\[\begin{align*}
\newcommand{\dif}{\mathop{}\!\mathrm{d}}\\
\newcommand{\p}{\partial}\\
\newcommand{\bd}{\boldsymbol}
\end{align*}\]
基础公式
库伦定律:$\vec{F}_{21} = \dfrac{q_1q_2}{4\pi\epsilon_0 R^3} \vec{R}$
电场强度与电位:$\vec{E}=-\nabla\phi$
电场强度 | 电位 | |
---|---|---|
点电荷 | \(\vec{E}(\vec{r})=\dfrac{q}{4\pi\varepsilon_0 R^3}\vec{R}=-\dfrac{q}{4\pi\varepsilon_0}\nabla\left(\dfrac{1}{R}\right)\) | \(\phi=\dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{R}+C\) |
体电荷 | \(\vec{E}(\vec{r})=\dfrac{1}{4\pi\varepsilon_0} \int_V \rho(r')\nabla(\dfrac{1}{R}) \dif V'\) | \(\phi =\dfrac{1}{4\pi\varepsilon_0} \int \dfrac{\rho(r')}{R} \dif V'+C\) |
面电荷 | \(\vec{E}(\vec{r})=\dfrac{1}{4\pi\varepsilon_0} \int_s \rho_s(r')\nabla(\dfrac{1}{R}) \dif S'\) | \(\phi =\dfrac{1}{4\pi\varepsilon_0} \int \dfrac{\rho_s(r')}{R} \dif S'+C\) |
线电荷 | \(\vec{E}(\vec{r})=\dfrac{1}{4\pi\varepsilon_0} \int_l \rho_l(r')\nabla(\dfrac{1}{R}) \dif l'\) | \(\phi =\dfrac{1}{4\pi\varepsilon_0} \int \dfrac{\rho_l(r')}{R} \dif l'+C\) |
静电场基本方程:
\[\begin{align} &微分形式 \begin{cases} \nabla \cdot \vec{D}=\rho\\ \nabla\times\vec{E}=0 \end{cases}\\ &积分形式 \begin{cases} \oint_s \vec{D}\cdot\dif \vec{S}=\int_V \rho \dif V=Q\\ \oint_C \vec{E}\cdot\dif \vec{l}=0 \end{cases} \end{align}\]本构关系:$\vec{D}=\varepsilon \vec{E}$
泊松方程:$\nabla^2 \phi = -\dfrac{\rho}{\varepsilon}$,特殊地,当无电荷密度时,$\nabla^2 \phi=0$
边界条件:
介质分界面 | 理想导体分界面 | |
---|---|---|
电场/电位移 | \(\begin{cases}D_{1n}-D{2n}=\rho_s\\E_{1t}-E_{2t}=0\end{cases}\) 介质1靠近正方向 |
\(\begin{cases}D_{n}=\rho_s\\E_t=0\end{cases}\) |
电位 | \(\begin{cases}\varepsilon_1 \dfrac{\p \phi_1}{\p n}-\varepsilon_2 \dfrac{\p \phi_2}{\p n}=-\rho_s\\ \phi_1=\phi_2\end{cases}\) | \(\begin{cases}\varepsilon\dfrac{\p\phi}{\p n}=-\rho_s\\ \phi=C\end{cases}\) |
场/源
由源求场:
- 定义:$\vec{E}(\vec{r})=\frac{1}{4\pi \varepsilon_0}\int_s \dfrac{\rho_s(\vec{r’}) \dif S’}{R^3}\vec{R}$
- 高斯定理:$\oint_s \vec{E}\cdot \dif \vec{S}=\dfrac{Q}{\varepsilon_0}$
- 电位:$\phi=\dfrac{1}{4\pi\varepsilon_0}\int_s \dfrac{\rho_s(r’)}{R}\dif S’+C$,电场 $\vec{E}=-\nabla\phi$
- 泊松方程+边界条件:$\nabla^2\phi=-\dfrac{\rho}{\varepsilon}$
由场求源:
高斯定理的微分形式:
\[\begin{cases} \vec{D}=\varepsilon_0\vec{E}\\ \nabla\cdot\vec{D}=\rho \end{cases}\]电容
假设电荷
假设电压
静电场能量
对源积分:
-
体、面、线电荷
\[W_e=\frac{1}{2}\int_V \rho \phi \dif V\]
对场积分