电磁学乱七八糟的符号(一)

@(study)[Maxe, markdown_study, LaTex_study]
author:何伟宝

chapter1 场量基础

通量 $\psi$

$\psi = \int_s \vec F \bullet \vec a_n d S$
$\psi = \oint_S \vec F \bullet d\vec S$

旋量 $\Gamma$

$\Gamma=\int_l \vec F \bullet d\vec l$
$\Gamma=\oint_l \vec F \bullet d\vec l$

矢性微分算符 $\nabla$

$\nabla =\vec a_x \frac{\partial }{\partial x}+\vec a_y \frac{\partial }{\partial y}+\vec a_z \frac{\partial }{\partial z}$

拉普拉斯算符 $\nabla^2$

$\nabla =\vec a_x \frac{\partial^2 }{\partial x^2}+\vec a_y \frac{\partial^2 }{\partial y^2}+\vec a_z \frac{\partial^2 }{\partial z^2}$

$\nabla \times (\nabla \times \vec F) = \nabla(\nabla \bullet \vec F) -\nabla^2 \vec F$

梯度 grad u

$grad u =\vec a_x \frac{\partial u}{\partial x}+\vec a_y \frac{\partial u}{\partial y}+\vec a_z \frac{\partial u}{\partial z}$
$gradu=\nabla u$

散度div F

$div \vec F \triangleq \lim_{\triangle V\to 0} \frac{\oint_S \vec F d \vec S}{\triangle V}$

$div \vec F=\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+ \frac{\partial F_z}{\partial z} =\nabla \bullet \vec F$

$\int_V \nabla \bullet \vec F d V =\oint_l \vec F d \vec S$

环量面密度 $\gamma_n$

$\gamma_n \triangleq \lim_{\triangle S\to 0} \frac{\oint_l \vec F d \vec l}{\triangle S}$

旋度 $R_m$

$\vec R_m \triangleq rot \vec F =\vec a_n \lgroup \lim_{\triangle S \to 0} \frac{\oint \vec F d \vec l }{\triangle S} \rgroup_{max}$

$rot \vec F =\nabla \times \vec F$

$\int_S \nabla \times \vec F \bullet d \vec S = \oint_l \vec F d \vec l$

chapter2 常量基本方程

电荷密度

体电荷密度:
$\rho (\vec r^{\bullet} ) = \lim_{\triangle V \to 0 } \frac{\triangle q}{\triangle V^\bullet} = \frac{d q}{d V^\bullet}$
$q= \int_V \rho(\vec r^\bullet) d V^\bullet$

面电荷密度:
$\rho_s (\vec r^{\bullet} ) = \lim_{\triangle S \to 0 } \frac{\triangle q}{\triangle S^\bullet} = \frac{d q}{d S^\bullet}$
$q= \int_S \rho_S(\vec r^\bullet) d S^\bullet$

线电荷密度:
$\rho_l (\vec r^{\bullet} ) = \lim_{\triangle l \to 0 } \frac{\triangle q}{\triangle l^\bullet} = \frac{d q}{d l^\bullet}$
$q= \int_l \rho_l(\vec r^\bullet) d l^\bullet$

点电荷:
$q(\vec r)= \sum_{i=1}^N q_i(\vec r_i)$

电流&&电流密度

电流:
$i = \lim_{\triangle t \to 0}\frac{\triangle q }{\triangle t}=\frac{d q}{d t}$

体电流密度矢量:

$\vec J= \vec a_n \lim_{\triangle S^\bullet \to 0} \frac{\triangle i}{\triangle S^\bullet}=\vec a_n \frac{di }{dS^\bullet}$

$i = \int_s \vec J \bullet d \vec S$

$\nabla \bullet \vec J=- \frac{\partial \rho}{\partial t}$

面电流密度:

$\vec J_s =\vec a_n \lim_{\triangle l^\bullet \to 0} \frac{\triangle i}{\triangle l^\bullet} = \vec a_n \frac{d i}{d l^\bullet}$

$i = \int_l \vec J_s \bullet (\vec n \times d \vec l^\bullet)$

由于静态场的麦克斯韦方程组还没有统一,这里就不写了

电场强度E:

$\vec E \triangleq \frac{\vec F}{q_0}$

磁感应强度B:

$\vec B \triangleq \frac{\mu}{4\pi}\oint_l \frac{I d \vec l \times a_R}{R^2}$

感应电动势 $\varepsilon_{in}$

$\varepsilon_{in} \triangleq -\frac{d \psi}{d t}$
其中 $\psi$ 为磁通量
$\psi \triangleq \int_S \vec B \bullet d \vec S$
所以:
$\varepsilon_{in} = \int_s \frac{\partial \vec B}{\partial t} \bullet d \vec S$

