabla}} \cdot {\bf{D}} =
\rho\)与\({\bf{D}}=\varepsilon{\bf{E}}\)，得\({\bf{\nabla}} \cdot {\bf{E}} =
\frac{\rho}{\varepsilon}\)
代入④可得\({\bf{\nabla}} \cdot
(-\frac{\partial {\bf{A}}}{\partial t} -{\bf{\nabla}}\Phi) =
\frac{\rho}{\varepsilon}\)
交换微分运算次序并整理得到\({\bf{\nabla}}^2
\Phi + \frac{\partial}{\partial t} ({\bf{\nabla}} \cdot {\bf{A}}) =
-\frac{\rho}{\varepsilon}\)⑨
为了保证\({\bf{A}}\)的唯一性，使⑧和⑨解耦，定义\({\bf{\nabla}} \cdot {\bf{A}} =
-\mu\sigma\Phi\)（电导率规范）
得到涡流场中电磁位满足的非齐次微分方程：
\({\bf{\nabla}}^2 {\bf{A}} -
\mu\sigma\frac{\partial {\bf{A}}}{\partial t} =
-\mu\bf{J_s}\)⑩
\({\bf{\nabla}}^2 \Phi -
\mu\sigma\frac{\partial\Phi}{\partial t} =
-\frac{\rho}{\varepsilon}\)⑪
高频电磁场中电磁位的引入及微分方程
假设研究区域为线性，各向同性，无导电媒质。
由电生磁的方程\({\bf{\nabla}} \times {\bf{H}}
= {\bf{J}} + \frac{\partial {\bf{D}}}{\partial t}\)①
考虑高斯磁场方程\({\bf{\nabla}} \cdot {\bf{B}}
= 0\)②
引入矢量磁位\({\bf{B}} = {\bf{\nabla}} \times
{\bf{A}}\)③
代入磁生电方程\({\bf{\nabla}} \times {\bf{E}}
= - \frac{\partial {\bf{B}}}{\partial t}\)④
\({\bf{\nabla}} \times {\bf{E}} = -
\frac{\partial {\bf{B}}}{\partial t} = - \frac{\partial ({\bf{\nabla}}
\times {\bf{A}})}{\partial t} = - {\bf{\nabla}} \times \frac{\partial
{\bf{A}}}{\partial t}\)⇒\({\bf{\nabla}}
\times ({\bf{E}} + \frac{\partial {\bf{A}}}{\partial
t})=0\)⑤
引入标量电位\({\bf{E}} = -
\frac{\partial{\bf{A}}}{\partial t} - {\bf{\nabla}}\Phi\)⑥
将关系式\({\bf{B}}=\mu{\bf{H}}\)，\({\bf{D}}=\varepsilon{\bf{E}}\)以及③与⑥代入①
得到\({\bf{\nabla}} \times ({\bf{\nabla}}
\times {\bf{A}}) + \mu\varepsilon\frac{\partial}{\partial
t}({\bf{\nabla}}\Phi) + \mu\varepsilon\frac{\partial^2
{\bf{A}}}{\partial t^2} = \mu {\bf{J_s}}\)⑦
由矢量恒等式\({\bf{\nabla}} \times
{\bf{\nabla}} \times {\bf{A}} = {\bf{\nabla}}({\bf{\nabla}} \cdot
{\bf{A}}) - {\bf{\nabla}}^2 {\bf{A}}\)
得到\({\bf{\nabla}}({\bf{\nabla}} \cdot
{\bf{A}}) - {\bf{\nabla}}^2 {\bf{A}} +
\mu\varepsilon\frac{\partial}{\partial t}({\bf{\nabla}}\Phi) +
\mu\varepsilon\frac{\partial^2 {\bf{A}}}{\partial t^2} = \mu
{\bf{J_s}}\)⑧
将关系式\({\bf{D}}=\varepsilon{\bf{E}}\)以及③与⑥代入高斯电场定律
\({\bf{\nabla}} \cdot {\bf{D}} =
\rho\)⇒\({\bf{\nabla}} \cdot
\varepsilon{\bf{E}} = \rho\)⇒\({\bf{\nabla}} \cdot
(-\frac{\partial{\bf{A}}}{\partial t} - {\bf{\nabla}}\Phi) =
\frac{\rho}{\varepsilon}\)⇒\({\bf{\nabla}}^2 \Phi + \frac{\partial}{\partial
t}({\bf{\nabla}} \cdot {\bf{A}}) =
-\frac{\rho}{\varepsilon}\)⑨
定义\({\bf{A}}\)的散度\({\bf{\nabla}} \cdot {\bf{A}} =
-\mu\varepsilon\frac{\partial \Phi}{\partial
t}\)（洛伦兹规范）
由⑧得到\({\bf{\nabla}}^2 {\bf{A}} -
\mu\varepsilon\frac{\partial^2 {\bf{A}}}{\partial t^2} = -\mu
{\bf{J_s}}\)⑩
由⑨得到\({\bf{\nabla}}^2 \Phi -
\mu\varepsilon\frac{\partial^2 \Phi}{\partial t^2} =
