静电场

数学

矢量：

\vec{r}=x \hat{x} + y \hat{y} +z \hat{z}

\vec{r} = r sin\theta cos\phi \hat{x} + r sin\theta sin\phi \hat{y} + r cos\theta \hat{z}

\vec{r} = \rho cos\theta \hat{x} + \rho sin\theta \hat{y} + z \hat{z}

内积：

\hat{x}\cdot\hat{y}=\hat{x}\cdot\hat{z}=\hat{y}\cdot\hat{z}=0

\hat{x}\cdot\hat{x}=\hat{y}\cdot\hat{y}=\hat{z}\cdot\hat{z}=1

\begin{aligned} \vec{A}=&A_x\hat{x}+A_y\hat{y}+A_z\hat{z}\\ \vec{B}=&B_x\hat{x}+B_y\hat{y}+B_z\hat{z}\\ \vec{A}\cdot\vec{B}=&(A_x\hat{x}+A_y\hat{y}+A_z\hat{z})\cdot(B_x\hat{x}+B_y\hat{y}+B_z\hat{z})\\ =&A_xB_x+A_yB_y+A_zB_z \end{aligned}

叉乘：

\hat{x}\times\hat{y}=\hat{z}

\hat{y}\times\hat{z}=\hat{x}

\hat{z}\times\hat{x}=\hat{y}

\begin{aligned} \vec{A}=&A_x\hat{x}+A_y\hat{y}+A_z\hat{z}\\ \vec{B}=&B_x\hat{x}+B_y\hat{y}+B_z\hat{z}\\ \vec{A}\times\vec{B}=&(A_x\hat{x}+A_y\hat{y}+A_z\hat{z})\times(B_x\hat{x}+B_y\hat{y}+B_z\hat{z})\\ =&A_xB_y\hat{x}\times\hat{y}+A_yB_x\hat{y}\times\hat{x}\\ +&A_yB_z\hat{y}\times\hat{z}+A_zB_y\hat{z}\times\hat{y}\\ +&A_zB_x\hat{z}\times\hat{x}+A_xB_z\hat{x}\times\hat{z}\\ =&(A_xB_y-A_yB_x)\hat{z}+(A_zB_y-A_yB_z)\hat{x}+(A_zB_x-A_xB_z)\hat{y} \end{aligned}

并矢：

\begin{aligned} \vec{A}=&A_x\hat{x}+A_y\hat{y}+A_z\hat{z}\\ \vec{B}=&B_x\hat{x}+B_y\hat{y}+B_z\hat{z}\\ \vec{A}\vec{B}=&(A_x\hat{x}+A_y\hat{y}+A_z\hat{z})(B_x\hat{x}+B_y\hat{y}+B_z\hat{z})\\ =&A_xB_x \hat{x}\hat{x} + A_xB_y \hat{x}\hat{y}+A_xB_z\hat{x}\hat{z}\\ +&A_yB_x \hat{y}\hat{x} + A_yB_y \hat{y}\hat{y}+ A_yB_z\hat{y}\hat{z}\\ +&A_zB_x \hat{z}\hat{x} + A_zB_y \hat{z}\hat{y}+ A_zB_z\hat{z}\hat{z}\\ \end{aligned}

梯度算符： $\nabla=\frac{\partial}{\partial x}\hat{x}+\frac{\partial}{\partial y}\hat{y}+\frac{\partial}{\partial z}\hat{z}$
球坐标下梯度算符： $\nabla = \frac{\partial}{\partial r}\hat{r}+\frac{1}{r}\frac{\partial}{\partial \theta} +\frac{1}{rsin\theta}\frac{\partial}{\partial \phi}\hat{\phi}$

梯度算符就相当于一个矢量，直角坐标的形式必须记住，球坐标可以记，考试记不住球坐标，换成直角坐标暴力求导不会扣分
梯度算符作用于标量：

\begin{aligned} \nabla f(\vec{r}) =& (\frac{\partial}{\partial x}\hat{x}+\frac{\partial}{\partial y}\hat{y}+\frac{\partial}{\partial z}\hat{z})f(\vec{r})\\ =&\frac{\partial f(\vec{r})}{\partial x}\hat{x}+\frac{\partial f(\vec{r})}{\partial y}\hat{y}+\frac{\partial f(\vec{r})}{\partial z}\hat{z} \end{aligned}

散度（梯度算符与矢量的内积）：

\vec{f}(\vec{r})=f_x(\vec{r})\hat{x}+f_y(\vec{r})\hat{y}+f_z(\vec{r})\hat{z}

\begin{aligned} \nabla \cdot \vec{f}(\vec{r})=&(\frac{\partial}{\partial x}\hat{x}+\frac{\partial}{\partial y}\hat{y}+\frac{\partial}{\partial z}\hat{z})\cdot(f_x(\vec{r})\hat x+f_y(\vec{r})\hat{y}+f_z(\vec{r})\hat{z})\\ =&(\frac{\partial f_x(\vec{r})}{\partial x} \hat{x}\cdot\hat{x}+\frac{\partial f_y(\vec{r})}{\partial y} \hat{y}\cdot\hat{y}+\frac{\partial f_z(\vec{r})}{\partial z} \hat{z}\cdot\hat{z})\\=&\frac{\partial f_x(\vec{r})}{\partial x}+\frac{\partial f_y(\vec{r})}{\partial y}+\frac{\partial f_z(\vec{r})}{\partial z} \end{aligned}

