电磁学乱七八糟的符号(一)

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author:何伟宝

文章目录

电磁学乱七八糟的符号(一)
- chapter1 场量基础
- - 通量$\psi$
  - 旋量$\Gamma$
  - 矢性微分算符$\nabla$
  - 拉普拉斯算符$\nabla^2$
  - 梯度 grad u
  - 散度div F
  - 环量面密度$\gamma_n$
  - 旋度$R_m$
- chapter2 常量基本方程
- - 电荷密度
  - 电流&&电流密度
  - 电场强度E:
  - 磁感应强度B:
  - 感应电动势$\varepsilon_{in}$
  - 本章的一些常数
- chapter3静态场
- - 标量电位$\Phi$
  - 矢量磁位(磁矢位) A
  - 极化强度矢量P
  - 电位移矢量D
  - 磁化强度矢量M
  - 磁化强度H
  - 欧姆定律微分形式
  - 热损耗功率
  - 边界条件
  - 能量
- chapter4 动态场
- - 麦克斯韦方程组
  - 标量电位更新
  - 波动方程
  - 坡印亭矢量
  - 复数表示
  - 复数形式麦克斯韦方程
  - 复波动方程
  - 波阻抗$\eta$
  - 时均坡印亭矢量$S_av$
  - 复坡印亭矢量$\dot{S}$
  - 复坡印亭定理
- 结语

chapter1 场量基础

通量ψ\psiψ

ψ=∫sF⃗∙a⃗ndS\psi = \int_s \vec F \bullet \vec a_n d Sψ=∫sF∙andS
ψ=∮SF⃗∙dS⃗\psi = \oint_S \vec F \bullet d\vec Sψ=∮SF∙dS

旋量Γ\GammaΓ

Γ=∫lF⃗∙dl⃗\Gamma=\int_l \vec F \bullet d\vec lΓ=∫lF∙dl
Γ=∮lF⃗∙dl⃗\Gamma=\oint_l \vec F \bullet d\vec lΓ=∮lF∙dl

矢性微分算符∇\nabla∇

∇=a⃗x∂∂x+a⃗y∂∂y+a⃗z∂∂z\nabla =\vec a_x \frac{\partial }{\partial x}+\vec a_y \frac{\partial }{\partial y}+\vec a_z \frac{\partial }{\partial z} ∇=ax∂x∂+ay∂y∂+az∂z∂

拉普拉斯算符∇2\nabla^2∇2

∇=a⃗x∂2∂x2+a⃗y∂2∂y2+a⃗z∂2∂z2\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} ∇=ax∂x2∂2+ay∂y2∂2+az∂z2∂2

∇×(∇×F⃗)=∇(∇∙F⃗)−∇2F⃗\nabla \times (\nabla \times \vec F) = \nabla(\nabla \bullet \vec F) -\nabla^2 \vec F ∇×(∇×F)=∇(∇∙F)−∇2F

梯度 grad u

gradu=a⃗x∂u∂x+a⃗y∂u∂y+a⃗z∂u∂zgrad 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=ax∂x∂u+ay∂y∂u+az∂z∂u
gradu=∇ugradu=\nabla ugradu=∇u

散度div F

divF⃗≜lim⁡△V→0∮SF⃗dS⃗△Vdiv \vec F \triangleq \lim_{\triangle V\to 0} \frac{\oint_S \vec F d \vec S}{\triangle V}divF≜△V→0lim△V∮SFdS

divF⃗=∂Fx∂x+∂Fy∂y+∂Fz∂z=∇∙F⃗div \vec F=\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+ \frac{\partial F_z}{\partial z} =\nabla \bullet \vec F divF=∂x∂Fx+∂y∂Fy+∂z∂Fz=∇∙F

∫V∇∙F⃗dV=∮lF⃗dS⃗\int_V \nabla \bullet \vec F d V =\oint_l \vec F d \vec S∫V∇∙FdV=∮lFdS

