5 时变电磁场

5.1 麦克斯韦方程组

5.1.1 麦克斯韦第一方程

• 电荷守恒定律（电流连续性方程）

  • 积分形式：${\displaystyle {\oint }_S\mathit{J}\mathbf{\cdot }\mathrm{d}\mathit{S}=-{\displaystyle \frac{\mathrm{d}Q}{\mathrm{d}t}}}$$\d\oint_S \bm J \vdot \dd{\bm S} = -\ddv{Q}{t}$.

  • 微分形式：$\mathbf{\nabla }\mathbf{\cdot }\mathit{J}=-{\displaystyle \frac{\partial \rho }{\partial t}}$$\div \bm J = -\dpdv{\rho}{t}$.

  • 媒质分界：${\mathbf{\nabla }}_t\mathbf{\cdot }{\mathit{J}}_S+({\mathit{J}}_{\text{1n}}-{\mathit{J}}_{\text{2n}})=-{\displaystyle \frac{\partial {\rho }_{\text{s}}}{\partial t}}$$\grad_t \vdot \bm J_S + (\bm J_\text{1n} - \bm J_\text{2n}) = -\dpdv{\rho_\text s}{t}$.

• 广义安培环路定理（麦克斯韦第一方程）

  • 微分形式：$\mathbf{\nabla }\times \mathit{H}=\mathit{J}+{\displaystyle \frac{\partial \mathit{D}}{\partial t}}$$\curl \bm H = \bm J + \d\pdv{\bm D}{t}$.

  • 积分形式：${\displaystyle {\oint }_l\mathit{H}\mathbf{\cdot }\mathrm{d}\mathit{l}={\int }_S(\mathit{J}+\frac{\partial \mathit{D}}{\partial t})\mathbf{\cdot }\mathrm{d}\mathit{S}}$$\d\oint_l \bm H \vdot \dd{\bm l} = \int_S \pqty{\bm J + \pdv{\bm D}{t}} \vdot \dd{\bm S}$.

• 全电流密度 $\mathit{J}+{\mathit{J}}_{\text{d}}$$\bm J + \bm J_\text d$.

  • $\mathit{J}={\mathit{J}}_{\mathrm{i}}+{\mathit{J}}_{\text{c}}+{\mathit{J}}_{\text{v}}$$\bm J = \bm J_\i + \bm J_\text c + \bm J_\text v$.

    • 外加电流密度 ${\mathit{J}}_{\mathrm{i}}$$\bm J_\i$.

    • 传导电流密度 ${\mathit{J}}_{\text{c}}=\sigma \mathit{E}$$\bm J_\text c = \sigma \bm E$.

    • 运流电流密度 ${\mathit{J}}_{\text{v}}=\rho \mathit{v}$$\bm J_\text v = \rho \bm v$.

  • 位移电流密度 ${\mathit{J}}_{\text{d}}={\displaystyle \frac{\partial \mathit{D}}{\partial t}}$$\bm J_\text d = \dpdv{\bm D}{t}$.

    • $\mathit{D}={\epsilon }_0\mathit{E}+\mathit{P}$$\bm D = \ve_0 \bm E + \bm P$.

    • ${\mathit{J}}_{\text{d}}={\epsilon }_0{\displaystyle \frac{\partial \mathit{E}}{\partial t}+\frac{\partial \mathit{P}}{\partial t}}$$\bm J_\text d = \ve_0 \dpdv{\bm E}{t} + \pdv{\bm P}{t}$.

    • 即由变化的电场和电偶极矩组成.

  • 传导电流和位移电流的比较

    • 频率越高，位移电流密度 ${\mathit{J}}_{\text{d}}$$\bm J_\text d$ 越大，而传导电流不受影响.

    • 因此水下无法用高频通信，海军使用长波通信.（微波怕水）

5.1.2 麦克斯韦第二方程

• 总电场 $\mathit{E}={\mathit{E}}_{\text{C}}+{\mathit{E}}_{\text{in}}$$\bm E = \bm E_\text{C} + \bm E_\text{in}$.

  • 库仑电场 ${\mathit{E}}_{\text{C}}$$\bm E_\text{C}$.

  • 感应电场 ${\mathit{E}}_{\text{in}}$$\bm E_\text{in}$.

• 法拉第电磁感应定律

  • ${\mathcal{E}}_{\text{in}}=-{\displaystyle \frac{\partial \mathrm{\Phi }}{\partial t}=-\frac{\mathrm{d}}{\mathrm{d}t}{\int }_S\mathit{B}\mathbf{\cdot }\mathrm{d}\mathit{S}}$$\scr E_\text{in} = -\d\pdv{\Phi}{t} = -\dv{t} \int_S \bm B \vdot \dd{\bm S}$.

  • ${\mathcal{E}}_{\text{in}}={\displaystyle {\oint }_l{\mathit{E}}_{\text{in}}\mathbf{\cdot }\mathrm{d}\mathit{l}}$$\scr E_\text{in} = \d\oint_l \bm E_\text{in} \vdot \dd{\bm l}$.

• 若导体与回路均变化

  • ${\mathcal{E}}_{\text{in}}=-{\displaystyle {\int }_S\frac{\partial \mathit{B}}{\partial t}\mathbf{\cdot }\mathrm{d}\mathit{S}+{\oint }_l(\mathit{v}\times \mathit{B})\mathbf{\cdot }\mathrm{d}\mathit{l}}$$\scr E_\text{in} = -\dint_S \pdv{\bm B}{t} \vdot \dd{\bm S} + \oint_l (\bm v \cp \bm B) \vdot \dd{\bm l}$.

  • $\mathbf{\nabla }\times {\mathit{E}}_{\text{in}}=-{\displaystyle \frac{\partial \mathit{B}}{\partial t}+\mathbf{\nabla }\times (\mathit{v}\times \mathit{B})}$$\curl \bm E_\text{in} = -\d\pdv{\bm B}{t} + \curl (\bm v \cp \bm B)$.

• 麦克斯韦第二方程

  • 积分形式：${\displaystyle {\oint }_l\mathit{E}\mathbf{\cdot }\mathrm{d}\mathit{l}=-{\int }_S\frac{\partial \mathit{B}}{\partial t}\mathbf{\cdot }\mathrm{d}\mathit{S}}$$\d\oint_l \bm E \vdot \dd{\bm l} = -\int_S \pdv{\bm B}{t} \vdot \dd{\bm S}$.

