矢量分析

三种常用坐标系

	直角	圆柱	球
基矢	\vec{e_x} \vec{e_y} \vec{e_z}	\vec{e_\rho} \vec{e_\varphi} \vec{e_z}	\vec{e_r} \vec{e_\theta} \vec{e_\varphi}
度量数h_i	1 1 1	1 \rho 1	1 r rsin\theta
体元	dxdydz	\rho d\rho d\varphi dz	r^2sin\theta drd\theta d\varphi

\nabla \cdot F = \frac{1}{h_1h_2h_3}[\frac{\partial }{\partial u_1}h_2h_3A_1 + \frac{\partial}{\partial u_2}h_1h_3A_2 + \frac{\partial}{\partial u_3}h_1h_2A_3]

\nabla \times F = \frac{1}{h_1h_2h_3}\left|\begin{matrix}
\vec{e_1}h_1 & \vec{e_2}h_2 & \vec{e_3}h_3 \\
\frac{\partial}{\partial u_1} & \frac{\partial}{\partial u_2} & \frac{\partial}{\partial u_3} \\
h_1A_1 & h_2A_2 & h_3A_3
\end{matrix}\right|

静电场

电偶极子

电偶极矩

\vec{p} = Q\vec{l}

电势

\phi = \frac{Ql}{4\pi \epsilon_0 r^2}cos\theta = \frac{\vec{p} \cdot \vec{r}}{4\pi\epsilon_0 r^3}

电场

\vec{E} = \frac{p}{4\pi \epsilon_0 r^3}(2cos\theta \vec{e_r} + sin\theta \vec{e_{\theta}})

电介质的极化

极化方式

• 电子极化（感生极化）
• 取向极化
• 离子极化

束缚电荷面密度

\rho_{Sb} = \vec{p} 
\cdot \vec{n}

束缚电荷密度

\rho_b = -\nabla \cdot \vec{P}

\rho_b = -\frac{\epsilon_r - 1}{\epsilon_r}\rho

静电场的边界条件

两种媒质之间

\begin{aligned}
\begin{cases}
\vec{n} \cdot (\vec{D_1} - \vec{D_2}) = \rho_S \\
-\epsilon_1 \frac{\partial \phi_1}{\partial n} + \epsilon_2 \frac{\partial \phi_2}{\partial n}         = \rho_S \\\\
\vec{n} \times \vec{E_1} = \vec{n} \times \vec{E_2}
\end{cases}
\end{aligned}

电位移矢量的法向分量不连续，其突变量为\rho_S，电场强度的切向分量连续。

两种介质

\begin{aligned}
\begin{cases}
\vec{n} \cdot (\vec{D_1} - \vec{D_2}) = 0 \\
-\epsilon_1 \frac{\partial \phi_1}{\partial n} + \epsilon_2 \frac{\partial \phi_2}{\partial n}         = 0 \\\\
\vec{n} \times \vec{E_1} = \vec{n} \times \vec{E_2}
\end{cases}
\end{aligned}

电位移的法向分量和电场强度的切向分量连续。电势连续。

介质和导体

材料1为介质，材料2为导体

\begin{aligned}
\begin{cases}
\vec{n} \cdot \vec{D} = \rho_S \\
-\epsilon\frac{\partial \phi}{\partial n} = \rho_S\\
\phi = C\\
\vec{n} \times \vec{E} = 0        
\end{cases}
\end{aligned}

\vec{E} = \frac{\rho_S}{\epsilon}\vec{n}

电容

平行板电容器

C = \frac{\epsilon S}{d}

同轴圆柱电容器

C = \frac{2\pi \epsilon l}{ln\frac{b}{a}}

同心球电容器

C = \frac{4\pi\epsilon ab}{b - a}

双根线

C = \frac{\pi\epsilon}{ln\frac{D}{a}}

电场的能量

W_e = \frac{1}{2}\sum_i \phi_iQ_i = \frac{1}{2}\sum_i \frac{Q_i^2}{C_i}

能量密度

w_e = \frac{1}{2} \vec{D} \cdot \vec{E}

C = \frac{2}{W_e}{U^2} = \frac{\int_V \epsilon E^2dV}{U^2}

稳恒电场与磁场

传导电流

\vec{J_c} = \sigma \vec{E}

运流电流

\vec{J_v} = \rho \vec{v}

电荷守恒定律

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

稳恒电场的边界条件

\begin{aligned}
\begin{cases}
\vec{n} \cdot (\vec{J_1} - \vec{J_2}) = 0 \\
\vec{n} \times \vec{E_1} = \vec{n} \times \vec{E_2} \\\\

