电磁学乱七八糟的符号(一)
电磁学乱七八糟的符号(一)
@(study)[DSP, markdown_study, LaTex_study] 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
\psi
ψ
ψ
=
∫
s
F
⃗
∙
a
⃗
n
d
S
\psi = \int_s \vec F \bullet \vec a_n d S
ψ=∫sF
∙a
ndS
ψ
=
∮
S
F
⃗
∙
d
S
⃗
\psi = \oint_S \vec F \bullet d\vec S
ψ=∮SF
∙dS
旋量
Γ
\Gamma
Γ
Γ
=
∫
l
F
⃗
∙
d
l
⃗
\Gamma=\int_l \vec F \bullet d\vec l
Γ=∫lF
∙dl
Γ
=
∮
l
F
⃗
∙
d
l
⃗
\Gamma=\oint_l \vec F \bullet d\vec l
Γ=∮lF
∙dl
矢性微分算符
∇
\nabla
∇
∇
=
a
⃗
x
∂
∂
x
+
a
⃗
y
∂
∂
y
+
a
⃗
z
∂
∂
z
\nabla =\vec a_x \frac{\partial }{\partial x}+\vec a_y \frac{\partial }{\partial y}+\vec a_z \frac{\partial }{\partial z}
∇=a
x∂x∂+a
y∂y∂+a
z∂z∂
拉普拉斯算符
∇
2
\nabla^2
∇2
∇
=
a
⃗
x
∂
2
∂
x
2
+
a
⃗
y
∂
2
∂
y
2
+
a
⃗
z
∂
2
∂
z
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}
∇=a
x∂x2∂2+a
y∂y2∂2+a
z∂z2∂2
∇
×
(
∇
×
F
⃗
)
=
∇
(
∇
∙
F
⃗
)
−
∇
2
F
⃗
\nabla \times (\nabla \times \vec F) = \nabla(\nabla \bullet \vec F) -\nabla^2 \vec F
∇×(∇×F
)=∇(∇∙F
)−∇2F
梯度 grad u
g
r
a
d
u
=
a
⃗
x
∂
u
∂
x
+
a
⃗
y
∂
u
∂
y
+
a
⃗
z
∂
u
∂
z
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=a
x∂x∂u+a
y∂y∂u+a
z∂z∂u
g
r
a
d
u
=
∇
u
gradu=\nabla u
gradu=∇u
散度div F
d
i
v
F
⃗
≜
lim
△
V
→
0
∮
S
F
⃗
d
S
⃗
△
V
div \vec F \triangleq \lim_{\triangle V\to 0} \frac{\oint_S \vec F d \vec S}{\triangle V}
divF
≜△V→0lim△V∮SF
dS
d
i
v
F
⃗
=
∂
F
x
∂
x
+
∂
F
y
∂
y
+
∂
F
z
∂
z
=
∇
∙
F
⃗
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
divF
=∂x∂Fx+∂y∂Fy+∂z∂Fz=∇∙F
∫
V
∇
∙
F
⃗
d
V
=
∮
l
F
⃗
d
S
⃗
\int_V \nabla \bullet \vec F d V =\oint_l \vec F d \vec S
∫V∇∙F
dV=∮lF
dS
环量面密度
γ
n
\gamma_n
γn
γ
n
≜
lim
△
S
→
0
∮
l
F
⃗
d
l
⃗
△
S
\gamma_n \triangleq \lim_{\triangle S\to 0} \frac{\oint_l \vec F d \vec l}{\triangle S}
γn≜△S→0lim△S∮lF
dl
旋度
R
m
R_m
Rm
R
⃗
m
≜
r
o
t
F
⃗
=
a
⃗
n
⟮
lim
△
S
→
0
∮
F
⃗
d
l
⃗
△
S
⟯
m
a
x
\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}
R
m≜rotF
=a
n⟮△S→0lim△S∮F
dl
⟯max
r
o
t
F
⃗
=
∇
×
F
⃗
rot \vec F =\nabla \times \vec F
rotF
=∇×F
∫
S
∇
×
F
⃗
∙
d
S
⃗
=
∮
l
F
⃗
d
l
⃗
\int_S \nabla \times \vec F \bullet d \vec S = \oint_l \vec F d \vec l
∫S∇×F
∙dS
=∮lF
dl
chapter2 常量基本方程
电荷密度
体电荷密度:
ρ
(
r
⃗
∙
)
=
lim
△
V
→
0
△
q
△
V
∙
=
d
q
d
V
∙
\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