  • 微分形式：$\mathbf{\nabla }\times \mathit{E}=-{\displaystyle \frac{\partial \mathit{B}}{\partial t}}$$\curl \bm E = -\dpdv{\bm B}{t}$.

  • 时变电场有旋，非保守场.

5.1.3 麦克斯韦第三方程

• 积分形式：${\displaystyle {\oint }_S\mathit{D}\cdot \mathrm{d}\mathit{S}=Q}$$\d\oint_S \bm D \cdot \dd{\bm S} = Q$.

• 微分形式：$\mathbf{\nabla }\mathbf{\cdot }\mathit{D}=\rho$$\div \bm D = \rho$.

5.1.4 麦克斯韦第四方程

• 积分形式：${\displaystyle {\oint }_S\mathit{B}\cdot \mathrm{d}\mathit{S}=0}$$\d\oint_S \bm B \cdot \dd{\bm S} = 0$.

• 微分形式：$\mathbf{\nabla }\mathbf{\cdot }\mathit{B}=0$$\div \bm B = 0$.

5.1.5 麦克斯韦方程组

• 麦克斯韦方程组的说明

  • 时变电场有旋有散（电场线可以闭合也可以不闭合），时变磁场有旋无散（磁感线闭合）.

  • 电流和时变电场产生磁场；电荷和时变磁场产生电场.

• 此外还有

  • 电流连续性方程积分形式：${\displaystyle {\oint }_S\mathit{J}\mathbf{\cdot }\mathrm{d}\mathit{S}=-\frac{\partial Q}{\partial t}}$$\d\oint_S \bm J \vdot \dd{\bm S} = -\pdv{Q}{t}$.

  • 电流连续性方程微分形式：$\mathbf{\nabla }\mathbf{\cdot }\mathit{J}=-{\displaystyle \frac{\partial \rho }{\partial t}}$$\div \bm J = -\d\pdv{\rho}{t}$.

  • ${\displaystyle {\oint }_S(\mathit{J}+\frac{\partial \mathit{D}}{\partial t})\mathbf{\cdot }\mathrm{d}\mathit{S}={\int }_V\mathbf{\nabla }\mathbf{\cdot }(\mathbf{\nabla }\times \mathit{H})\mathrm{d}V=0}$$\d\oint_S \pqty{\bm J + \pdv{\bm D}{t}} \bm\cdot \dd{\bm S} = \int_V \div (\curl \bm H) \dd{V} = 0$.

• 后两个散度方程，可以由前两个旋度方程和电流连续性方程推出.

5.1.6 媒质的本构方程

• 三个基本的媒质参数 (本构参数)

  • 介电常数：$\epsilon ={\epsilon }_0{\epsilon }_{\text{r}}$$\ve = \ve_0 \ve_\text r$.

  • 磁导率：$\mu ={\mu }_0{\mu }_{\text{r}}$$\mu = \mu_0 \mu_\text r$.

  • 电导率：$\sigma$$\sigma$.

• 媒质的三种分类

  • 若媒质参数与空间位置无关, 则称为 均匀媒质.

  • 若媒质参数与场强大小无关, 则称为 线性媒质.

  • 若媒质参数与场强方向无关, 则称为 各向同性媒质.

5.2 时变电磁场的边界条件

5.2.1 法向场的边界条件

• 电位移矢量

  • ${\mathit{e}}_{\text{n}}\mathbf{\cdot }({\mathit{D}}_1-{\mathit{D}}_2)={\rho }_{\text{s}}$$\bm e_\text n \bm\cdot (\bm D_1 - \bm D_2) = \rho_\text s$.

  • $D_{\text{1n}}-D_{\text{2n}}={\rho }_{\text{s}}$$D_\text{1n} - D_\text{2n} = \rho_\text s$.

  • 若面电荷密度 ${\rho }_{\text{s}}$$\rho_\text s$ 非零，则不连续.

• 电场强度

  • ${\epsilon }_1{E}_{\text{1n}}-{\epsilon }_2{E}_{\text{2n}}={\rho }_{\text{s}}$$\ve_1 E_\text{1n} - \ve_2 E_\text{2n} = \rho_\text{s}$.

  • 即使面电荷密度 ${\rho }_{\text{s}}$$\rho_\text s$ 非零，仍然不连续.

• 磁感应强度

  • ${\mathit{e}}_{\text{n}}\mathbf{\cdot }({\mathit{B}}_1-{\mathit{B}}_2)=0$$\bm e_\text n \bm\cdot (\bm B_1 - \bm B_2) = 0$.

  • $B_{\text{1n}}=B_{\text{2n}}$$B_\text{1n} = B_\text{2n}$.

  • 磁感应强度的法向分量连续.

• 磁场强度

  • ${\mu }_1{H}_{\text{1n}}={\mu }_2{H}_{\text{2n}}$$\mu_1 H_\text{1n} = \mu_2 H_\text{2n}$.

  • 磁场强度的法向分量不连续.

5.2.2 切向场的边界条件

• 电场强度

  • ${\mathit{e}}_{\text{n}}\times ({\mathit{E}}_1-{\mathit{E}}_2)=\mathbf{0}$$\bm e_\text n \cp (\bm E_1 - \bm E_2) = \bm 0$.

  • $E_{\text{1t}}=E_{\text{2t}}$$E_\text{1t} = E_\text{2t}$.

  • 电场强度的切向分量连续.

• 电位移矢量

  • ${\displaystyle \frac{D_{\text{1t}}}{{\epsilon }_1}}={\displaystyle \frac{D_{\text{2t}}}{{\epsilon }_2}}$$\dfrac{D_\text{1t}}{\ve_1} = \dfrac{D_\text{2t}}{\ve_2}$.

  • 电位移矢量的切向分量不连续.

• 磁场强度

  • ${\mathit{e}}_{\text{n}}\times ({\mathit{H}}_1-{\mathit{H}}_2)={\mathit{J}}_{\text{s}}$$\bm e_\text n \cp (\bm H_1 - \bm H_2) = \bm J_\text s$.

  • $H_{\text{1t}}-H_{\text{2t}}=J_{\text{s}}$$H_\text{1t} - H_\text{2t} = J_\text s$.

  • 若媒质分界面没有自由电流, 则磁场强度切向分量连续.