环量面密度γn\gamma_nγn

γn≜lim⁡△S→0∮lF⃗dl⃗△S\gamma_n \triangleq \lim_{\triangle S\to 0} \frac{\oint_l \vec F d \vec l}{\triangle S}γn≜△S→0lim△S∮lFdl

旋度RmR_mRm

R⃗m≜rotF⃗=a⃗n⟮lim⁡△S→0∮F⃗dl⃗△S⟯max\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} Rm≜rotF=an⟮△S→0lim△S∮Fdl⟯max

rotF⃗=∇×F⃗rot \vec F =\nabla \times \vec F rotF=∇×F

∫S∇×F⃗∙dS⃗=∮lF⃗dl⃗\int_S \nabla \times \vec F \bullet d \vec S = \oint_l \vec F d \vec l∫S∇×F∙dS=∮lFdl

chapter2 常量基本方程

电荷密度

体电荷密度:
ρ(r⃗∙)=lim⁡△V→0△q△V∙=dqdV∙\rho (\vec r^{\bullet} ) = \lim_{\triangle V \to 0 } \frac{\triangle q}{\triangle V^\bullet} = \frac{d q}{d V^\bullet} ρ(r∙)=△V→0lim△V∙△q=dV∙dq
q=∫Vρ(r⃗∙)dV∙q= \int_V \rho(\vec r^\bullet) d V^\bullet q=∫Vρ(r∙)dV∙

面电荷密度:
ρs(r⃗∙)=lim⁡△S→0△q△S∙=dqdS∙\rho_s (\vec r^{\bullet} ) = \lim_{\triangle S \to 0 } \frac{\triangle q}{\triangle S^\bullet} = \frac{d q}{d S^\bullet} ρs(r∙)=△S→0lim△S∙△q=dS∙dq
q=∫SρS(r⃗∙)dS∙q= \int_S \rho_S(\vec r^\bullet) d S^\bullet q=∫SρS(r∙)dS∙

线电荷密度:
ρl(r⃗∙)=lim⁡△l→0△q△l∙=dqdl∙\rho_l (\vec r^{\bullet} ) = \lim_{\triangle l \to 0 } \frac{\triangle q}{\triangle l^\bullet} = \frac{d q}{d l^\bullet} ρl(r∙)=△l→0lim△l∙△q=dl∙dq
q=∫lρl(r⃗∙)dl∙q= \int_l \rho_l(\vec r^\bullet) d l^\bullet q=∫lρl(r∙)dl∙

点电荷:
q(r⃗)=∑i=1Nqi(r⃗i)q(\vec r)= \sum_{i=1}^N q_i(\vec r_i)q(r)=i=1∑Nqi(ri)

电流&&电流密度

电流:
i=lim⁡△t→0△q△t=dqdti = \lim_{\triangle t \to 0}\frac{\triangle q }{\triangle t}=\frac{d q}{d t} i=△t→0lim△t△q=dtdq

体电流密度矢量:

J⃗=a⃗nlim⁡△S∙→0△i△S∙=a⃗ndidS∙\vec J= \vec a_n \lim_{\triangle S^\bullet \to 0} \frac{\triangle i}{\triangle S^\bullet}=\vec a_n \frac{di }{dS^\bullet}J=an△S∙→0lim△S∙△i=andS∙di

i=∫sJ⃗∙dS⃗i = \int_s \vec J \bullet d \vec Si=∫sJ∙dS

∇∙J⃗=−∂ρ∂t\nabla \bullet \vec J=- \frac{\partial \rho}{\partial t}∇∙J=−∂t∂ρ

面电流密度:

J⃗s=a⃗nlim⁡△l∙→0△i△l∙=a⃗ndidl∙\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}Js=an△l∙→0lim△l∙△i=andl∙di

i=∫lJ⃗s∙(n⃗×dl⃗∙)i = \int_l \vec J_s \bullet (\vec n \times d \vec l^\bullet)i=∫lJs∙(n×dl∙)

