Download 3) Boundary conditions of time-dependent electromagnetic field

《电磁波基础及应用》沈建其讲义
Chapter 2.
Maxwell equations
1) Displacement current
2) Maxwell equations
3) Boundary conditions of time-dependent electromagnetic field
4) Poynting’s Theorem and Poynting’s Vector
5) The generalized definition of conductors and insulators
6) The Lorentz potential
1
1) Displacement current
1. Basic principles of time-dependent electric and magnetic fields
（1）Gauss’ law in electrostatic field



Dd S  q
Active field
s
（2）The loop theorem of the electrostatic field

 
E dl  0
l
Conservative field
(保守场)
2
（3）Gauss’ law for magnetic field



Bd S  0
Passive field
s
（4）The loop theorem of the magnetostatic field

l
 
H dl  I
H是涡旋场,因为
它的旋度不为零
In the above four equations, D, E, B, H are the fields produced by rest
charges or steady current. q is the sum of charges enclosed by Gauss’
surface, and I is algebraic sum of conduction current through the closed
loop.
3
（5）Faraday’s law of electromagnetic induction
d m
i  
dt
The relationship between the circulation (环量) of rotational field
and the varying magnetic field is:


 ( 2) 
d m

B
LE  d l   d t    s t  d S
The equation indicates that the varying magnetic field can
produce a rotional electric field. Then a new question could be
asked: can the varying electric field produce a magnetic field?
4
2. Displacement current
S1
C
I
q
C A
S1
S2
I
S2

 
H  dl  I

q

E
B
 
H  dl  ?
C
C
Assume that the capacitor is charged. The charges on plates A and B
is +q and –q, respectively. The charge densities are ＋ and －  ,
respectively. So, one can readily obtain
q
D  
S
5
Due to the current continuity equation
S1
I



S
S2
On the polar
plate
 
dq
J  dS  
S
dt
d

dt
q
C A

d
dV  
V
dt


D  dS
q

E
B
S2 surface
S


J D  dS
where


D
JD 
t
JD is a displacement current density
6

 
J  dS 
S1
Now Ampere’s law can be
rewritten as:

 
H  dl 
C

S2


J d  dS



D  
J 
  dS
S
t 


The differential form of Ampere’s law can be
expressed as

  D
 H  J 
t

D
t

H


D
in the loop C obey the right-hand screw rule
H and
t
7
3. The relationship between the displacement current and the conduction
current (I.e., the connection and difference between… )
(1) They can both produce the magnetic field with the
same strength, provided that the displacement current and
the conduction current have the same current density.
位移电流与传导电流在产生磁效应上是等价的.
(2)They are produced in different ways (They originate from
different sources): specifically, the conduction current is caused by
the motion of the free charges, while the physical essence of the
displacement current is the varying electric field.
(3)They will exhibit different effects when passing through the
metal conductor: the conduction current can produce the Joule heat
while the displacement current cannot.
8
Example: the conductivity of sea water is 4S/m and its relative dielectric
constant is 81, determine the ratio of the displacement current to the
conduction current at 1MHz frequency.
We assume that the electric field is of the
sinusoidal form,
E  E m cos t
The density of the displacement current
is
D
Jd 
t
  r  0 E m sint
The amplitude is
given by
J dm   r  0 E m  2  106  81 
1
4  9  109
E m  4.5  10 3 E m
The density of the conduction current is
with the
amplitude
So,
J c  E m cos t
J cm  E m  4 E m
J dm
 1.125  10 3
J cm
9
2) Maxwell equations
1. Maxwell equation set in integral and differential forms
(1) The characteristic of electric field
(电场特性)
Integral form



Dd S 
S

dV
V
Gauss’ law
The dielectric flux through a closed surface equals the total charges Q
inside the closed surface.
The source of a electric field is the free charge

 D 

Differential form
10
(2) The characteristic of magnetic field

 
B  dS  0
Continuity of magnetic flux
S
Magnetic field is passive field. There is no free magnetic charge in nature.