• 磁感应强度

  • ${\displaystyle \frac{B_{\text{1t}}}{{\mu }_1}}-{\displaystyle \frac{B_{\text{2t}}}{{\mu }_2}}=J_{\text{s}}$$\dfrac{B_\text{1t}}{\mu_1} - \dfrac{B_\text{2t}}{\mu_2} = J_\text{s}$.

  • 即使分界面没有自由电流, 磁感应强度的切向分量也不连续.

5.2.3 边界条件公式总结

备注 理想介质 $\sigma =0$$\sigma = 0$，理想导体 $\sigma =\mathrm{\infty }$$\sigma = \infty$.

5.3 坡应廷定理

5.3.1 能量密度与损耗功率

• 能量密度

  • 电场：$w_{\mathrm{e}}={\displaystyle \frac{1}{2}}\mathit{D}(\mathit{r},t)\mathbf{\cdot }\mathit{E}(\mathit{r},t)={\displaystyle \frac{1}{2}}\epsilon E^2(\mathit{r},t)$$w_\e = \dfrac{1}{2} \bm D(\bm r, t) \vdot \bm E(\bm r, t) = \dfrac{1}{2} \ve E^2(\bm r, t)$.

  • 磁场：$w_{\text{m}}={\displaystyle \frac{1}{2}}\mathit{B}(\mathit{r},t)\mathbf{\cdot }\mathit{H}(\mathit{r},t)={\displaystyle \frac{1}{2}}\mu H^2(\mathit{r},t)$$w_\text m = \dfrac{1}{2} \bm B(\bm r, t) \vdot \bm H(\bm r, t) = \dfrac{1}{2} \mu H^2(\bm r, t)$.

  • 电磁场：$w=w_{\text{e}}+w_{\text{m}}={\displaystyle \frac{1}{2}}\epsilon E^2(\mathit{r},t)+{\displaystyle \frac{1}{2}}\mu H^2(\mathit{r},t)$$w = w_\text e + w_\text m = \dfrac{1}{2} \ve E^2(\bm r, t) + \dfrac{1}{2} \mu H^2(\bm r, t)$.

• 损耗功率密度：$p=\mathit{J}(\mathit{r},t)\mathbf{\cdot }\mathit{E}(\mathit{r},t)=\sigma E^2(\mathit{r},t)$$p = \bm J(\bm r, t) \vdot \bm E(\bm r, t) = \sigma E^2(\bm r, t)$.

5.3.2 坡应廷定理物理含义

• 损耗功率

  $$\begin{array}{rl}P& ={\int }_V\mathit{J}\mathbf{\cdot }\mathit{E}\mathrm{d}V={\int }_V[(\mathbf{\nabla }\times \mathit{H})\mathbf{\cdot }\mathit{E}-\frac{\partial \mathit{D}}{\partial t}\mathbf{\cdot }\mathit{E}]\mathrm{d}V\\ & ={\int }_V[(\mathbf{\nabla }\times \mathit{E})\mathbf{\cdot }\mathit{H}-\mathbf{\nabla }\mathbf{\cdot }(\mathit{E}\times \mathit{H})-\frac{\partial \mathit{D}}{\partial t}\mathbf{\cdot }\mathit{E}]\mathrm{d}V\\ & =-{\int }_V[\frac{\partial \mathit{B}}{\partial t}\mathbf{\cdot }\mathit{H}+\frac{\partial \mathit{D}}{\partial t}\mathbf{\cdot }\mathit{E}+\mathbf{\nabla }\mathbf{\cdot }(\mathit{E}\times \mathit{H})]\mathrm{d}V\\ & =-{\int }_V[{\displaystyle \frac{\mu }{2}}\frac{\partial }{\partial t}(\mathit{H}\mathbf{\cdot }\mathit{H})+{\displaystyle \frac{\epsilon }{2}}\frac{\partial }{\partial t}(\mathit{E}\mathbf{\cdot }\mathit{E})+\mathbf{\nabla }\mathbf{\cdot }(\mathit{E}\times \mathit{H})]\mathrm{d}V\\ & =-\frac{\partial }{\partial t}{\int }_V({\displaystyle \frac{\mathit{B}\mathbf{\cdot }\mathit{H}}{2}}+{\displaystyle \frac{\mathit{D}\mathbf{\cdot }\mathit{E}}{2}})\mathrm{d}V-{\oint }_{S^{\prime }}(\mathit{E}\times \mathit{H})\mathbf{\cdot }\mathrm{d}{\mathit{S}}^{\prime }.\end{array}$$

• 坡应廷定理

  $${\displaystyle \frac{\partial }{\partial t}{\int }_V({\displaystyle \frac{\mathit{B}\mathbf{\cdot }\mathit{H}}{2}}+{\displaystyle \frac{\mathit{D}\mathbf{\cdot }\mathit{H}}{2}})\mathrm{d}V+{\int }_V\mathit{J}\mathbf{\cdot }\mathit{E}\mathrm{d}V+{\oint }_{S^{\prime }}(\mathit{E}\times \mathit{H})\mathbf{\cdot }\mathrm{d}{\mathit{S}}^{\prime }=0.}$$

  • 第一项表示体积 $V$$V$ 内单位时间内存储的总电磁能量 (电磁功率).

  • 第二项表示传导电流 $\mathit{J}$$\bm J$ 引起的损耗功率.

  • 第三项表示通过闭曲面 ${\mathit{S}}^{\prime }$$\bm S'$ 离开体积 $V$$V$ 的总功率.

5.3.3 坡应廷矢量的瞬时值

• 坡印廷矢量的瞬时值 $\mathit{S}=\mathit{E}\times \mathit{H}$$\bm S = \bm E \cp \bm H$.

  • 表示闭曲面上通过单位面积的功率.

  • 方向是能量流动的方向，单位是 $\mathrm{W}/{\mathrm{m}}^2$$\mathrm{W / m^2}$.

  • 坡印廷矢量又称为能量流密度或功率密度.

• 特例

  • 在静电场和恒定磁场中，没有电磁能量流动.

  • 在恒定电场和恒定磁场中，$-\mathit{S}$$-\bm S$ 表示 $V$$V$ 内的损耗功率.

  • 在时变电场中，$\mathit{S}$$\bm S$ 表示瞬时功率密度.

5.4 时谐电磁场与复数形式

5.4.1 时谐电磁场的复数形式

任意时变场可分解为正弦时间函数表示的时谐场.