-\frac{\rho}{\varepsilon}\)⑪
通过引入洛伦兹规范，电场与磁场实现了解耦，解耦后的方程被称作达朗贝尔方程。
矢量磁位的物理意义
计算磁通\(\Phi\)
由斯托克斯定理\(\Phi=\int_S{
{\bf{B}}d{\bf{s}}}=\int_S{({\bf{\nabla}}\times{\bf{A}})d{\bf{s}}}=\oint_l{
{\bf{A}}d{\bf{l}}}\)
对于二维场\(\bf{A}\)只有\(A_z\)分量，有\(\Phi=\oint_l{
{\bf{A}}d{\bf{l}}}=(A_{1z}-A_{2z})L\)
使用\(\bf{A}\)而非\(\bf{B}\)计算磁通，可以确保计算的准确性。
计算感生电动势
\(e=-\frac{\partial \Phi}{\partial t}=-\frac{\partial}{\partial t}\oint_l{ {\bf{A}}d{\bf{l}}}=-\oint_l{\frac{\partial {\bf{A}}}{\partial t}d{\bf{l}}}\)
画二维磁场（证明磁力线是等A线）
二维磁场如下图所示，其拥有\(B_x\)、\(B_y\)、\(A_z\)分量。\({\bf{B}}\times d{\bf{l}}=0\)
\({\bf{B} } = {\bf{\nabla} } \times {\bf{A} } = \left| {\begin{array}{*{20}{c}}{\bf{i}}&{\bf{j}}&{\bf{k}} \\ {\frac{\partial }{ {\partial x}}}&{\frac{\partial }{ {\partial y}}}&{\frac{\partial}{ {\partial z}}} \\ {0}&{0}&{A_z} \end{array}} \right| = \frac{\partial A_z}{\partial y}{\bf{i}} - \frac{\partial A_z}{\partial x}{\bf{j}} = B_x{\bf{i}} + B_y{\bf{j}}\)
\(d{\bf{l}}=dx{\bf{i}}+dy{\bf{j}}\)
\({\bf{B}}\times d{\bf{l}} = \left| {\begin{array}{*{20}{c}}{\bf{i}}&{\bf{j}}&{\bf{k}} \\ {\frac{\partial A_z}{ {\partial y}}}&{\frac{\partial A_z}{ {\partial x}}}&{0} \\ {dx}&{dy}&{0} \end{array}} \right| = 0{\bf{i}} + 0{\bf{j}} + (\frac{\partial A_z}{ {\partial y}}dy+\frac{\partial A_z}{ {\partial x}}dx){\bf{k}} =0\)
由\(\bf{A}\)只有\(A_z\)分量，可知\(d{\bf{A}}=\frac{\partial A}{ {\partial x}}dx+\frac{\partial A}{ {\partial y}}dy=0\)
即A是常量，得证二维磁场的磁力线是等A线。
图1
证明磁力线是等A线
计算磁场能量
磁场能量密度\(\omega_m=\frac{1}{2}{\bf{H}}{\bf{B}}\)
由\({\bf{B} } = {\bf{\nabla} } \times {\bf{A}
}\)与\({\bf{\nabla}} \times {\bf{H}} =
{\bf{J}}\)可知
\(\omega_m=\frac{1}{2}{\bf{H}}{\bf{B}}=\frac{1}{2}{\bf{H}}\cdot({\bf{\nabla}}\times{\bf{A}})=\frac{1}{2}{\bf{\nabla}}\cdot({\bf{A}}\times{\bf{H}})+\frac{1}{2}{\bf{A}}\cdot({\bf{\nabla}}\times{\bf{H}})=\frac{1}{2}{\bf{\nabla}}\cdot({\bf{A}}\times{\bf{H}})+\frac{1}{2}{\bf{A}}\cdot{\bf{J}}\)
注：使用了运算公式\({\bf{\nabla}} \cdot
({\bf{A}} \times {\bf{B}}) = {\bf{B}} \cdot ({\bf{\nabla}} \times
{\bf{A}}) - {\bf{A}} \cdot ({\bf{\nabla}} \times
{\bf{B}})\)
对整个空间积分，得到磁场总能量\(W_m=\int_V{\omega_mdV}=\frac{1}{2}\int_V{
{\bf{\nabla}}\cdot({\bf{A}}\times{\bf{H}})dV}+\frac{1}{2}\int_V{
{\bf{A}}\cdot{\bf{J}}dV}\)
由散度定理，得等式右侧第一项\(\frac{1}{2}\int_V{
{\bf{\nabla}}\cdot({\bf{A}}\times{\bf{H}})dV}=\frac{1}{2}\oint_S{({\bf{A}}\times{\bf{H}})dS}\)
当\(R \to \infty\)时，\(A \propto \frac{1}{R}\)，\(H \propto \frac{1}{R^2}\)，\(S \propto R^2\)
可知\({\bf{A}}\times{\bf{H}}dS \propto
\frac{1}{R}\)，\(\frac{1}{2}\oint_S{({\bf{A}}\times{\bf{H}})dS} \to
0\)
得\(W_m=\frac{1}{2}\int_V{
{\bf{A}}\cdot{\bf{J}}dV}\)