# 静电场的散度与旋度

\begin{align*} \newcommand{\dif}{\mathop{}\!\mathrm{d}} \newcommand{\p}{\partial} \end{align*}

# 预备知识

## 立体角

$\dif \Omega = \frac{\dif \vec{S}\cdot \vec{a}_R}{R^2}$

# 高斯定理

$\vec{E} = \frac{q}{4\pi\varepsilon_0} \frac{\vec{a}_R}{R^2}$

$\oint_S \vec{E}\cdot\dif \vec{S} = \frac{q}{4\pi\varepsilon_0} \oint_S \frac{\vec{a}_R}{R^2}\cdot\dif\vec{S}=\frac{q}{4\pi\varepsilon_0}\oint_S \dif \Omega\\ =\begin{cases} q/\varepsilon_0 & q在S内\\ 0 & q在S外 \end{cases}$

\begin{align} \oint_S \vec{E}\cdot\dif\vec{S}&=\frac{\sum q}{\varepsilon_0}\\ &\Downarrow\\ \int_V \nabla\cdot\vec{E}\dif V&=\frac{1}{\varepsilon_0}\int_V\rho\dif V\\ &\Downarrow\\ \nabla\cdot\vec{E}&=\rho/\varepsilon_0 \end{align}

$\oint_S \vec{D}\cdot\dif \vec{S} \oint_S \varepsilon_0\vec{E}\cdot\dif \vec{S}=Q\\ \nabla\cdot\vec{D}=\rho$

# 电场的旋度

$\vec{E}(\vec{r})=-\frac{1}{4\pi\varepsilon_0}\int_{V'} \rho(\vec{r}') \nabla (\frac{1}{R})\dif V'$

$\vec{E}(\vec{r})=- \nabla \left[ \frac{1}{4\pi\varepsilon_0}\int_{V'} \rho(\vec{r}')\frac{1}{R}\dif V' \right]$

$\nabla\times\vec{E}(\vec{r}) = \nabla\times \left\{ - \nabla \left[ \frac{1}{4\pi\varepsilon_0}\int_{V'} \rho(\vec{r}')\frac{1}{R}\dif V' \right] \right\} \equiv 0$

$\oint_c\vec{E}\cdot\dif \vec{l} =\int_S \nabla\times\vec{E}\cdot\dif\vec{S}=\int_S \vec{0}\cdot\dif \vec{S}=0$

$U_{PQ}=\int_P^Q \vec{E}\cdot\dif\vec{l}$

$\frac{1}{4\pi\varepsilon_0}\int_{V'} \rho(\vec{r}')\frac{1}{R}\dif V' \Leftrightarrow \frac{q}{4\pi\varepsilon_0}\left( \frac{1}{R_P}-\frac{1}{R_Q} \right)$