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:

E⃗≜F⃗q0\vec E \triangleq \frac{\vec F}{q_0} E≜q0F

磁感应强度B:

B⃗≜μ4π∮lIdl⃗×aRR2\vec B \triangleq \frac{\mu}{4\pi}\oint_l \frac{I d \vec l \times a_R}{R^2}B≜4πμ∮lR2Idl×aR

感应电动势εin\varepsilon_{in}εin

εin≜−dψdt\varepsilon_{in} \triangleq -\frac{d \psi}{d t}εin≜−dtdψ
其中ψ\psiψ为磁通量
ψ≜∫SB⃗∙dS⃗\psi \triangleq \int_S \vec B \bullet d \vec S ψ≜∫SB∙dS
所以:
εin=∫s∂B⃗∂t∙dS⃗\varepsilon_{in} = \int_s \frac{\partial \vec B}{\partial t} \bullet d \vec Sεin=∫s∂t∂B∙dS

本章的一些常数

$\varepsilon_0 自由空间的电容率 (介电常数) $
μ0\mu_0μ0真空磁导率

chapter3静态场

标量电位Φ\PhiΦ

E⃗(r⃗)≜−△Φ(r⃗)\vec E(\vec r) \triangleq -\triangle\Phi(\vec r)E(r)≜−△Φ(r)

Φ(r⃗)=Wq\Phi(\vec r)=\frac{W}{q}Φ(r)=qW

电位的标量泊松方程:
∇2Φ(r⃗)=−ρ(r⃗)ε0\nabla^2 \Phi(\vec r) = - \frac{\rho(\vec r)}{\varepsilon_0} ∇2Φ(r)=−ε0ρ(r)

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

矢量磁位(磁矢位) A

B⃗(r⃗)≜∇×A⃗(r⃗)\vec B (\vec r )\triangleq \nabla \times \vec A(\vec r)B(r)≜∇×A(r)

库仑规范:
∇∙A⃗=0\nabla \bullet \vec A = 0∇∙A=0

磁矢位的矢量泊松方程:
∇2A⃗(r⃗)=−μ0J⃗(r⃗)\nabla^2 \vec A (\vec r )=- \mu_0 \vec J (\vec r)∇2A(r)=−μ0J(r)

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

磁矩m:
m⃗≜I⃗S⃗\vec m \triangleq \vec I \vec S m≜IS

极化强度矢量P

P⃗(r⃗)=lim⁡△V→0∑ip⃗i△V\vec P(\vec r)=\lim_{\triangle V \to 0} \frac{\sum_i \vec p_i}{\triangle V}P(r)=△V→0lim△V∑ipi
P⃗=χeε0E⃗\vec P = \chi_e \varepsilon_0 \vec EP=χeε0E
其中χe\chi_eχe为电极化率

电位移矢量D

D⃗(r⃗)≜ε0E⃗(r⃗)+P⃗(r⃗)\vec D(\vec r) \triangleq \varepsilon_0 \vec E(\vec r)+\vec P(\vec r)D(r)≜ε0E(r)+P(r)
所以有:

∫sD⃗(r⃗)∙dS⃗=q\int_s \vec D(\vec r) \bullet d \vec S =q ∫sD(r)∙dS=q

∇∙D⃗(r⃗)=ρ(r⃗)\nabla \bullet \vec D(\vec r) = \rho(\vec r) ∇∙D(r)=ρ(r)

D⃗=εE⃗\vec D = \varepsilon \vec E D=εE

磁化强度矢量M

M⃗(r⃗)=lim⁡△V→0∑im⃗i△V\vec M(\vec r)=\lim_{\triangle V \to 0} \frac{\sum_i \vec m_i}{\triangle V}M(r)=△V→0lim△V∑imi
M⃗=χmH\vec M = \chi_m HM=χmH
其中χm\chi_mχm为磁化率

