# 泊松方程与边界条件

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

# 泊松方程

$\begin{cases} \nabla \cdot \vec{D} =\rho\\ \vec{D}=\varepsilon\vec{E}\\ \vec{E}=-\nabla \phi \end{cases} \Longrightarrow \nabla^2 \phi=-\frac{\rho}{\varepsilon}$

直角坐标

$\nabla^2 \phi= \frac{\p \phi}{\p x^2}+\frac{\p \phi}{\p y^2}+\frac{\p \phi}{\p z^2}$

柱坐标

$\nabla^2 \phi=\frac{1}{r}\frac{\p}{\p r}\left(r \frac{\p \phi}{\p r} \right)+\frac{1}{r^2}\frac{\p^2 \phi}{\p \varphi^2}+\frac{\p \phi}{\p z^2}$

球坐标

$\nabla^2 \phi= \frac{1}{r^2}\frac{\p}{\p r}\left(r \frac{\p \phi}{\p r} \right)+\frac{1}{r^2\sin\theta}\frac{\p}{\p \theta}\left(\sin\theta\frac{\p \phi}{\p \theta}\right)+\frac{1}{r^2\sin^2\theta}\frac{\p^2 \phi}{\p \varphi^2}$

$\nabla^2 = \frac{1}{h_1h_2h_3} \sum_{i=1}^3 \frac{\p}{\p u_i} \left( \frac{h_1h_2h_3}{h_i} \frac{\p}{h_i \p u_i} \right)$

$\nabla^2 \vec{E} = (\nabla^2 E_1) \hat{a}_1+(\nabla^2 E_2) \hat{a}_2+(\nabla^2 E_3) \hat{a}_3$

• 分配律：$\nabla^2(u+v)=\nabla^2 u + \nabla^2 v$
• $\nabla^2(uv)=u\nabla^2 v+2\nabla u \cdot \nabla v +v\nabla^2 u$
• $\nabla \times (\nabla\times \vec{F}) = \nabla(\nabla\cdot\vec{F}) - \nabla^2\vec{F}$

## 边界条件

$$\nabla^2\phi = \frac{1}{r^2}\frac{\p}{\p r}\left(r \frac{\p \phi}{\p r} \right)=0\\ \phi=-\frac{C_1}{r}+C_2$$

$$\begin{cases} \phi(a)=U\\ \phi(\infty)=0 \end{cases} \Rightarrow \begin{cases} C_2=0\\ C_1=-aU \end{cases}$$ 最终解出： $$\phi(r)=\frac{aU}{r} \; (a<r<\infty)$$

# 分界面上的边界条件

## 电位移的边界条件

$\oint_S \vec{D}\cdot\dif \vec{S}=\vec{D}_1\cdot\vec{n}\Delta S - \vec{D}_2\cdot\vec{n}\Delta S=\rho_s\Delta S\\ 即：\vec{n}\cdot(D_1 - D_2)=\rho_s\\ 或：D_{1n}-D_{2n}=\rho_s$

$\vec{n}\cdot\vec{D}_1=\vec{n}\cdot\vec{D}_2\\ D_{1n}=D_{2n}$

$D_n=\varepsilon E_n=\varepsilon \vec{E}\cdot\vec{n}=\varepsilon(-\nabla\phi)\cdot\vec{n}=-\varepsilon\frac{\p \phi}{\p \vec{n}}$

$\varepsilon_1 E_{n1}-\varepsilon_2 E_{n2}=\rho_s\\ -\varepsilon_1\frac{\p \phi_1}{\p \vec{n}}+\varepsilon_2\frac{\p \phi_2}{\p \vec{n}}=\rho_2$

## 电场强度的边界条件

$\oint_c \vec{E} \cdot \dif \vec{l}=\vec{E}_1 \cdot \vec{a}_l \Delta l - \vec{E}_2\cdot \vec{a}_l \Delta l =0$

$\bd{A}\cdot(\bd{B}\times\bd{C})=\bd{B}\cdot(\bd{C}\times\bd{A})\\ \vec{E}\cdot(\vec{a}_s\times \vec{n})=\vec{a}_s\cdot(\vec{n}\times \vec{E})$

$\vec{n}\times \vec{E}_1-\vec{n}\times \vec{E}_2=0\\ E_{1t}=E_1\cos\theta_1=E_2\cos\theta_2=E_{2t}$

## 理想介质分界面

$\begin{cases} \vec{n}\times \vec{E}_1=\vec{n}\times \vec{E}_2\\ \varepsilon_1 \vec{E}_1\cdot\vec{n}=\varepsilon_2 \vec{E}_2\cdot\vec{n} \end{cases}\\ 即 \begin{cases} E_1\sin\theta_1=E_2\sin\theta_2\\ \varepsilon_1 E_1\cos\theta_1=\varepsilon_2E_2\cos\theta_2 \end{cases}$

$\frac{\tan\theta_1}{\tan\theta_2}=\frac{\varepsilon_2}{\varepsilon_1}$