附录
磁场高斯定理的推导¶
本节中，我们基于毕奥-萨法尔定律推导磁场的高斯定理
如下图所示，激发磁场的电流元和要测量磁场的目标点相对于坐标原点 \(O\) 的位矢分别为 \(\vec{q}\) 和 \(\vec{r}\)，目标点相对于电流元的位矢 \(\vec{k} = \vec{r} - \vec{q}\)
根据毕奥-萨法尔定律
\[
\vec{B} = \frac{\mu_0}{4\pi} \int_V \frac{\vec{j}\big(\vec{q}\tm\big) \times \vec{k}}{k^3} \,\d{v}
= \frac{\mu_0}{4\pi} \int_V \vec{j}\big(\vec{q}\tm\big) \times \frac{\vec{r} - \vec{q}}{\abs{\vec{r} - \vec{q}\tm}^3} \,\d{v}
\]
\[
\begin{align}
\vec{B} & = \frac{\mu_0}{4\pi} \int_V \frac{\vec{j}\big(\vec{q}\tm\big) \times \vec{k}}{k^3} \,\d{v} \\
& = \frac{\mu_0}{4\pi} \int_V \vec{j}\big(\vec{q}\tm\big) \times \frac{\vec{r} - \vec{q}}{\abs{\vec{r} - \vec{q}\tm}^3} \,\d{v}
\end{align}
\]
这里需要明确的是，\(\vec{r}\) 和 \(\vec{q}\) 是两套不同的坐标，上式中对体积求积分是对坐标 \(\vec{q}\) 求积分，得到的磁感应强度为依赖 \(\vec{r}\) 的场函数 \(\vec{B}\big(\vec{r}\tm\big)\)，其散度
\[
\nabla \cdot \vec{B}
= \frac{\mu_0}{4\pi} \nabla \cdot \int_V \vec{j}\big(\vec{q}\tm\big) \times \frac{\vec{r} - \vec{q}}{\abs{\vec{r} - \vec{q}\tm}^3} \,\d{v}
= \frac{\mu_0}{4\pi} \int_V \nabla \cdot \sb{
\vec{j}\big(\vec{q}\tm\big) \times \frac{\vec{r} - \vec{q}}{\abs{\vec{r} - \vec{q}\tm}^3}
} \,\d{v}
\]
\[
\begin{align}
\nabla \cdot \vec{B}
& = \frac{\mu_0}{4\pi} \nabla \cdot \int_V \vec{j}\big(\vec{q}\tm\big) \times \frac{\vec{r} - \vec{q}}{\abs{\vec{r} - \vec{q}\tm}^3} \,\d{v} \\
& = \frac{\mu_0}{4\pi} \int_V \nabla \cdot \sb{
\vec{j}\big(\vec{q}\tm\big) \times \frac{\vec{r} - \vec{q}}{\abs{\vec{r} - \vec{q}\tm}^3}
} \,\d{v}
\end{align}
\]
由公式
\[
\nabla \cdot \rb{\vec{a} \times \vec{b}\tm} = \vec{b} \cdot \rb{\nabla \times \vec{a}\tm} - \vec{a} \cdot \rb{\nabla \times \vec{b}\tm}
\]
可得
\[
\nabla \cdot \rb{
\vec{j}\big(\vec{q}\tm\big) \times \frac{\vec{r} - \vec{q}}{\abs{\vec{r} - \vec{q}\tm}^3}
}
= \underbrace{\frac{\vec{r}-\vec{q}}{\abs{\vec{r}-\vec{q}\tm}^3} \cdot \sb{\nabla \times \vec{j}\big(\vec{q}\tm\big)}}_{\displaystyle \rb{1}}
-
\underbrace{\vec{j}\big(\vec{q}\tm\big) \cdot \rb{\nabla \times \frac{\vec{r} - \vec{q}}{\abs{\vec{r} - \vec{q}\tm}^3}}}_{\displaystyle \rb{2}}
\]
\[
\begin{align}
& \nabla \cdot \rb{
\vec{j}\big(\vec{q}\tm\big) \times \frac{\vec{r} - \vec{q}}{\abs{\vec{r} - \vec{q}\tm}^3}
}
= \underbrace{\frac{\vec{r}-\vec{q}}{\abs{\vec{r}-\vec{q}\tm}^3} \cdot \sb{\nabla \times \vec{j}\big(\vec{q}\tm\big)}}_{\displaystyle \rb{1}} \\
& -
\underbrace{\vec{j}\big(\vec{q}\tm\big) \cdot \rb{\nabla \times \frac{\vec{r} - \vec{q}}{\abs{\vec{r} - \vec{q}\tm}^3}}}_{\displaystyle \rb{2}}
\end{align}
\]
对 \(\vec{B}\) 求散度和旋度，\(\nabla\) 只对坐标 \(\vec{r}\) 求导，与坐标 \(\vec{q}\) 及其场函数无关。因此，上式中
- \(\nabla \times \vec{j}\rb{\vec{q}\tm} = \vec{0}\)，因此，上式中 \(\rb{1} = 0\)
-
在 \(\rb{2}\) 中，\(\displaystyle \frac{\vec{r}-\vec{q}}{\abs{\vec{r}-\vec{q}\tm}^3} = - \nabla \frac{1}{\abs{\vec{r}-\vec{q}\tm}}\)，因此
\[ \rb{2} = \vec{j}\rb{\vec{q}\tm} \cdot \nabla \times \rb{\nabla \frac{1}{\abs{\vec{r}-\vec{q}\tm}}} \]由于标量场的梯度是无旋场，上式中 \(\rb{2} = 0\)
综上所述，基于毕奥-萨法尔定律，磁感应强度的散度为零，即
\[
\nabla \cdot \vec{B} = 0
\]
另外，根据高斯定理，磁通量的积分形式
\[
\oint_S \vec{B} \cdot \d{\vec{s}} = \int_{V} \nabla \cdot \vec{B} = 0
\]
其中，\(S\) 为包围空间区域 \(V\) 的边缘曲面