静电的基本现象和规律

几个重要定律

库仑定律

$$ \begin{gather} \begin{aligned} \pmb F_{12} &= k\frac{q_1q_2}{r^2}\pmb r_{12} \\ e &= 1.602\times 10^{-19} (C) \\ k &=\frac{1}{4\pi \varepsilon_0} = 8.99\times 10^9(N\cdot m^2/C^2) \\ \varepsilon_0 &= 8.85\times 10^{-12} \ C/(N\cdot m^2) \\ 1C &= 1A \cdot s \end{aligned} \end{gather} $$

电场与高斯定律

场强公式

$$ \begin{gather} \begin{aligned} \pmb E=\frac{\pmb F}{q_0}=\frac{1}{4\pi\varepsilon_0}\frac{q_0}{r^2}\pmb{\widehat r} \end{aligned} \end{gather} $$

引入立体角 !!!

$$ \begin{aligned} dA_2 &= r d \theta \cdot r sin \theta d \phi \end{aligned} $$

$$ \begin{gather} \begin{aligned} d\Omega &=\frac{dS}{r^2} \Rightarrow d\Omega = \frac{\widehat{\pmb r}\cdot d\pmb S}{r^2} \\ \Omega &= \int_{ }^{ } d \Omega \\ &= \int_{ 0 }^{ \pi } sin \theta d \theta \int_{ 0 }^{ 2 \pi } d \phi \\ &= 4 \pi \end{aligned} \end{gather} $$

• 其中，$\pmb{ r }$是单位方向向量

高斯定理的表述和证明

1. 表述

通过任意闭合曲面的电通量等于该面包围的所有电量的代数和除以$\varepsilon_0$

2. 证明

$$ \begin{aligned} d\Phi_E &= \pmb E\cdot d \pmb S=\frac{q}{4\pi\varepsilon_0} \frac{\widehat{\pmb r}\cdot d\pmb S}{r^2} = \frac{q}{4\pi\varepsilon_0}d\Omega \\ \oiint d\Phi_E &= \frac{q}{4\pi\varepsilon_0}\oiint d\Omega = \frac{q}{\varepsilon_0} \end{aligned} $$

$$ \begin{gather} \begin{aligned} \Phi_E &= \int \pmb E \cdot d\pmb S = \frac{Q_{in}}{\varepsilon_0} \end{aligned} \end{gather} $$