• 时谐电磁场

  $$\begin{array}{cc}\begin{array}{rcl}\mathit{E}(\mathit{r},t)& =& {\displaystyle {\mathit{E}}_{\text{m}}(\mathit{r})\cos [\omega t+\varphi (\mathit{r})]}\\ & =& {\displaystyle \mathrm{Re}[{\mathit{E}}_{\text{m}}(\mathit{r}){\mathrm{e}}^{\mathrm{j}[\omega t+\varphi (\mathit{r})]}]}\\ & =& {\displaystyle \mathrm{Re}[{\dot{\mathit{E}}}_{\text{m}}(\mathit{r}){\mathrm{e}}^{\mathrm{j}\omega t}].}\end{array}& \begin{array}{rcl}\mathit{E}(\mathit{r},t)& =& {\displaystyle \sqrt{2}\mathit{E}(\mathit{r})\cos [\omega t+\varphi (\mathit{r})]}\\ & =& {\displaystyle \mathrm{Re}[\sqrt{2}\mathit{E}(\mathit{r}){\mathrm{e}}^{\mathrm{j}[\omega t+\varphi (\mathit{r})]}]}\\ & =& {\displaystyle \mathrm{Re}[\sqrt{2}\dot{\mathit{E}}(\mathit{r}){\mathrm{e}}^{\mathrm{j}\omega t}].}\end{array}\end{array}$$

  • 概念

    • 空间相位 $\varphi (\mathit{r})$$\phi(\bm r)$.

    • 时间因子 ${\mathrm{e}}^{\mathrm{j}\omega t}$$\e^{\j\omega t}$.

  • 时域

    • 振幅值矢量 ${\mathit{E}}_{\text{m}}(\mathit{r})=\sqrt{2}\mathit{E}(\mathit{r})$$\bm E_\text m(\bm r) = \sqrt 2 \bm E(\bm r)$.

    • 有效值矢量 $\mathit{E}(\mathit{r})={\displaystyle \frac{\sqrt{2}}{2}}{\mathit{E}}_{\text{m}}(\mathit{r})$$\bm E(\bm r) = \dfrac{\sqrt 2}{2} \bm E_\text m (\bm r)$.

  • 频域

    • 复振幅矢量 ${\dot{\mathit{E}}}_{\text{m}}(\mathit{r})={\mathit{E}}_{\text{m}}(\mathit{r}){\mathrm{e}}^{\mathrm{j}\varphi (\mathit{r})}$$\dot{\bm E}_\text m(\bm r) = \bm E_\text m(\bm r) \e^{\j \phi(\bm r)}$.

    • 复有效矢量 $\dot{\mathit{E}}(\mathit{r})={\displaystyle \frac{\sqrt{2}}{2}}{\mathit{E}}_{\text{m}}(\mathit{r}){\mathrm{e}}^{\mathrm{j}\varphi (\mathit{r})}$$\dot{\bm E}(\bm r) = \dfrac{\sqrt 2}{2} \bm E_\text m(\bm r) \e^{\j\phi(\bm r)}$.

• 电场矢量的瞬时值形式与复数形式的转换

  $$\begin{array}{rl}\frac{\partial }{\partial t}\mathit{E}(x,y,z,t)& =-{\mathit{E}}_{\text{m}}(x,y,z)\omega \sin [\omega t+\varphi (x,y,z)]\\ & =\mathrm{Re}[\mathrm{j}\omega {\mathit{E}}_{\text{m}}(x,y,z){\mathrm{e}}^{\mathrm{j}\omega t}].\end{array}$$

  • $\mathit{E}(x,y,z,t)\leftrightarrow {\dot{\mathit{E}}}_{\text{m}}(x,y,z)$$\bm E(x, y, z, t) \rla \dot{\bm E}_\text m (x, y, z)$.

  • ${\displaystyle \frac{\partial }{\partial t}\mathit{E}(x,y,z,t)\leftrightarrow \mathrm{j}\omega {\dot{\mathit{E}}}_{\text{m}}(x,y,z)}$$\dpdv{t} \bm E(x, y, z, t) \rla \j\omega \dot{\bm E}_\text m (x, y, z)$.

  • ${\displaystyle {\int }_{-\mathrm{\infty }}^t\mathit{E}(x,y,z,t)\mathrm{d}t\leftrightarrow {\displaystyle \frac{{\dot{\mathit{E}}}_{\text{m}}(x,y,z)}{\mathrm{j}\omega }}}$$\dint_{-\infty}^t \bm E(x, y, z, t) \dt \rla \dfrac{\dot{\bm E}_\text m (x, y, z)}{\j\omega}$.

5.4.2 麦克斯方程组复数形式

• 麦克斯韦方程组为

电流连续性方程为

备注

• 有的教材习惯上将 ${\dot{\mathit{E}}}_{\text{m}}$$\dot{\bm E}_\text m$ 简写为 $\mathit{E}$$\bm E$，诸如此类. 这里为了清晰起见，不采用该记号.

• 由于幅值与有效值满足 ${\dot{\mathit{E}}}_{\text{m}}(\mathit{r})=\sqrt{2}\dot{\mathit{E}}(\mathit{r})$$\dot{\bm E}_\text m (\bm r) = \sqrt 2 \dot{\bm E} (\bm r)$，也可以将上式改成有效值的形式.

5.4.3 能量密度的复数表示法

• 瞬时值

  • $w_{\mathrm{e}}(\mathit{r},t)={\displaystyle \frac{1}{2}}\epsilon E^2(\mathit{r},t)$$w_\e(\bm r, t) = \dfrac{1}{2} \ve E^2(\bm r, t)$.

  • $w_{\text{m}}(\mathit{r},t)={\displaystyle \frac{1}{2}}\mu H^2(\mathit{r},t)$$w_\text m (\bm r, t) = \dfrac{1}{2} \mu H^2(\bm r, t)$.

  • $w(\mathit{r},t)=w_{\mathrm{e}}(\mathit{r},t)+w_{\text{m}}(\mathit{r},t)$$w(\bm r, t) = w_\e(\bm r, t) + w_\text m(\bm r, t)$.

  • $p(\mathit{r},t)=\sigma E^2(\mathit{r},t)$$p(\bm r, t) = \sigma E^2(\bm r, t)$.