=
∫
V
ρ
(
r
⃗
∙
)
d
V
∙
q= \int_V \rho(\vec r^\bullet) d V^\bullet
q=∫Vρ(r
∙)dV∙
面电荷密度:
ρ
s
(
r
⃗
∙
)
=
lim
△
S
→
0
△
q
△
S
∙
=
d
q
d
S
∙
\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
=
∫
S
ρ
S
(
r
⃗
∙
)
d
S
∙
q= \int_S \rho_S(\vec r^\bullet) d S^\bullet
q=∫SρS(r
∙)dS∙
线电荷密度:
ρ
l
(
r
⃗
∙
)
=
lim
△
l
→
0
△
q
△
l
∙
=
d
q
d
l
∙
\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
=
∫
l
ρ
l
(
r
⃗
∙
)
d
l
∙
q= \int_l \rho_l(\vec r^\bullet) d l^\bullet
q=∫lρl(r
∙)dl∙
点电荷:
q
(
r
⃗
)
=
∑
i
=
1
N
q
i
(
r
⃗
i
)
q(\vec r)= \sum_{i=1}^N q_i(\vec r_i)
q(r
)=i=1∑Nqi(r
i)
电流&&电流密度
电流:
i
=
lim
△
t
→
0
△
q
△
t
=
d
q
d
t
i = \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:
J
⃗
=
a
⃗
n
lim
△
S
∙
→
0
△
i
△
S
∙
=
a
⃗
n
d
i
d
S
∙
\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
=a
n△S∙→0lim△S∙△i=a
ndS∙di
i
=
∫
s
J
⃗
∙
d
S
⃗
i = \int_s \vec J \bullet d \vec S
i=∫sJ
∙dS
∇
∙
J
⃗
=
−
∂
ρ
∂
t
\nabla \bullet \vec J=- \frac{\partial \rho}{\partial t}
∇∙J
=−∂t∂ρ
面电流密度:
J
⃗
s
=
a
⃗
n
lim
△
l
∙
→
0
△
i
△
l
∙
=
a
⃗
n
d
i
d
l
∙
\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}
J
s=a
n△l∙→0lim△l∙△i=a
ndl∙di
i
=
∫
l
J
⃗
s
∙
(
n
⃗
×
d
l
⃗
∙
)
i = \int_l \vec J_s \bullet (\vec n \times d \vec l^\bullet)
i=∫lJ
s∙(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:
E
⃗
≜
F
⃗
q
0
\vec E \triangleq \frac{\vec F}{q_0}
E
≜q0F
磁感应强度B:
B
⃗
≜
μ
4
π
∮
l
I
d
l
⃗
×
a
R
R
2
\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
ε
i
n
\varepsilon_{in}
εin
ε
i
n
≜
−
d
ψ
d
t
\varepsilon_{in} \triangleq -\frac{d \psi}{d t}
εin≜−dtdψ 其中
ψ
\psi
ψ为磁通量
ψ
≜
∫
S
B
⃗
∙
d
S
⃗
\psi \triangleq \int_S \vec B \bullet d \vec S
ψ≜∫SB
∙dS
所以:
ε
i
n
=
∫
s
∂
B
⃗
∂
t
∙
d
S
⃗
\varepsilon_{in} = \int_s \frac{\partial \vec B}{\partial t} \bullet d \vec S
εin=∫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) $
μ
0
\mu_0
μ0真空磁导率
chapter3静态场
标量电位
Φ
\Phi
Φ
E
⃗
(
r
⃗
)
≜
−
△
Φ
(
r
⃗
)
\vec E(\vec r) \triangleq -\triangle\Phi(\vec r)
E
(r
)≜−△Φ(r
)
Φ
(
r
⃗
)
=
W
q
\Phi(\vec r)=\frac{W}{q}
Φ(r
)=qW
电位的标量泊松方程:
∇
2
Φ
(
r
⃗
)
=
−
ρ
(
r
⃗
)
ε
0
\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:
∇
2
Φ
(
r
⃗
)
=
0
\nabla^2 \Phi(\vec r) = 0
∇2Φ(r
)=0