磁化强度H

H⃗(r⃗)=B⃗(r⃗)μ0−M⃗(r⃗)\vec H(\vec r)=\frac{\vec B(\vec r)}{\mu_0}-\vec M(\vec r)H(r)=μ0B(r)−M(r)
∮lH⃗∙dl⃗=I\oint_l \vec H\bullet d\vec l=I∮lH∙dl=I
∇×H⃗(r⃗)=J⃗(r⃗)\nabla \times \vec H (\vec r )=\vec J(\vec r)∇×H(r)=J(r)
B⃗=μH⃗\vec B=\mu \vec HB=μH

欧姆定律微分形式

J⃗(r⃗)=σE⃗(r⃗)\vec J(\vec r)=\sigma \vec E(\vec r)J(r)=σE(r)
其中σ\sigmaσ为电导率

热损耗功率

p(r⃗)=J⃗(r⃗)∙E⃗(r⃗)=σE2(r⃗)p(\vec r)=\vec J(\vec r)\bullet \vec E(\vec r)=\sigma E^2(\vec r)p(r)=J(r)∙E(r)=σE2(r)

边界条件

a⃗n×(E⃗1−E⃗2)=0,E1t=E2t\vec a_n \times (\vec E_1 -\vec E_2)=0,\quad \quad E_{1t}=E_{2t}an×(E1−E2)=0,E1t=E2t
a⃗n×(H⃗1−H⃗2)=J⃗s,H1t=H2t\vec a_n \times (\vec H_1 -\vec H_2)=\vec J_s,\quad \quad H_{1t}=H_{2t}an×(H1−H2)=Js,H1t=H2t
a⃗n∙(D⃗1−D⃗2)=ρs,D1n−D2n=ρs\vec a_n \bullet (\vec D_1 -\vec D_2) =\rho_s, \quad D_{1n}-D_{2n}=\rho_s an∙(D1−D2)=ρs,D1n−D2n=ρs
a⃗n∙(B⃗1−B⃗2)=0,B1n=B2n\vec a_n \bullet (\vec B_1 - \vec B_2)=0,\quad \quad B_{1n}=B_{2n} an∙(B1−B2)=0,B1n=B2n

能量

静电场能量密度:
ωe=12εE2\omega_e = \frac 12 \varepsilon E^2ωe=21εE2
ωe=12D⃗(r⃗)∙E⃗(r⃗)\omega_e = \frac 12 \vec D(\vec r )\bullet \vec E(\vec r)ωe=21D(r)∙E(r)
静磁场能量密度:
ωm=12μH2\omega_m = \frac 12 \mu H^2ωm=21μH2
ωm=12H⃗(r⃗)∙B⃗(r⃗)\omega_m = \frac 12 \vec H(\vec r )\bullet \vec B(\vec r)ωm=21H(r)∙B(r)

chapter4 动态场

麦克斯韦方程组

{∮lE⃗(r⃗,t)∙dl⃗=−∫S∂B⃗(r⃗,t)∂t∙dS⃗,∇×E⃗(r⃗,t)=−∂B⃗(r⃗,t)∂t∮lH⃗(r⃗,t)∙dl⃗=∫S(J⃗(r⃗,t)+∂D⃗(r⃗,t)∂t),∇×H⃗(r⃗,t)=J⃗(r⃗,t)+∂D⃗(r⃗,t)∂t∮SD⃗(r⃗,t)∙dS⃗=∫Vρ(r⃗,t)dV,∇∙D⃗(r⃗,t)=ρ(r⃗,t)∮SB⃗(r⃗,t)∙dS⃗=0,∇∙B⃗(r⃗,t)=0\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}⎩⎪⎪⎪⎪⎨⎪⎪⎪⎪⎧∮lE(r,t)∙dl=−∫S∂t∂B(r,t)∙dS,∇×E(r,t)=−∂t∂B(r,t)∮lH(r,t)∙dl=∫S(J(r,t)+∂t∂D(r,t)),∇×H(r,t)=J(r,t)+∂t∂D(r,t)∮SD(r,t)∙dS=∫Vρ(r,t)dV,∇∙D(r,t)=ρ(r,t)∮SB(r,t)∙dS=0,∇∙B(r,t)=0