\sigma_1\frac{\partial \phi_1}{\partial n} = \sigma_2\frac{\partial \phi_2}{\partial n} \\
\phi_1 = \phi_2 \\
\rho_S = J_n(\frac{\epsilon_1}{\sigma_1} - \frac{\epsilon_2}{\sigma_2})
\end{cases}
\end{aligned}

焦耳定律

p = \vec{E} \cdot \vec{J} = \frac{J^2}{\sigma} = \sigma E^2

毕-萨定律

d\vec{B} = \frac{\mu_0}{4\pi}\int_V\frac{\vec{J} \times \vec{e_R}}{R^2}dV'

洛伦兹力

F = q(\vec{E} + \vec{v} \times \vec{B}) \\
\vec{f} = \rho\vec{E} + \vec{J} \times \vec{B}

磁通连续性定理

\oint \vec{B} \cdot d\vec{S} = 0

磁矢势

\vec{A} = \frac{\mu_0}{4\pi}\int_V\frac{\vec{J}}{R} dV'

库伦规范\nabla \cdot \vec{A} = 0

磁偶极子

\vec{A} = \frac{\mu_0IS}{4\pi r^2}sin\theta\vec{e_\varphi} = \frac{\mu_0}{4\pi}\frac{\vec{m} \times \vec{e_R}}{R^2}

\vec{B} = \frac{\mu_0m}{4\pi r^3}(2cos\theta \vec{e_r} + sin\theta\vec{e_\theta})

束缚电流

\vec{J}_m = \nabla \times \vec{M} = (\mu_r - 1)\vec{J}

\vec{J}_{Sm} = \vec{M} \times \vec{n}

磁场的边界条件

两种磁介质之间

磁感应强度的法向分量连续

\vec{n} \cdot ({\vec{B_1} - \vec{B_2}}) = 0

磁场强度的切向分量突变

\vec{n} \times (\vec{H_1} - \vec{H_2}) = \vec{J}_S

磁矢势连续

\vec{A_1} = \vec{A_2}

如果分界面上没有自由电流分布

\vec{n} \times (\vec{H_1} - \vec{H_2}) = 0

磁标势

\phi_{m1} = \phi_{m2}

\mu_1\frac{\partial \phi_{m1}}{\partial n} = \mu_2\frac{\partial \phi_{m2}}{\partial n}

磁介质和理想导磁体

即\mu = \infty

电感

L = \frac{\Psi}{I}

互感

M_{12} = M_{21} = \frac{\Psi_{12}}{I} = \frac{\Psi_{21}}{I}

磁场的能量

W_m = \frac{1}{2}\sum_i L_i\Psi_i

能量密度

w_m = \frac{1}{2} \vec{H} \cdot \vec{B}

电感

L = \frac{2W_m}{I^2}

静态场的解

唯一性定理

对于任意静电场，当空间各点的电荷分布与整个边界上的边界条件已知时，空间各部分的场就唯一确定了。

镜像法

平面镜像

球面镜像

q' = \frac{-a}{d}q

b = \frac{a^2}{d}

圆柱面镜像

q' = q

b = \frac{a^2}{d}

分离变量法

圆柱坐标系

球坐标系

\phi(r, \theta, \varphi) = \sum_{n=0}^{\infty}\sum_{m=0}^n[A_nr^n + B_nr^{-(n+1)}]P_n^m(cos\theta)(C_msinm\varphi + D_mcosm\varphi)

轴对称

\phi(r, \varphi) = \sum_{n=0}^{\infty}[A_nr^n + B_nr^{-(n + 1)}]P_n(cos\theta)

时变电磁场

法拉第电磁感应定律

{\epsilon} = -\frac{d\Psi}{dt} = -\int_S \frac{\partial \vec{B}}{\partial t} \cdot d\vec{S} + \oint (\vec{v}\times\vec{B}) \cdot d\vec{l}

\nabla \times \vec{E} = -\frac{\partial\vec{B}}{\partial t}

位移电流

\vec{J}_d = \frac{\partial \vec{D}}{\partial t}

麦克斯韦方程组

\begin{aligned}
\begin{cases}
\nabla \times \vec{H} = \vec{J} + \frac{\partial \vec D}{\partial t}        \\
\nabla \times \vec{E} = -\frac{\partial\vec{B}}{\partial t}\\
\nabla \cdot \vec{B} = 0\\
\nabla \cdot \vec{D} = \rho
\end{cases}
\end{aligned}

坡印廷定理

\nabla \cdot (\vec{E} \times \vec{H}) + \frac{J^2}{\sigma} = -\frac{\partial w}{\partial t} + \vec{J} \cdot \vec{E}_e