矢量磁位(磁矢位) A
B
⃗
(
r
⃗
)
≜
∇
×
A
⃗
(
r
⃗
)
\vec B (\vec r )\triangleq \nabla \times \vec A(\vec r)
B
(r
)≜∇×A
(r
)
库仑规范:
∇
∙
A
⃗
=
0
\nabla \bullet \vec A = 0
∇∙A
=0
磁矢位的矢量泊松方程:
∇
2
A
⃗
(
r
⃗
)
=
−
μ
0
J
⃗
(
r
⃗
)
\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
∇
2
A
⃗
(
r
⃗
)
=
0
\nabla^2 \vec A (\vec r )=0
∇2A
(r
)=0
磁矩m:
m
⃗
≜
I
⃗
S
⃗
\vec m \triangleq \vec I \vec S
m
≜I
S
极化强度矢量P
P
⃗
(
r
⃗
)
=
lim
△
V
→
0
∑
i
p
⃗
i
△
V
\vec P(\vec r)=\lim_{\triangle V \to 0} \frac{\sum_i \vec p_i}{\triangle V}
P
(r
)=△V→0lim△V∑ip
i
P
⃗
=
χ
e
ε
0
E
⃗
\vec P = \chi_e \varepsilon_0 \vec E
P
=χeε0E
其中
χ
e
\chi_e
χe为电极化率
电位移矢量D
D
⃗
(
r
⃗
)
≜
ε
0
E
⃗
(
r
⃗
)
+
P
⃗
(
r
⃗
)
\vec D(\vec r) \triangleq \varepsilon_0 \vec E(\vec r)+\vec P(\vec r)
D
(r
)≜ε0E
(r
)+P
(r
) 所以有:
∫
s
D
⃗
(
r
⃗
)
∙
d
S
⃗
=
q
\int_s \vec D(\vec r) \bullet d \vec S =q
∫sD
(r
)∙dS
=q
∇
∙
D
⃗
(
r
⃗
)
=
ρ
(
r
⃗
)
\nabla \bullet \vec D(\vec r) = \rho(\vec r)
∇∙D
(r
)=ρ(r
)
D
⃗
=
ε
E
⃗
\vec D = \varepsilon \vec E
D
=εE
磁化强度矢量M
M
⃗
(
r
⃗
)
=
lim
△
V
→
0
∑
i
m
⃗
i
△
V
\vec M(\vec r)=\lim_{\triangle V \to 0} \frac{\sum_i \vec m_i}{\triangle V}
M
(r
)=△V→0lim△V∑im
i
M
⃗
=
χ
m
H
\vec M = \chi_m H
M
=χmH 其中
χ
m
\chi_m
χm为磁化率
磁化强度H
H
⃗
(
r
⃗
)
=
B
⃗
(
r
⃗
)
μ
0
−
M
⃗
(
r
⃗
)
\vec H(\vec r)=\frac{\vec B(\vec r)}{\mu_0}-\vec M(\vec r)
H
(r
)=μ0B
(r
)−M
(r
)
∮
l
H
⃗
∙
d
l
⃗
=
I
\oint_l \vec H\bullet d\vec l=I
∮lH
∙dl
=I
∇
×
H
⃗
(
r
⃗
)
=
J
⃗
(
r
⃗
)
\nabla \times \vec H (\vec r )=\vec J(\vec r)
∇×H
(r
)=J
(r
)
B
⃗
=
μ
H
⃗
\vec B=\mu \vec H
B
=μH
欧姆定律微分形式
J
⃗
(
r
⃗
)
=
σ
E
⃗
(
r
⃗
)
\vec J(\vec r)=\sigma \vec E(\vec r)
J
(r
)=σE
(r
) 其中
σ
\sigma
σ为电导率
热损耗功率
p
(
r
⃗
)
=
J
⃗
(
r
⃗
)
∙
E
⃗
(
r
⃗
)
=
σ
E
2
(
r
⃗
)
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
a
⃗
n
×
(
E
⃗
1
−
E
⃗
2
)
=
0
,
E
1
t
=
E
2
t
\vec a_n \times (\vec E_1 -\vec E_2)=0,\quad \quad E_{1t}=E_{2t}
a
n×(E
1−E
2)=0,E1t=E2t
a
⃗
n
×
(
H
⃗
1
−
H
⃗
2
)
=
J
⃗
s
,
H
1
t
=
H
2
t
\vec a_n \times (\vec H_1 -\vec H_2)=\vec J_s,\quad \quad H_{1t}=H_{2t}
a
n×(H
1−H
2)=J
s,H1t=H2t
a
⃗
n
∙
(
D
⃗
1
−
D
⃗
2
)
=
ρ
s
,
D
1
n
−
D
2
n
=
ρ
s
\vec a_n \bullet (\vec D_1 -\vec D_2) =\rho_s, \quad D_{1n}-D_{2n}=\rho_s
a
n∙(D
1−D
2)=ρs,D1n−D2n=ρs