• 极大值

  • $w_{\mathrm{e}max}(\mathit{r})={\displaystyle \frac{1}{2}}\epsilon E_{\text{m}}^2(\mathit{r})={\displaystyle \frac{\epsilon }{2}}{\dot{\mathit{E}}}_{\text{m}}(\mathit{r})\mathbf{\cdot }{\dot{\mathit{E}}}_{\text{m}}^{\ast }(\mathit{r})$$w_{\e \max}(\bm r) = \dfrac{1}{2} \ve E^2_\text m(\bm r) = \dfrac{\ve}{2} \dot{\bm E}_\text m (\bm r) \vdot \dot{\bm E}_\text m^* (\bm r)$.

  • $w_{\text{m}max}(\mathit{r})={\displaystyle \frac{1}{2}}\mu H_{\text{m}}^2(\mathit{r})={\displaystyle \frac{\mu }{2}}{\dot{\mathit{H}}}_{\text{m}}(\mathit{r})\mathbf{\cdot }{\dot{\mathit{H}}}_{\text{m}}^{\ast }(\mathit{r})$$w_{\text m \max}(\bm r) = \dfrac{1}{2} \mu H^2_\text m(\bm r) = \dfrac{\mu}{2} \dot{\bm H}_\text m(\bm r) \vdot \dot{\bm H}_\text m^* (\bm r)$.

  • $w_{max}(\mathit{r})=w_{\mathrm{e}max}(\mathit{r})+w_{\text{m}max}(\mathit{r})$$w_\max(\bm r) = w_{\e \max}(\bm r) + w_{\text m \max}(\bm r)$.

  • $p_{max}(\mathit{r})=\sigma E_{\text{m}}^2(\mathit{r})=\sigma {\dot{\mathit{E}}}_{\text{m}}(\mathit{r})\mathbf{\cdot }{\dot{\mathit{E}}}_{\text{m}}(\mathit{r})$$p_\max(\bm r) = \sigma E^2_\text m(\bm r) = \sigma \dot{\bm E}_\text m (\bm r) \vdot \dot{\bm E}_\text m (\bm r)$.

• 平均值

  • $w_{\mathrm{e}\mathrm{avg}}(\mathit{r})={\displaystyle \frac{\epsilon }{4}}{\dot{\mathit{E}}}_{\text{m}}(\mathit{r})\mathbf{\cdot }{\dot{\mathit{E}}}_{\text{m}}^{\ast }(\mathit{r})={\displaystyle \frac{\epsilon }{2}}\dot{\mathit{E}}(\mathit{r})\mathbf{\cdot }{\dot{\mathit{E}}}^{\ast }(\mathit{r})$$w_{\e \avg}(\bm r) = \dfrac{\ve}{4} \dot{\bm E}_\text m (\bm r) \vdot \dot{\bm E}_\text m^* (\bm r) = \dfrac{\ve}{2} \dot{\bm E}(\bm r) \vdot \dot{\bm E}^* (\bm r)$.

  • $w_{\text{m}\mathrm{avg}}(\mathit{r})={\displaystyle \frac{\mu }{4}}{\dot{\mathit{H}}}_{\text{m}}(\mathit{r})\mathbf{\cdot }{\dot{\mathit{H}}}_{\text{m}}^{\ast }(\mathit{r})={\displaystyle \frac{\mu }{2}}\dot{\mathit{H}}(\mathit{r})\mathbf{\cdot }{\dot{\mathit{H}}}^{\ast }(\mathit{r})$$w_{\text m \avg}(\bm r) = \dfrac{\mu}{4} \dot{\bm H}_\text m(\bm r) \vdot \dot{\bm H}_\text m^* (\bm r) = \dfrac{\mu}{2} \dot{\bm H}(\bm r) \vdot \dot{\bm H}^*(\bm r)$.

  • $w_{\mathrm{avg}}(\mathit{r})=w_{\mathrm{e}\mathrm{avg}}(\mathit{r})+w_{\text{m}\mathrm{avg}}(\mathit{r})$$w_\avg(\bm r) = w_{\e \avg}(\bm r) + w_{\text m \avg}(\bm r)$.

  • $p_{\mathrm{avg}}(\mathit{r})={\displaystyle \frac{\sigma }{2}}{\dot{\mathit{E}}}_{\text{m}}(\mathit{r})\mathbf{\cdot }{\dot{\mathit{E}}}_{\text{m}}^{\ast }(\mathit{r})={\dot{\mathit{J}}}^{\ast }(\mathit{r})\mathbf{\cdot }\dot{\mathit{E}}(\mathit{r})$$p_\avg(\bm r) = \dfrac{\sigma}{2} \dot{\bm E}_\text m (\bm r) \vdot \dot{\bm E}_\text m^* (\bm r) = \dot{\bm J}^*(\bm r) \vdot \dot{\bm E}(\bm r)$.

5.4.4 坡应廷定理的复数形式

• 损耗功率的平均值

  $$\begin{array}{rl}P& ={\int }_V{\dot{\mathit{J}}}^{\ast }\mathbf{\cdot }\dot{\mathit{E}}\mathrm{d}V={\int }_V[(\mathbf{\nabla }\times {\dot{\mathit{H}}}^{\ast })\mathbf{\cdot }\dot{\mathit{E}}+\mathrm{j}\omega {\dot{\mathit{D}}}^{\ast }\mathbf{\cdot }\dot{\mathit{E}}]\mathrm{d}V\\ & ={\int }_V[(\mathbf{\nabla }\times \dot{\mathit{E}})\mathbf{\cdot }{\dot{\mathit{H}}}^{\ast }-\mathbf{\nabla }\mathbf{\cdot }(\dot{\mathit{E}}\times {\dot{\mathit{H}}}^{\ast })+\mathrm{j}\omega {\dot{\mathit{D}}}^{\ast }\mathbf{\cdot }\dot{\mathit{E}}]\mathrm{d}V\\ & =-{\int }_V[\mathrm{j}\omega \dot{\mathit{B}}\mathbf{\cdot }{\dot{\mathit{H}}}^{\ast }-\mathrm{j}\omega \dot{\mathit{E}}\mathbf{\cdot }{\dot{\mathit{D}}}^{\ast }+\mathbf{\nabla }\mathbf{\cdot }(\dot{\mathit{E}}\times {\dot{\mathit{H}}}^{\ast })]\mathrm{d}V\\ & =-\mathrm{j}\omega {\int }_V(\mu \dot{\mathit{H}}\mathbf{\cdot }{\dot{\mathit{H}}}^{\ast }-\epsilon \dot{\mathit{E}}\mathbf{\cdot }{\dot{\mathit{E}}}^{\ast })\mathrm{d}V-{\oint }_S(\dot{\mathit{E}}\times {\dot{\mathit{H}}}^{\ast })\mathbf{\cdot }\mathrm{d}S.\end{array}$$