本章的一些常数

$\varepsilon_0 自由空间的电容率 (介电常数)$
$\varepsilon_0 =8.85\times 10^{-12}\approx \frac {10^{-9}}{36\pi} F/m$
$\mu_0$ 真空磁导率
$\mu_0=4\pi \times 10^{-7}H/m$

chapter3静态场

标量电位 $\Phi$

$\vec E(\vec r) \triangleq -\triangle\Phi(\vec r)$

$\Phi(\vec r)=\frac{W}{q}$

电位的标量泊松方程:
$\nabla^2 \Phi(\vec r) = - \frac{\rho(\vec r)}{\varepsilon_0}$

电位的标量拉普拉斯方程:
$\nabla^2 \Phi(\vec r) = 0$

矢量磁位(磁矢位) A

$\vec B (\vec r )\triangleq \nabla \times \vec A(\vec r)$

库仑规范:
$\nabla \bullet \vec A = 0$

磁矢位的矢量泊松方程:
$\nabla^2 \vec A (\vec r )=- \mu_0 \vec J (\vec r)$

磁矢位的矢量拉普拉斯方程
$\nabla^2 \vec A (\vec r )=0$

磁矩m:
$\vec m \triangleq \vec I \vec S$

极化强度矢量P

$\vec P(\vec r)=\lim_{\triangle V \to 0} \frac{\sum_i \vec p_i}{\triangle V}$
$\vec P = \chi_e \varepsilon_0 \vec E$
其中 $\chi_e$ 为电极化率

电位移矢量D

$\vec D(\vec r) \triangleq \varepsilon_0 \vec E(\vec r)+\vec P(\vec r)$
所以有:

$\int_s \vec D(\vec r) \bullet d \vec S =q$

$\nabla \bullet \vec D(\vec r) = \rho(\vec r)$

$\vec D = \varepsilon \vec E$

磁化强度矢量M

$\vec M(\vec r)=\lim_{\triangle V \to 0} \frac{\sum_i \vec m_i}{\triangle V}$
$\vec M = \chi_m H$
其中 $\chi_m$ 为磁化率

磁化强度H

$\vec H(\vec r)=\frac{\vec B(\vec r)}{\mu_0}-\vec M(\vec r)$
$\oint_l \vec H\bullet d\vec l=I$
$\nabla \times \vec H (\vec r )=\vec J(\vec r)$
$\vec B=\mu \vec H$

欧姆定律微分形式

$\vec J(\vec r)=\sigma \vec E(\vec r)$
其中 $\sigma$ 为电导率

热损耗功率

$p(\vec r)=\vec J(\vec r)\bullet \vec E(\vec r)=\sigma E^2(\vec r)$

边界条件

$\vec a_n \times (\vec E_1 -\vec E_2)=0,\quad \quad E_{1t}=E_{2t}$
$\vec a_n \times (\vec H_1 -\vec H_2)=\vec J_s,\quad \quad H_{1t}-H_{2t}=J_s$
$\vec a_n \bullet (\vec D_1 -\vec D_2) =\rho_s, \quad D_{1n}-D_{2n}=\rho_s$
$\vec a_n \bullet (\vec B_1 - \vec B_2)=0,\quad \quad B_{1n}=B_{2n}$

能量

静电场能量密度:
$\omega_e = \frac 12 \varepsilon E^2$
$\omega_e = \frac 12 \vec D(\vec r )\bullet \vec E(\vec r)$
静磁场能量密度:
$\omega_m = \frac 12 \mu H^2$
$\omega_m = \frac 12 \vec H(\vec r )\bullet \vec B(\vec r)$

chapter4 动态场

麦克斯韦方程组

$\begin{cases} \oint_l \vec E(\vec r,t)\bullet d \vec l = -\int_S \frac{\partial \vec B(\vec r,t)}{\partial t} \bullet d \vec S , \quad\quad \nabla \times \vec E(\vec r,t) = - \frac{\partial \vec B(\vec r,t)}{\partial t} \\ \oint_l \vec H(\vec r,t)\bullet d\vec l = \int_S (\vec J(\vec r,t)+\frac{\partial \vec D(\vec r,t)}{\partial t}),\quad \nabla \times \vec H(\vec r,t)=\vec J(\vec r,t)+\frac{\partial \vec D(\vec r,t)}{\partial t}\\ \oint_S \vec D(\vec r,t)\bullet d \vec S = \int_V \rho(\vec r,t)dV,\quad\quad\quad\quad \nabla \bullet \vec D(\vec r,t)=\rho(\vec r,t)\\ \oint_S \vec B(\vec r ,t)\bullet d \vec S =0 ,\quad \quad\quad\quad\quad\quad\quad\quad\nabla \bullet \vec B(\vec r,t)=0 \end{cases}$