a
⃗
n
∙
(
B
⃗
1
−
B
⃗
2
)
=
0
,
B
1
n
=
B
2
n
\vec a_n \bullet (\vec B_1 - \vec B_2)=0,\quad \quad B_{1n}=B_{2n}
a
n∙(B
1−B
2)=0,B1n=B2n
能量
静电场能量密度:
ω
e
=
1
2
ε
E
2
\omega_e = \frac 12 \varepsilon E^2
ωe=21εE2
ω
e
=
1
2
D
⃗
(
r
⃗
)
∙
E
⃗
(
r
⃗
)
\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:
ω
m
=
1
2
μ
H
2
\omega_m = \frac 12 \mu H^2
ωm=21μH2
ω
m
=
1
2
H
⃗
(
r
⃗
)
∙
B
⃗
(
r
⃗
)
\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
{
∮
l
E
⃗
(
r
⃗
,
t
)
∙
d
l
⃗
=
−
∫
S
∂
B
⃗
(
r
⃗
,
t
)
∂
t
∙
d
S
⃗
,
∇
×
E
⃗
(
r
⃗
,
t
)
=
−
∂
B
⃗
(
r
⃗
,
t
)
∂
t
∮
l
H
⃗
(
r
⃗
,
t
)
∙
d
l
⃗
=
∫
S
(
J
⃗
(
r
⃗
,
t
)
+
∂
D
⃗
(
r
⃗
,
t
)
∂
t
)
,
∇
×
H
⃗
(
r
⃗
,
t
)
=
J
⃗
(
r
⃗
,
t
)
+
∂
D
⃗
(
r
⃗
,
t
)
∂
t
∮
S
D
⃗
(
r
⃗
,
t
)
∙
d
S
⃗
=
∫
V
ρ
(
r
⃗
,
t
)
d
V
,
∇
∙
D
⃗
(
r
⃗
,
t
)
=
ρ
(
r
⃗
,
t
)
∮
S
B
⃗
(
r
⃗
,
t
)
∙
d
S
⃗
=
0
,
∇
∙
B
⃗
(
r
⃗
,
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
E
⃗
=
−
∇
Φ
−
∂
A
⃗
∂
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):
∇
∙
A
⃗
=
−
μ
ε
∂
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):
∇
2
Φ
(
r
⃗
,
t
)
−
μ
ε
∂
2
Φ
(
r
⃗
,
t
)
∂
t
2
=
−
ρ
(
r
⃗
,
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)
∇
2
A
(
r
⃗
,
t
)
−
μ
ε
∂
2
A
(
r
⃗
,
t
)
∂
t
2
=
−
μ
J
⃗
(
r
⃗
,
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
S
⃗
(
r
⃗
,
t
)
≜
E
⃗
(
r
⃗
,
t
)
×
H
⃗
(
r
⃗
,
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)
−
∇
∙
S
⃗
=
∂
ω
∂
t
+
p
-\nabla \bullet \vec S=\frac{\partial\omega}{\partial t}+p
−∇∙S
=∂t∂ω+p
−
∮
S
S
⃗
(
r
⃗
,
t
)
∙
d
S
⃗
=
∂
∂
t
∫
V
ω
(
r
⃗
,
t
)
d
V
+
∫
V
p
(
r
⃗
,
t
)
d
V
-\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
)
=
R
e
{
[
U
0
(
z
)
e
j
ϕ
]
e
j
ω
t
}
=
R
e
{
U
˙
(
z
)
e
j
ω
t
}
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
)
=
U
0
(
z
)
e
j
ϕ
\dot{U}(z)=U_0(z)e^{j\phi}
U˙(z)=U0(z)ejϕ
复数形式麦克斯韦方程
∇
×
E
⃗
=
j
ω
B
⃗
\nabla \times \vec E=j\omega \vec B
∇×E
=jωB
∇
×
H
⃗
=
J
⃗
+
j
ω
D
⃗
\nabla \times \vec H =\vec J + j \omega \vec D
∇×H
=J
+jωD
E
⃗
˙
=
a
⃗
x
E
x
˙
(
r
⃗
)
+
a
⃗
y
E
y
˙
(
r
⃗
)
+
a
⃗
z
E
z
˙
(
r
⃗
)
\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
˙=a
xEx˙(r
)+a
yEy˙(r
)+a
zEz˙(r
)
复波动方程
∇
∙
A
⃗
(
r
⃗
)
=
−
j
