电磁学乱七八糟的符号(一)
author:何伟宝
[TOC]
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