标量电位更新

E⃗=−∇Φ−∂A⃗∂t\vec E=-\nabla\Phi -\frac{\partial \vec A}{\partial t}E=−∇Φ−∂t∂A

波动方程

洛伦兹条件(洛伦兹规范):
∇∙A⃗=−με∂Φ∂t\nabla \bullet \vec A=-\mu \varepsilon \frac{\partial \Phi}{\partial t}∇∙A=−με∂t∂Φ
非齐次波动方程(动态退化可以得到其他规范):
∇2Φ(r⃗,t)−με∂2Φ(r⃗,t)∂t2=−ρ(r⃗,t)ε\nabla^2 \Phi(\vec r,t)-\mu\varepsilon\frac{\partial^2\Phi(\vec r,t)}{\partial t^2}=- \frac{\rho(\vec r,t)}{\varepsilon}∇2Φ(r,t)−με∂t2∂2Φ(r,t)=−ερ(r,t)

∇2A(r⃗,t)−με∂2A(r⃗,t)∂t2=−μJ⃗(r⃗,t)\nabla^2 A(\vec r,t)-\mu\varepsilon\frac{\partial^2 A(\vec r,t)}{\partial t^2}= -\mu \vec J(\vec r,t)∇2A(r,t)−με∂t2∂2A(r,t)=−μJ(r,t)

坡印亭矢量

S⃗(r⃗,t)≜E⃗(r⃗,t)×H⃗(r⃗,t)\vec S (\vec r,t) \triangleq \vec E(\vec r,t)\times \vec H(\vec r,t)S(r,t)≜E(r,t)×H(r,t)

−∇∙S⃗=∂ω∂t+p-\nabla \bullet \vec S=\frac{\partial\omega}{\partial t}+p−∇∙S=∂t∂ω+p

−∮SS⃗(r⃗,t)∙dS⃗=∂∂t∫Vω(r⃗,t)dV+∫Vp(r⃗,t)dV-\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−∮SS(r,t)∙dS=∂t∂∫Vω(r,t)dV+∫Vp(r,t)dV

复数表示

u(z,t)=Re{[U0(z)ejϕ]ejωt}=Re{U˙(z)ejωt}u(z,t)=Re\{ [U_0(z)e^{j\phi}]e^{j\omega t} \} = Re \{ \dot{U}(z) e^{j\omega t} \}u(z,t)=Re{[U0(z)ejϕ]ejωt}=Re{U˙(z)ejωt}
U˙(z)=U0(z)ejϕ\dot{U}(z)=U_0(z)e^{j\phi}U˙(z)=U0(z)ejϕ

复数形式麦克斯韦方程

∇×E⃗=jωB⃗\nabla \times \vec E=j\omega \vec B ∇×E=jωB
∇×H⃗=J⃗+jωD⃗\nabla \times \vec H =\vec J + j \omega \vec D ∇×H=J+jωD
E⃗˙=a⃗xEx˙(r⃗)+a⃗yEy˙(r⃗)+a⃗zEz˙(r⃗)\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)E˙=axEx˙(r)+ayEy˙(r)+azEz˙(r)

复波动方程

∇∙A⃗(r⃗)=−jωμεΦ(r⃗)\nabla \bullet \vec A(\vec r) = -j\omega \mu\varepsilon \Phi(\vec r) ∇∙A(r)=−jωμεΦ(r)