旋度（梯度算符与矢量的叉乘）：

\begin{aligned} \nabla \times \vec{f}(\vec{r})=&(\frac{\partial}{\partial x}\hat{x}+\frac{\partial}{\partial y}\hat{y}+\frac{\partial}{\partial z}\hat{z})\times(f_x(\vec{r})\hat x+f_y(\vec{r})\hat{y}+f_z(\vec{r})\hat{z})\\ =&(\frac{\partial f_y(\vec{r})}{\partial x}\hat{x}\times\hat{y}+\frac{\partial f_z(\vec{r})}{\partial x}\hat{x}\times\hat{z}+\frac{\partial f_x(\vec{r})}{\partial y}\hat{y}\times\hat{x}+\frac{\partial f_z(\vec{r})}{\partial y}\hat{y}\times\hat{z}+\frac{\partial f_x(\vec{r})}{\partial z}\hat{z}\times\hat{x}+\frac{\partial f_y(\vec{r})}{\partial z}\hat{z}\times\hat{y})\\ =&(\frac{\partial f_y(\vec{r})}{\partial x}-\frac{\partial f_x(\vec{r})}{\partial y})\hat{z}+(\frac{\partial f_x(\vec{r})}{\partial z}-\frac{\partial f_z(\vec{r})}{\partial x})\hat{y}+(\frac{\partial f_z(\vec{r})}{\partial y}-\frac{\partial f_y(\vec{r})}{\partial z})\hat{x} \end{aligned}

并（梯度算符与矢量的叉乘）：

\begin{aligned} \nabla\vec{f}(\vec{r})=&(\frac{\partial}{\partial x}\hat{x}+\frac{\partial}{\partial y}\hat{y}+\frac{\partial}{\partial z}\hat{z})(f_x(\vec{r})\hat x+f_y(\vec{r})\hat{y}+f_z(\vec{r})\hat{z})\\ \end{aligned}

拉普拉斯算子 $\nabla^2$

\begin{aligned} \nabla^2f(\vec{r})=&\nabla\cdot\nabla f(\vec{r})\\ =&(\frac{\partial}{\partial x}\hat{x}+\frac{\partial}{\partial y}\hat{y}+\frac{\partial}{\partial z}\hat{z})\cdot(\frac{\partial}{\partial x}\hat{x}+\frac{\partial}{\partial y}\hat{y}+\frac{\partial}{\partial z}\hat{z})f(\vec{r})\\ =&(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2})f(\vec{r}) \end{aligned}

注意顺序，符号含义，你可以计算任意的东西！

体积分：

\iiint_{V^{'}}dV^{'}=\iiint{dxdydz}=\iiint{dr d\theta d\phi} r^2sin\theta=\iiint d\rho d\phi dz\rho

面积分：
x-y平面:

\iint_{S}dS=\iint dxdy

球面： $r = R$

\iint_{S}dS=\iint d\phi d\theta R^2sin\theta d\theta d\phi

圆柱侧面： $\rho = R$

\iint_{S}dS=\iint R d\phi dz

圆柱底面：

\iint_{S}dS=\iint \rho d\rho d\phi

积分：

\begin{aligned} \int_P^Qd\vec{l}\cdot\nabla [\quad]=&[\quad]|_P^Q\\ \int_VdV\nabla[\quad]= &\oint_{\partial V}d\vec{\sigma}[\quad]\\ \int_{\Sigma}(d\vec{\sigma}\times\nabla)[\quad]=& \oint_{\partial \Sigma}d\vec{l}[\quad]\\ \end{aligned}

泰勒展开：

f(x) = \sum \frac{1}{n!}f^{n}(x_0)(x-x_0)^n

可以直接展开，但也建议小量展开一般往几种模板上展开： $sin(x),cos(x),e^{x},ln(1+x)，\frac{1}{(1+x)^a}$ 这通常需要把一些部分进行换元（例如作业1.17）需要对 $\frac{1}{1+x^2-2xcos\theta}$ 做展开，这里 $x<<1$ 是小量，当 $x\to 0$ 时 $(x^2-2xcos\theta) \to 0$ ,对 $(x^2-2xcos\theta)$ 整体先展开，展开到需要的阶数，注意由于是对 $(x^2-2xcos\theta)$ 展开，一阶项也会包含二阶项，因此计算二阶项时不要漏了。

库仑定律

点电荷 $q_1$ 对点电荷 $q_2$ 的力 $\vec{F}=\frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r_{12}^2}\hat{r}_{12}$
点电荷 $q$ 在 $\vec{r}$ 处的电场强度 $\vec{E}=\frac{1}{4\pi\epsilon}\frac{q}{r^2}\hat{r}$
体电荷系统，面电荷系统，线电荷系统：

\vec{E}(\vec{r})=\frac{1}{4{\pi}{\epsilon}_0}\iiint_{V^{'}}dV^{'}\frac{\rho_e(\vec{r}^{'})}{|\vec{r}-\vec{r}^{'}|^2}(\hat{r}-\hat{r}^{'})

\vec{E}(\vec{r})=\frac{1}{4{\pi}{\epsilon}_0}\iint_{S^{'}}dS^{'}\frac{\sigma_e(\vec{r}^{'})}{|\vec{r}-\vec{r}^{'}|^2}(\hat{r}-\hat{r}^{'})

\vec{E}(\vec{r})=\frac{1}{4{\pi} {\epsilon}_0}\int_{L^{'}}dl^{'}\frac{\lambda_e(\vec{r}^{'})}{|\vec{r}-\vec{r}^{'}|^2}(\hat{r}-\hat{r}^{'})

只求空间中一个比较特殊的点（这个点和电荷分布一起具有某种特殊性时）的电场时，用这几个式子,习题1.5，1.6（由于1.6相当于求x方向电场，求AC上的电势再对x求导也行，注意因为只求AC上的电势，你取了y=0，最后的电势中不含有y了，这不代表没有y方向电场，如果要求y方向电场，你就要任取（x,y），求该店的电势，对y求导得到y方向电场。最经典的例子就是1.7，先求电势再求电场会很麻烦