标量电位更新

$\vec E=-\nabla\Phi -\frac{\partial \vec A}{\partial t}$

波动方程

洛伦兹条件(洛伦兹规范):
$\nabla \bullet \vec A=-\mu \varepsilon \frac{\partial \Phi}{\partial t}$
非齐次波动方程(动态退化可以得到其他规范):
$\nabla^2 \Phi(\vec r,t)-\mu\varepsilon\frac{\partial^2\Phi(\vec r,t)}{\partial t^2}=- \frac{\rho(\vec r,t)}{\varepsilon}$

$\nabla^2 A(\vec r,t)-\mu\varepsilon\frac{\partial^2 A(\vec r,t)}{\partial t^2}= -\mu \vec J(\vec r,t)$

坡印亭矢量

$\vec S (\vec r,t) \triangleq \vec E(\vec r,t)\times \vec H(\vec r,t)$

$-\nabla \bullet \vec S=\frac{\partial\omega}{\partial t}+p$

$-\oint_S \vec S(\vec r,t)\bullet d \vec S=\frac{\partial}{\partial t}\int_V \omega(\vec r,t)d V+\int_Vp(\vec r,t)dV$

复数表示

$u(z,t)=Re\{ [U_0(z)e^{j\phi}]e^{j\omega t} \} = Re \{ \dot{U}(z) e^{j\omega t} \}$
$\dot{U}(z)=U_0(z)e^{j\phi}$

复数形式麦克斯韦方程

$\nabla \times \vec E=j\omega \vec B$
$\nabla \times \vec H =\vec J + j \omega \vec D$
$\dot{\vec E}=\vec a_x\dot{E_x}(\vec r)+\vec a_y\dot{E_y}(\vec r)+\vec a_z\dot{E_z}(\vec r)$

复波动方程

$\nabla \bullet \vec A(\vec r) = -j\omega \mu\varepsilon \Phi(\vec r)$

$\nabla^2\Phi(\vec r)+\omega^2\mu\varepsilon\Phi(\vec r)=-\frac{\rho(\vec r)}{\varepsilon}$
$\nabla^2 \vec A(\vec r)+\omega^2\mu\varepsilon \vec A(\vec r)=-\mu \vec J(\vec r)$
令 $k^2=\omega^2\mu\varepsilon$ 有:
非齐次亥姆霍兹方程:
$\nabla^2\Phi(\vec r)+k^2\Phi(\vec r)=-\frac{\rho(\vec r)}{\varepsilon}$
$\nabla^2 \vec A(\vec r)+k^2 \vec A(\vec r)=-\mu \vec J(\vec r)$
齐次亥姆霍兹方程:
$\nabla^2\Phi(\vec r)+k^2\Phi(\vec r)=0$
$\nabla^2 \vec A(\vec r)+k^2 \vec A(\vec r)=0$

波阻抗 $\eta$

$\eta_0=\sqrt{\frac{\mu_0}{\varepsilon_0}}$

时均坡印亭矢量 $S_{av}$

$\vec S_av(\vec r)=\frac 1T\int_0^T\vec S(\vec r,t)dt=\frac 12 [\vec E_0(\vec r)\times \vec H_0(\vec r)]cos(\phi_e-\phi_n)$

复坡印亭矢量 $\dot{S}$

$\dot{S}(\vec r)=\frac 12 \vec E(\vec r) \times \vec H^*(\vec r)=\frac 12 \vec E_0(\vec r)e^{-j\phi_e}\times \vec H_0(\vec r )e^{j\phi_n}=\frac 12[\vec E_0(\vec r)\times \vec H_0(\vec r)]e^{\phi_e-\phi_n}$

其中:
$\vec S_av(\vec r)=Re\{ \dot{S}(\vec r) \}$

复坡印亭定理

$-\oint_s \dot{S}(\vec r)\bullet d \dot{S} =j2\omega \int_V[\omega_{m-av}(\vec r)-\omega_{e-av}(\vec r)]dV +\int_V p_{av}(\vec r)dV$

其中:
$\omega_av (\vec r)=\frac 14[\vec E(\vec r)\bullet \vec D^*(\vec r)+\vec B(\vec r)\bullet \vec H^*(\vec r)]=\frac 14[\varepsilon|\vec E(\vec r)|^2 + \mu|\vec H(\vec r)|^2 ]=Re\omega(\vec r)$
$p_{av}(\vec r)=\frac 12 \vec E(\vec r )\bullet \vec J^*(\vec r) =\frac 12 \sigma |\vec E(\vec r)|^2 =Rep(\vec r)$

结语

天书虽然可怕,但,他还是你爸爸
也就,100条公式而已,前四章
想我尽早更新的方法之一