• 坡应廷定理

  $$-{\oint }_S(\dot{\mathit{E}}\times {\dot{\mathit{H}}}^{\ast })\mathbf{\cdot }\mathrm{d}\mathit{S}={\int }_V\sigma \dot{\mathit{E}}\mathbf{\cdot }{\dot{\mathit{E}}}^{\ast }\mathrm{d}V+\mathrm{j}\omega {\int }_V(\mu \dot{\mathit{H}}\mathbf{\cdot }{\dot{\mathit{H}}}^{\ast }-\epsilon \dot{\mathit{E}}\mathbf{\cdot }{\dot{\mathit{E}}}^{\ast })\mathrm{d}V.$$

  • 右式

    • 第一项表示损耗功率的平均值（有功功率）

    • 第二项表示交换的电磁场功率（无功功率）

    • 第二项的积分表示电磁场能量（不是功率）

  • 左式

    • 左式的实部表示流入的有功功率（用于补偿损耗功率）

    • 左式的虚部表示流入的无功功率（与电磁场功率交换）

5.4.5 复坡应廷矢量物理含义

• 平均能流密度矢量

  • $\mathit{S}(\mathit{r},t)=\mathit{E}(\mathit{r},t)\times \mathit{H}(\mathit{r},t)=\mathrm{Re}(\dot{\mathit{E}}\times {\dot{\mathit{H}}}^{\ast })+\mathrm{Re}(\dot{\mathit{E}}\times \dot{\mathit{H}}{\mathrm{e}}^{2\mathrm{j}\omega t})$$\bm S(\bm r, t) = \bm E(\bm r, t) \cp \bm H(\bm r, t) = \Re\pqty{\dot{\bm E} \cp \dot{\bm H}^*} + \Re\pqty{\dot{\bm E} \cp \dot{\bm H} \e^{2\j\omega t}}$.

  • ${\mathit{S}}_{\mathrm{avg}}(\mathit{r})={\displaystyle \frac{1}{T}}{\displaystyle {\int }_0^T\mathit{S}(\mathit{r},t)\mathrm{d}t=\mathrm{Re}(\dot{\mathit{E}}\times {\dot{\mathit{H}}}^{\ast })}$$\bm S_\avg(\bm r) = \dfrac{1}{T} \dint_0^T \bm S(\bm r, t) \dt = \Re\pqty{\dot{\bm E} \cp \dot{\bm H}^*}$.

• 复坡应廷矢量（复数能流密度矢量）

  • $\tilde{\mathit{S}}(\mathit{r})=\dot{\mathit{E}}(\mathit{r})\times {\dot{\mathit{H}}}^{\ast }(\mathit{r})={\displaystyle \frac{1}{2}}{\dot{\mathit{E}}}_{\text{m}}(\mathit{r})\times {\dot{\mathit{H}}}_{\text{m}}^{\ast }(\mathit{r})$$\wt{\bm S}(\bm r) = \dot{\bm E}(\bm r) \cp \dot{\bm H}^*(\bm r) = \dfrac{1}{2} \dot{\bm E}_\text m(\bm r) \cp \dot{\bm H}_\text m^*(\bm r)$.

  • 其角频率为 $\dot{\mathit{E}}$$\dot{\bm E}$ 的两倍，因此不能由瞬时值方程取复数得到.

  • 有功功率密度：$\mathrm{Re}[\tilde{\mathit{S}}(\mathit{r})]={\mathit{S}}_{\mathrm{avg}}(\mathit{r})$$\Re\bqty{\wt{\bm S}(\bm r)} = \bm S_\avg(\bm r) %= \dfrac{1}{T} \dint_0^T \bm S(\bm r, t) \dt$.

  • 无功功率密度：$\mathrm{Im}[\tilde{\mathit{S}}(\mathit{r})]$$\Im\bqty{\wt{\bm S}(\bm r)}$.

5.4.6 各种形式的记号的总结

• 有坐标有时间

  • 粗体 $\mathit{E}(\mathit{r},t)$$\bm E(\bm r, t)$ 和 ${\mathit{E}}_x(\mathit{r},t)$$\bm E_x(\bm r, t)$ 表示瞬时值向量.

  • 斜体 $E(\mathit{r},t)$$E(\bm r, t)$ 和 $E_x(\mathit{r},t)$$E_x(\bm r, t)$ 表示瞬时值模长.

• 有坐标无时间

  • ${\mathit{E}}_{\text{m}}(\mathit{r}),E_{x\text{m}}(\mathit{r}),{\mathit{E}}_{x\text{m}}(\mathit{r})$$\bm E_\text m(\bm r), E_{x \text m}(\bm r), \bm E_{x \text m}(\bm r)$ 表示振幅值.

  • $\mathit{E}(\mathit{r}),E_x(\mathit{r}),{\mathit{E}}_x(\mathit{r})$$\bm E(\bm r), E_x(\bm r), \bm E_x(\bm r)$ 表示有效值（非平均值）

  • $w_{max}(\mathit{r}),w_{\mathrm{avg}}(\mathit{r})$$w_\max(\bm r), w_\avg(\bm r)$ 分别表示极大值与平均值.

• 无坐标无时间

  • 如向量 $\mathit{E},{\mathit{E}}_x$$\bm E, \bm E_x$，或标量 $E,E_x$$E, E_x$ 等.

  • 一般是瞬时值的简写（而非振幅值或有效值）

  • 不一定表示与坐标与时间无关，只是为了方便.

• 频域（相量）

  • ${\dot{\mathit{E}}}_{\text{m}}(\mathit{r}),{\dot{E}}_{x\text{m}}(\mathit{r}),{\dot{\mathit{E}}}_{x\text{m}}(\mathit{r})$$\dot{\bm E}_\text m(\bm r), \dot{E}_{x \text m}(\bm r), \dot{\bm E}_{x \text m}(\bm r)$ 表示复振幅.