坡印廷矢量

\vec{S} = \vec{E}\times\vec{H}

其数值为通过与能量流动垂直方向的单位面积的功率，又叫能流密度

复坡印廷矢量

\vec{S}_c = \frac{1}{2}\vec{E}_m \times \vec{H}_m^*

似稳条件

R << \frac{\lambda}{2\pi} 

f << \frac{v}{2\pi R}

矢量势和标量势

\vec{E} + \frac{\partial \vec{A}}\partial{t} = -\nabla \phi

洛伦兹规范

\nabla \cdot \vec{A} + \epsilon\mu\frac{\partial \phi}{\partial t} = 0

\begin{aligned}
\begin{cases}
    \nabla^2 \vec{A} - \epsilon\mu\frac{\partial^2 \vec{A}}{\partial t^2} = -\mu \vec{J} \\
    \nabla^2 \phi - \epsilon\mu\frac{\partial^2 \phi}{\partial t^2} = -\frac{\rho}{\epsilon}
\end{cases}
\end{aligned}

相速度

v_p = \frac{dz}{dt} = \frac{\omega}{\beta}

群速度

v_g = v_p - \lambda\frac{dv_p}{d\lambda}

无损耗介质的电磁场

\vec{H} = \frac{\vec{e_k}}{Z_0} \times \vec{E}

\vec{E} = Z_0\vec{H} \times \vec{e_k}

均匀平面波的能流密度

\vec{S} = \frac{E^2}{Z}\vec{e_k} = ZH^2\vec{e_k}

有耗媒质的电磁场

\alpha = \omega\sqrt{\frac{\epsilon\mu}{2}(\sqrt{1 + (\frac{\sigma}{\omega\epsilon})^2}) - 1}

\beta = \omega\sqrt{\frac{\epsilon\mu}{2}(\sqrt{1 + (\frac{\sigma}{\omega\epsilon})^2}) + 1}

\vec{k} = \vec{\beta} - j\vec{\alpha}

Z = \sqrt{\frac{\mu}{\epsilon}} = \sqrt{\frac{|\mu|}{|\epsilon|}}e^{j\frac{\delta_e - \delta_m}{2}}

设传播方向为\vec{e}_z

\vec{E} = \vec{E}_{om}e^{-\alpha z}e^{-j\beta z}

良导体

对于导电媒质

\epsilon_c = \epsilon - j\frac{\sigma}{\omega} 

对于良导体

\alpha \approx \beta = \sqrt{\frac{\omega\mu\sigma}{2}} = \sqrt{\pi f \mu \sigma}

Z_c = (1 + j)(\frac{\pi f \mu}{\sigma})

\vec{\overline{S}} = {\sqrt\frac{\sigma}{8\omega\mu}}E_{0m}^2e^{-2\alpha z}\vec{e}_k = {\sqrt\frac{\omega\sigma}{8\sigma}}H_{0m}^2e^{-2\alpha z}\vec{e}_k

集肤深度

\delta = \frac{1}{\alpha} = \sqrt{\frac{1}{\pi f \mu \sigma}}

不良导体

\frac{\sigma}{\omega\epsilon} \approx =1

介质

\frac{\sigma}{\omega\epsilon} \approx << 1

反射和折射

理想导体垂直入射

\Gamma = -1, T = 0, S = \infty

理想介质垂直入射

\Gamma = \frac{Z_2 - Z_1}{Z_2 + Z_1}

T = \frac{2Z_2}{Z_2 + Z_1}

S \geq 1 (取等时\Gamma = 0)

斜入射

平行极化波

\Gamma_{//} = \frac{Z_1cos\theta_i - Z_2cos\theta_t}{Z_1cos\theta_i + Z_2cos\theta_t} = \frac{tan(\theta_i - \theta_i)}{tan(\theta_i + \theta_t)}

T_{//} = \frac{2Z_2cos\theta_i}{Z_1cos\theta_i + Z_2cos\theta_t}

1 + \Gamma_{//} = \sqrt{\frac{\epsilon_2}{\epsilon_1}}T_{//}

全透射，布儒斯特角

\theta_B = arcsin\sqrt{\frac{\epsilon_2}{\epsilon_1 + \epsilon_2}} = arctan\sqrt{\frac{\epsilon_2}{\epsilon_1}}

全反射1 -> 2且\epsilon_2 < \epsilon_1

arcsin\sqrt{\frac{\epsilon_2}{\epsilon_1}} = \theta_C \leq \theta_i \leq \frac{\pi}{2}