ω
μ
ε
Φ
(
r
⃗
)
\nabla \bullet \vec A(\vec r) = -j\omega \mu\varepsilon \Phi(\vec r)
∇∙A
(r
)=−jωμεΦ(r
)
∇
2
Φ
(
r
⃗
)
+
ω
2
μ
ε
Φ
(
r
⃗
)
=
−
ρ
(
r
⃗
)
ε
\nabla^2\Phi(\vec r)+\omega^2\mu\varepsilon\Phi(\vec r)=-\frac{\rho(\vec r)}{\varepsilon}
∇2Φ(r
)+ω2μεΦ(r
)=−ερ(r
)
∇
2
A
⃗
(
r
⃗
)
+
ω
2
μ
ε
A
⃗
(
r
⃗
)
=
−
μ
J
⃗
(
r
⃗
)
\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
k
2
=
ω
2
μ
ε
k^2=\omega^2\mu\varepsilon
k2=ω2με有: 非齐次亥姆霍兹方程:
∇
2
Φ
(
r
⃗
)
+
k
2
Φ
(
r
⃗
)
=
−
ρ
(
r
⃗
)
ε
\nabla^2\Phi(\vec r)+k^2\Phi(\vec r)=-\frac{\rho(\vec r)}{\varepsilon}
∇2Φ(r
)+k2Φ(r
)=−ερ(r
)
∇
2
A
⃗
(
r
⃗
)
+
k
2
A
⃗
(
r
⃗
)
=
−
μ
J
⃗
(
r
⃗
)
\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:
∇
2
Φ
(
r
⃗
)
+
k
2
Φ
(
r
⃗
)
=
0
\nabla^2\Phi(\vec r)+k^2\Phi(\vec r)=0
∇2Φ(r
)+k2Φ(r
)=0
∇
2
A
⃗
(
r
⃗
)
+
k
2
A
⃗
(
r
⃗
)
=
0
\nabla^2 \vec A(\vec r)+k^2 \vec A(\vec r)=0
∇2A
(r
)+k2A
(r
)=0
波阻抗
η
\eta
η
η
0
=
μ
0
ε
0
\eta_0=\sqrt{\frac{\mu_0}{\varepsilon_0}}
η0=ε0μ0
时均坡印亭矢量
S
a
v
S_av
Sav
S
⃗
a
v
(
r
⃗
)
=
1
T
∫
0
T
S
⃗
(
r
⃗
,
t
)
d
t
=
1
2
[
E
⃗
0
(
r
⃗
)
×
H
⃗
0
(
r
⃗
)
]
c
o
s
(
ϕ
e
−
ϕ
n
)
\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)
S
av(r
)=T1∫0TS
(r
,t)dt=21[E
0(r
)×H
0(r
)]cos(ϕe−ϕn)
复坡印亭矢量
S
˙
\dot{S}
S˙
S
˙
(
r
⃗
)
=
1
2
E
⃗
(
r
⃗
)
×
H
⃗
∗
(
r
⃗
)
=
1
2
E
⃗
0
(
r
⃗
)
e
−
j
ϕ
e
×
H
⃗
0
(
r
⃗
)
e
j
ϕ
n
=
1
2
[
E
⃗
0
(
r
⃗
)
×
H
⃗
0
(
r
⃗
)
]
e
ϕ
e
−
ϕ
n
\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
)=21E
0(r
)e−jϕe×H
0(r
)ejϕn=21[E
0(r
)×H
0(r
)]eϕe−ϕn
其中:
S
⃗
a
v
(
r
⃗
)
=
R
e
{
S
˙
(
r
⃗
)
}
\vec S_av(\vec r)=Re\{ \dot{S}(\vec r) \}
S
av(r
)=Re{S˙(r
)}
复坡印亭定理
−
∮
s
S
˙
(
r
⃗
)
∙
d
S
˙
=
j
2
ω
∫
V
[
ω
m
−
a
v
(
r
⃗
)
−
ω
e
−
a
v
(
r
⃗
)
]
d
V
+
∫
V
p
a
v
(
r
⃗
)
d
V
-\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:
ω
a
v
(
r
⃗
)
=
1
4
[
E
⃗
(
r
⃗
)
∙
D
⃗
∗
(
r
⃗
)
+
B
⃗
(
r
⃗
)
∙
H
⃗
∗
(
r
⃗
)
]
=
1
4
[
ε
∣
E
⃗
(
r
⃗
)
∣
2
+
μ
∣
H
⃗
(
r
⃗
)
∣
2
]
=
R
e
ω
(
r
⃗
)
\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
)
p
a
v
(
r
⃗
)
=
1
2
E
⃗
(
r
⃗
)
∙
J
⃗
∗
(
r
⃗
)
=
1
2
σ
∣
E
⃗
(
r
⃗
)
∣
2
=
R
e
p
(
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)
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