  • $\dot{\mathit{E}}(\mathit{r}),{\dot{E}}_x(\mathit{r}),{\dot{\mathit{E}}}_x(\mathit{r})$$\dot{\bm E}(\bm r), \dot{E}_{x}(\bm r), \dot{\bm E}_{x}(\bm r)$ 表示复有效值.

  • $\tilde{\mathit{S}}(\mathit{r})$$\wt{\bm S}(\bm r)$ 表示另一个频率的复数.

• 注意：

  • 时域与频域的转换方式有两种

    1. 以余弦为基准，复数取实部得到瞬时值.

    2. 以正弦为基准，复数取虚部得到瞬时值.

    二者不可混用，若取实部则虚部无实际含义.

    一般以余弦为基准，如果用正弦请事先说明.

  • 区分 $0,\mathbf{0},\dot{0},\dot{\mathbf{0}}$$0, \bm 0, \dot0, \dot{\bm 0}$，不过它们常简写为 $0$$0$ 或 $\mathbf{0}$$\bm 0$.

  • 有时候无坐标无时间的量可能表示其它含义，可能只是某个函数的记号，以具体情况中的说明为准.

5.5 波动方程

5.5.1 非齐次波动方程

• 非齐次波动方程

  • ${\displaystyle {\mathbf{\nabla }}^2\mathit{E}-\mu \epsilon \frac{{\partial }^2\mathit{E}}{\partial t^2}=\mu \frac{\partial \mathit{J}}{\partial t}+{\displaystyle \frac{\mathbf{\nabla }\rho }{\epsilon }}}$$\d \grad^2 \bm E - \mu\ve \pdv[2]{\bm E}{t} = \mu \pdv{\bm J}{t} + \dfrac{\grad\rho}{\ve}$.

  • ${\displaystyle {\mathbf{\nabla }}^2\mathit{H}-\mu \epsilon \frac{{\partial }^2\mathit{H}}{\partial t^2}=-\mathbf{\nabla }\times \mathit{J}}$$\d \grad^2 \bm H - \mu \ve \pdv[2]{\bm H}{t} = -\curl \bm J$.

• 波动方程的说明

  • $\mathit{J}={\mathit{J}}^{\prime }+\sigma \mathit{E}$$\bm J = \bm J' + \sigma \bm E$，${\mathit{J}}^{\prime }$$\bm J'$ 为非电性外加源等效电流.

  • 总电流满足电流连续性方程：$\mathbf{\nabla }\mathbf{\cdot }\mathit{J}=-{\displaystyle \frac{\partial \rho }{\partial t}}$$\div \bm J = -\dpdv{\rho}{t}$.

• 波动方程的标准形式

  • ${\displaystyle {\mathbf{\nabla }}^{2}u(\mathit{r},t)-{\displaystyle \frac{1}{v^2}}\frac{{\partial }^{2}u(\mathit{r},t)}{\partial t^2}=-f(\mathit{r},t)}$$\d \grad^2 u(\bm r, t) - \dfrac{1}{v^2} \pdv[2]{u(\bm r, t)}{t} = - f(\bm r, t)$.

  • $u$$u$ 为待求场量，$v$$v$ 为波速，$f$$f$ 为自由项.

5.5.2 齐次波动方程

对于无源理想介质，$\mathit{J}={\mathit{J}}^{\prime }=\mathbf{0},\sigma =0,\epsilon ,\mu \in \mathbb{R}$$\bm J = \bm J' = \bm 0, \sigma = 0, \ve, \mu \in \R$，

• ${\mathbf{\nabla }}^2\mathit{E}-\mu \epsilon {\displaystyle \frac{{\partial }^2\mathit{E}}{\partial t^2}=\mathbf{0}}$$\grad^2 \bm E - \mu \ve \dpdv[2]{\bm E}{t} = \bm 0$.

• ${\mathbf{\nabla }}^2\mathit{H}-\mu \epsilon {\displaystyle \frac{{\partial }^2\mathit{H}}{\partial t^2}=\mathbf{0}}$$\grad^2 \bm H - \mu \ve \dpdv[2]{\bm H}{t} = \bm 0$.

5.5.3 齐次亥姆霍兹方程

对于无源理想介质中的时谐场，有

• 齐次亥姆霍兹方程

  • ${\mathbf{\nabla }}^2\dot{\mathit{E}}+k^2\dot{\mathit{E}}=\dot{\mathbf{0}}$$\grad^2 \dot{\bm E} + k^2 \dot{\bm E} = \dot{\bm 0}$.

  • ${\mathbf{\nabla }}^2\dot{\mathit{H}}+k^2\dot{\mathit{H}}=\dot{\mathbf{0}}$$\grad^2 \dot{\bm H} + k^2 \dot{\bm H} = \dot{\bm 0}$.

• 符号说明

  • $k^2={\omega }^2\mu \epsilon ={\left({\displaystyle \frac{\omega }{v}}\right)}^2$$k^2 = \omega^2 \mu \ve = \pqty{\dfrac{\omega}{v}}^2$.

  • 波速：$v={\displaystyle \frac{1}{\sqrt{\mu \epsilon }}}$$v = \dfrac{1}{\sqrt{\mu \ve}}$.

5.6 时变场的标量位与矢量位

5.6.1 标量位与矢量位及其多值性

• 矢量磁位

  • 由于 $\mathbf{\nabla }\mathbf{\cdot }\mathit{B}=0$$\div \bm B = 0$，

  • 定义 $\mathit{B}\equiv \mathbf{\nabla }\times \mathit{A}$$\bm B \equiv \curl \bm A$.

  • 单位为 $\mathrm{Wb}/\mathrm{m}$$\rm{Wb/m}$.

• 标量磁位

  • 由于 ${\displaystyle \mathbf{\nabla }\times (\mathit{E}+\frac{\partial \mathit{A}}{\partial t})=\mathbf{0}}$$\d \curl (\bm E + \pdv{\bm A}{t}) = \bm 0$，

  • 定义 $-\mathbf{\nabla }\varphi =\mathit{E}+{\displaystyle \frac{\partial \mathit{A}}{\partial t}}$$-\grad \phi = \bm E + \dpdv{\bm A}{t}$.

  • 单位为 $\mathrm{V}$$\rm V$.

• 多值性：对于 $\mathrm{\forall }f$$\forall f$，有

  • $\mathbf{\nabla }\times (\mathit{A}+\mathbf{\nabla }f)=\mathbf{0}$$\curl (\bm A + \grad f) = \bm 0$.