垂直极化波

\Gamma_{\perp} = \frac{Z_2cos\theta_i - Z_1cos\theta_t}{Z_2cos\theta_i + Z_1cos\theta_t} = -\frac{sin(\theta_i - \theta_i)}{sin(\theta_i + \theta_t)}

T_{\perp} = \frac{2Z_2cos\theta_i}{Z_2cos\theta_i + Z_1cos\theta_t}

1 + \Gamma_{\perp} = T_{\perp}

反射率\rho = \Gamma^2

透射率\tau = \frac{Z_1cos\theta_t}{Z_2cos\theta_i}T^2

驻波

\vec{E} = 2E_{im}sin\beta z sin\omega t \vec{e}_x

\vec{H} = 2H_{im}cos\beta z cos\omega t \vec{e}_y

\vec{S} = E_{im}H_{im}sin2\beta z sin 2\omega z \vec{e}_z

趋肤深度

表面电阻

R_S = \frac{1}{\delta\sigma}

无反射条件

Z_1 = Z_3 \neq Z_2

则sink_2d = 0

d = n\frac{\lambda_2}{2}, n = 0, 1, 2, ...

Z_1 \neq Z_3

则cosk_2d = 0

d = (2n + 1)\frac{\lambda_2}{2}, n = 0, 1, 2, ...

且

\Z_2 = \sqrt{Z_1Z_2}

电磁波的辐射

方向图函数

F(\theta, \varphi) = \frac{|E(\theta, \varphi)|}{|E_{max}|}

方向性系数

D(\theta, \varphi) = \frac{2\pi F^2}{\int d\varphi\int F^2 sin\theta d\theta}

D_{max} = \frac{E^2_{max}}{E_0^2}\vert_{P_r = P_{r0}} = \frac{P_{r0}}{P_r}\vert_{E = E_0}

在产生最大场强相等的情况下，定义天线增益

G = \frac{P_{i0}}{P_i} = \frac{P_{r0}}{P_i} = \frac{P_r}{P_i}\frac{P_{r0}}{P_r} = \eta\cdot D

电偶极子的辐射场

\vec{A} = \frac{\mu_0I\Delta l}{4\pi r}e^{-jk_0r}\vec{e}_z = \frac{\mu_0I\Delta l}{4\pi r}e^{-jk_0r}(cos\theta \vec{e}_r - sin\theta \vec{e}_\theta)

近区场

平均坡印廷矢量为0，电场和磁场相位差90°，无法辐射出去，称为感应场。

远区场

\vec{E} = j\frac{I\delta l Z_0}{2\lambda} \frac{e^{-j_0kr}}{r}sin\theta \vec{e}_\theta

\vec{H} = j\frac{I\delta l }{2\lambda} \frac{e^{-j_0kr}}{r}sin\theta \vec{e}_\varphi

相位相同，方向垂直

电参数

方向图函数

F(\theta) = sin\theta

辐射功率

\vec{\overline{S}} = Z_0(\frac{I\Delta l}{2\lambda_0 r})^2sin^2\theta \vec{e}_r

P_r = \int \overline{S} dS = 80\pi^2I^2(\frac{\Delta l}{\lambda_0})^2

辐射电阻

R_r = \frac{P_r}{I^2} = 80\pi^2(\frac{\Delta l}{\lambda_0})^2

方向性系数

D = 1.5sin^2\theta

主瓣宽度

2\theta_{0.5} = 90°

磁偶极子的辐射场

远区场

\vec{E} = \frac{\pi ISZ_0}{\lambda_0^2} \frac{e^{-jk_0r}}{r}sin\theta \vec{e}_\varphi

\vec{H} = -\frac{\pi IS}{\lambda_0^2} \frac{e^{-jk_0r}}{r}sin\theta \vec{e}_\theta

电参数

方向图函数

F(\theta) = sin\theta

辐射功率

\vec{\overline{S}} = \frac{\pi^2I^2S^2}{\lambda_0^4r^2}Z_0sin^2\theta \vec{e}_r

P_r = \int \overline{S} dS = \frac{320\pi^4I^2S^2}{\lambda_0^4}

辐射电阻

R_r = \frac{320\pi^4S^2}{\lambda_0^4}

方向性系数

D = 1.5

主瓣宽度

2\theta_{0.5} = 90°

对称半波振子天线

电参数

方向图函数

E = \frac{cos(\frac{\pi}{2}cos\theta)}{sin\theta}

主瓣宽度

2\theta_{0.5} = 78°

辐射功率

P_r = 73.13I^2

辐射电阻

R_r = 73.13\Omega