∇2Φ(r⃗)+ω2μεΦ(r⃗)=−ρ(r⃗)ε\nabla^2\Phi(\vec r)+\omega^2\mu\varepsilon\Phi(\vec r)=-\frac{\rho(\vec r)}{\varepsilon}∇2Φ(r)+ω2μεΦ(r)=−ερ(r)
∇2A⃗(r⃗)+ω2μεA⃗(r⃗)=−μJ⃗(r⃗)\nabla^2 \vec A(\vec r)+\omega^2\mu\varepsilon \vec A(\vec r)=-\mu \vec J(\vec r)∇2A(r)+ω2μεA(r)=−μJ(r)
令k2=ω2μεk^2=\omega^2\mu\varepsilonk2=ω2με有:
非齐次亥姆霍兹方程:
∇2Φ(r⃗)+k2Φ(r⃗)=−ρ(r⃗)ε\nabla^2\Phi(\vec r)+k^2\Phi(\vec r)=-\frac{\rho(\vec r)}{\varepsilon}∇2Φ(r)+k2Φ(r)=−ερ(r)
∇2A⃗(r⃗)+k2A⃗(r⃗)=−μJ⃗(r⃗)\nabla^2 \vec A(\vec r)+k^2 \vec A(\vec r)=-\mu \vec J(\vec r)∇2A(r)+k2A(r)=−μJ(r)
齐次亥姆霍兹方程:
∇2Φ(r⃗)+k2Φ(r⃗)=0\nabla^2\Phi(\vec r)+k^2\Phi(\vec r)=0∇2Φ(r)+k2Φ(r)=0
∇2A⃗(r⃗)+k2A⃗(r⃗)=0\nabla^2 \vec A(\vec r)+k^2 \vec A(\vec r)=0∇2A(r)+k2A(r)=0

波阻抗η\etaη

η0=μ0ε0\eta_0=\sqrt{\frac{\mu_0}{\varepsilon_0}}η0=ε0μ0

时均坡印亭矢量SavS_avSav

S⃗av(r⃗)=1T∫0TS⃗(r⃗,t)dt=12[E⃗0(r⃗)×H⃗0(r⃗)]cos(ϕe−ϕn)\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)Sav(r)=T1∫0TS(r,t)dt=21[E0(r)×H0(r)]cos(ϕe−ϕn)

复坡印亭矢量S˙\dot{S}S˙

S˙(r⃗)=12E⃗(r⃗)×H⃗∗(r⃗)=12E⃗0(r⃗)e−jϕe×H⃗0(r⃗)ejϕn=12[E⃗0(r⃗)×H⃗0(r⃗)]eϕe−ϕn\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}S˙(r)=21E(r)×H∗(r)=21E0(r)e−jϕe×H0(r)ejϕn=21[E0(r)×H0(r)]eϕe−ϕn

其中:
S⃗av(r⃗)=Re{S˙(r⃗)}\vec S_av(\vec r)=Re\{ \dot{S}(\vec r) \}Sav(r)=Re{S˙(r)}

复坡印亭定理

−∮sS˙(r⃗)∙dS˙=j2ω∫V[ωm−av(r⃗)−ωe−av(r⃗)]dV+∫Vpav(r⃗)dV-\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 −∮sS˙(r)∙dS˙=j2ω∫V[ωm−av(r)−ωe−av(r)]dV+∫Vpav(r)dV

其中:
ωav(r⃗)=14[E⃗(r⃗)∙D⃗∗(r⃗)+B⃗(r⃗)∙H⃗∗(r⃗)]=14[ε∣E⃗(r⃗)∣2+μ∣H⃗(r⃗)∣2]=Reω(r⃗)\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)ωav(r)=41[E(r)∙D∗(r)+B(r)∙H∗(r)]=41[ε∣E(r)∣2+μ∣H(r)∣2]=Reω(r)
pav(r⃗)=12E⃗(r⃗)∙J⃗∗(r⃗)=12σ∣E⃗(r⃗)∣2=Rep(r⃗)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) pav(r)=21E(r)∙J∗(r)=21σ∣E(r)∣2=Rep(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

如果你想请我吃个南五的话