电磁学乱七八糟的符号(一)

@(study)[DSP, markdown_study, LaTex_study] author:何伟宝

ψ = ∫ s F ⃗ ∙ a ⃗ n d S \psi = \int_s \vec F \bullet \vec a_n d S ψ=∫s​F ∙a n​dS ψ = ∮ S F ⃗ ∙ d S ⃗ \psi = \oint_S \vec F \bullet d\vec S ψ=∮S​F ∙dS

Γ = ∫ l F ⃗ ∙ d l ⃗ \Gamma=\int_l \vec F \bullet d\vec l Γ=∫l​F ∙dl Γ = ∮ l F ⃗ ∙ d l ⃗ \Gamma=\oint_l \vec F \bullet d\vec l Γ=∮l​F ∙dl

∇ = 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∂​

∇ = 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

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

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∮S​F 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=∮l​F dS

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∮l​F dl ​

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 =∮l​F dl

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=∫V​ρ(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=∫S​ρS​(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=∫l​ρl​(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∑N​qi​(r i​)

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 n​dS∙di​

i = ∫ s J ⃗ ∙ d S ⃗ i = \int_s \vec J \bullet d \vec S i=∫s​J ∙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 n​dl∙di​

i = ∫ l J ⃗ s ∙ ( n ⃗ × d l ⃗ ∙ ) i = \int_l \vec J_s \bullet (\vec n \times d \vec l^\bullet) i=∫l​J 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

E ⃗ ≜ F ⃗ q 0 \vec E \triangleq \frac{\vec F}{q_0} E ≜q0​F ​

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​∮l​R2Idl ×aR​​

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≜∫S​B ∙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

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 )=−ε0​ρ(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

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 )=−μ0​J (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

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∑i​p ​i​​ P ⃗ = χ e ε 0 E ⃗ \vec P = \chi_e \varepsilon_0 \vec E P =χe​ε0​E 其中 χ e \chi_e χe​为电极化率

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 )≜ε0​E (r )+P (r ) 所以有:

∫ s D ⃗ ( r ⃗ ) ∙ d S ⃗ = q \int_s \vec D(\vec r) \bullet d \vec S =q ∫s​D (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

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∑i​m i​​ M ⃗ = χ m H \vec M = \chi_m H M =χm​H 其中 χ m \chi_m χm​为磁化率

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 )=μ0​B (r )​−M (r ) ∮ l H ⃗ ∙ d l ⃗ = I \oint_l \vec H\bullet d\vec l=I ∮l​H ∙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

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

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 )

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​

静电场能量密度: ω e = 1 2 ε E 2 \omega_e = \frac 12 \varepsilon E^2 ωe​=21​εE2 ω e = 1 2 D ⃗ ( r ⃗ ) ∙ E ⃗ ( r ⃗ ) \omega_e = \frac 12 \vec D(\vec r )\bullet \vec E(\vec r) ωe​=21​D (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​=21​μH2 ω m = 1 2 H ⃗ ( r ⃗ ) ∙ B ⃗ ( r ⃗ ) \omega_m = \frac 12 \vec H(\vec r )\bullet \vec B(\vec r) ωm​=21​H (r )∙B (r )

{ ∮ 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} ⎩⎪⎪⎪⎪⎨⎪⎪⎪⎪⎧​∮l​E (r ,t)∙dl =−∫S​∂t∂B (r ,t)​∙dS ,∇×E (r ,t)=−∂t∂B (r ,t)​∮l​H (r ,t)∙dl =∫S​(J (r ,t)+∂t∂D (r ,t)​),∇×H (r ,t)=J (r ,t)+∂t∂D (r ,t)​∮S​D (r ,t)∙dS =∫V​ρ(r ,t)dV,∇∙D (r ,t)=ρ(r ,t)∮S​B (r ,t)∙dS =0,∇∙B (r ,t)=0​

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

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)

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 −∮S​S (r ,t)∙dS =∂t∂​∫V​ω(r ,t)dV+∫V​p(r ,t)dV

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

∇ × 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 x​Ex​˙​(r )+a y​Ey​˙​(r )+a z​Ez​˙​(r )

∇ ∙ 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

η 0 = μ 0 ε 0 \eta_0=\sqrt{\frac{\mu_0}{\varepsilon_0}} η0​=ε0​μ0​​ ​

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 a​v(r )=T1​∫0T​S (r ,t)dt=21​[E 0​(r )×H 0​(r )]cos(ϕe​−ϕn​)

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 )=21​E (r )×H ∗(r )=21​E 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 a​v(r )=Re{S˙(r )}

− ∮ 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 −∮s​S˙(r )∙dS˙=j2ω∫V​[ωm−av​(r )−ωe−av​(r )]dV+∫V​pav​(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) ωa​v(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 )=21​E (r )∙J ∗(r )=21​σ∣E (r )∣2=Rep(r )

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