  • ${\displaystyle -\mathbf{\nabla }(\varphi -\frac{\partial f}{\partial t})=\mathit{E}+\frac{\partial }{\partial t}(\mathit{A}+\mathbf{\nabla }f)}$$\d -\grad (\phi - \pdv{f}{t}) = \bm E + \pdv{t} (\bm A + \grad f)$.

5.6.2 洛伦兹规范与达朗贝尔方程

• 洛伦兹规范：$\mathbf{\nabla }\mathbf{\cdot }\mathit{A}=-\mu \epsilon {\displaystyle \frac{\partial \varphi }{\partial t}}$$\div \bm A = -\mu \ve \dpdv{\phi}{t}$.

• 达朗贝尔方程

  • ${\mathbf{\nabla }}^2\mathit{A}-\mu \epsilon {\displaystyle \frac{{\partial }^2\mathit{A}}{\partial t^2}=-\mu \mathit{J}}$$\grad^2 \bm A - \mu \ve \d\pdv[2]{\bm A}{t} = -\mu \bm J$.

  • ${\mathbf{\nabla }}^2\varphi -\mu \epsilon {\displaystyle \frac{{\partial }^2\varphi }{\partial t^2}=-{\displaystyle \frac{\rho }{\epsilon }}}$$\grad^2 \phi - \mu \ve \d\pdv[2]{\phi}{t} = -\dfrac{\rho}{\ve}$.

5.6.3 达朗贝尔方程解的形式

• 标量磁位达朗贝尔方程的特例

  • 当 $\varphi$$\phi$ 与时间无关时，成为泊松方程：${\mathbf{\nabla }}^2\varphi =-{\displaystyle \frac{\rho }{\epsilon }}$$\grad^2 \phi = -\dfrac{\rho}{\ve}$.

  • 当场点在源点 $\rho$$\rho$ 外时，为齐次波动方程：${\mathbf{\nabla }}^2\varphi -\mu \epsilon {\displaystyle \frac{{\partial }^2\varphi }{\partial t^2}=0}$$\grad^2 \phi - \mu\ve \dpdv[2]{\phi}{t} = 0$.

• 解的形式

  • $\varphi (\mathit{r},t)={\displaystyle \frac{1}{4\pi \epsilon }}{\displaystyle {\int }_{V^{\prime }}{\displaystyle \frac{\rho ({\mathit{r}}^{\prime },t-\frac{R}{v})}{R}}\mathrm{d}{V}^{\prime }}$$\phi(\bm r, t) = \dfrac{1}{4\pi\ve} \dint_{V'} \dfrac{\rho\pqty{\bm r', t - \cfrac{R}{v}}}{R} \dd{V'}$.

  • $\mathit{A}(\mathit{r},t)={\displaystyle \frac{\mu }{4\pi }}{\displaystyle {\int }_{V^{\prime }}{\displaystyle \frac{\mathit{J}({\mathit{r}}^{\prime },t-\frac{R}{v})}{R}}\mathrm{d}{V}^{\prime }}$$\bm A(\bm r, t) = \dfrac{\mu}{4\pi} \dint_{V'} \dfrac{\bm J\pqty{\bm r', t - \cfrac{R}{v}}}{R} \dd{V'}$.

5.6.4 达朗贝尔方程复数形式

在时谐场中，

• 洛伦兹规范：$\mathbf{\nabla }\mathbf{\cdot }\dot{\mathit{A}}=-\mathrm{j}\omega \mu \epsilon \dot{\varphi }$$\div \dot{\bm A} = -\j\omega \mu \ve \dot\phi$.

• 达朗贝尔方程

  • ${\mathbf{\nabla }}^2\dot{\mathit{A}}+k^2\dot{\mathit{A}}=-\mu \dot{\mathit{J}}$$\grad^2 \dot{\bm A} + k^2 \dot{\bm A} = -\mu \dot{\bm J}$.

  • ${\mathbf{\nabla }}^2\dot{\varphi }+k^2\dot{\varphi }=-{\displaystyle \frac{\dot{\rho }}{\epsilon }}$$\grad^2 \dot\phi + k^2 \dot\phi = -\dfrac{\dot\rho}{\ve}$.

  • $k^2={\omega }^2\mu \epsilon ,v={\displaystyle \frac{1}{\sqrt{\mu \epsilon }}}$$k^2 = \omega^2 \mu \ve, v = \dfrac{1}{\sqrt{\mu\ve}}$.

• 解的形式

  • $\dot{\varphi }(\mathit{r})={\displaystyle \frac{1}{4\pi \epsilon }}{\displaystyle {\int }_{V^{\prime }}{\displaystyle \frac{\dot{\rho }({\mathit{r}}^{\prime }){\mathrm{e}}^{-\mathrm{j}kR}}{R}}\mathrm{d}{V}^{\prime }}$$\dot\phi(\bm r) = \dfrac{1}{4\pi\ve} \dint_{V'} \dfrac{\dot\rho(\bm r') \e^{-\j k R}}{R} \dd{V'}$.

  • $\dot{\mathit{A}}(\mathit{r},t)={\displaystyle \frac{\mu }{4\pi }}{\displaystyle {\int }_{V^{\prime }}{\displaystyle \frac{\dot{\mathit{J}}({\mathit{r}}^{\prime }){\mathrm{e}}^{-\mathrm{j}kR}}{R}}\mathrm{d}{V}^{\prime }}$$\dot{\bm A}(\bm r, t) = \dfrac{\mu}{4\pi} \dint_{V'} \dfrac{\dot{\bm J}(\bm r') \e^{-\j k R}}{R} \dd{V'}$.

  • $R=|\mathit{r}-{\mathit{r}}^{\prime }|$$R = \vqty{\bm r - \bm r'}$.

• 求解方法

  • 若已知 $\mathit{A}$$\bm A$，则可求解 $\varphi$$\phi$，反之亦然.

  • $\mathit{B}=\mathbf{\nabla }\times \mathit{A}$$\bm B = \curl \bm A$.

  • $\mathit{E}=-\mathbf{\nabla }\varphi -{\displaystyle \frac{\partial \mathit{A}}{\partial t}}$$\bm E = -\grad \phi - \d\pdv{\bm A}{t}$.

  • $\mathit{S}=\mathit{E}\times \mathit{H}$$\bm S = \bm E \cp \bm H$.