B  0
11
(3) The relationship between the varying electric field and the magnetic
field

 
H  dl 
C



D  
 J 
  dS
S
t 

General Ampere law
The integral of magnetic field strength H along closed loop C equals
the sum of conduction current and displacement current

  D
 H  J 
t
The vorticity source of a magnetic field is the conduction current and
displacement current.
12
(4) The relationship between the varying magnetic field and the electric
field

 
E  dl  
C

B 
 dS
S t

Faraday’s law of electromagnetic induction
Time-dependent magnetic flux can produce electromotive force,
the vorticity of an electric field is the time-dependent magnetic
field


B
 E  
t
13
Maxwell equation set
(需熟记Maxwell方程组，并明确各个方程的物理含义)

Integral form



Dd S 

 
B  dS  0

 
H  dl 

 
E  dl  
S

dV
V
S
C
C

  D  
 J 
  dS
S
t 


B 
 dS
S t


Differential form

D  

 B  0


 D
 H  J 
t


B
 E  
t
14
2. Constitutive equation
材料的本构关系（方程）




D  E   0 r E



B  H  0 r H


J  E
J   E是微观欧姆( Ohm)定律.
下面用宏观欧姆定律I  U / R推导微观欧姆定律：
一段金属电阻, 截面S，长L， 电导率 ，
R=（1/ ）L/ S， U=EL， 电流密度J=I / S
J=I / S=( U / R) / S＝( EL（（1/
/
）L/ S）) / S＝ E.
15
3. The relationship between electric field and magnetic field
charge
motion
Agitation
（电荷能激发
电场）
Electric
field
current
Agitation
varying
varying
（电流能激发
磁场）
Magnetic
field
16
3) Boundary conditions of time-dependent electromagnetic field
(电磁场边界条件,具体讨论可见谢处方、饶克谨《电磁场与电磁波》pp. 74-79)
1. Boundary condition of magnetic field strength H
 
D
H  dl  H 1t l  H 2t l  J ST l 
hl
C
t

JST is the component of J vertical
to l. When h0, the second term
in the right equation is 0. Then we
have
H1t  H 2t  J ST
or
1, 1
2, 2
Rectangle loop in the interface
n  H 1  H 2   J S
When JS＝0
H 1t  H 2 t  0
h
or n  H1  H 2   0
tangential:
切向的
normal:
法向的
17
2. Boundary condition of electric field strength E
 
B
E  dl  E1t l  E 2t l  
hl
C
t

When h0, the right term in the above equation is 0
E1t  E 2 t
1, 1
2, 2
or
n   E1  E2   0
h
Rectangle loop in the interface
tangential:
切向的
normal:
法向的
18
3. Boundary condition of magnetic field strength B
B1n  B2n
or



 
n  B1  B2  0
The normal component of the magnetic flux density B in
the interface is continuous
4. Boundary condition of dielectric flux density D
D1n  D2n   S
or

n  D1  D2    S
or

n  D1  D2   0
When S＝0
D1n  D2n  0
19
5. Summary of the boundary conditions

E1t  E 2 t
B1n  B2n
D1n  D2n   S

n  H 1  H 2   J S
H 1t  H 2 t  J S
or
n   E1  E2   0

 
n  B1  B2  0



n  D1  D2    S
20

Interface of two passive media
无源，交界面上的边界条件
H 1t  H 2 t  0
E1t  E 2t  0
B1n  B2n  0
D1n  D2n  0
or

n  H 1  H 2   0
n   E1  E2   0

 
n  B1  B2  0



n  D1  D2   0
21
Ideal medium 1 and ideal conductor 2
理想介质１与理想导体２
E2  0,

D2  0, H 2  0, B2  0
H 1t  J S
E1t  E 2t  0
B1n  B2n  0
D1n   S
or


n  H1  J S
n  E1  0
 
n  B1  0
 
n  D1   S
22
4) Poynting’s Theorem and Poynting’s Vector
Poynting’s theorem is the mathematical expression for the law of
conservation of energy of the electromagnetic fields. Poynting’s vector
describes the flow of electromagnetic energy.
Poynting定律是电磁能量的守恒定律，其中Poynting矢量的物理意义是：电
磁能流密度。
23
1. Poynting’s theorem
From Maxwell equation set, we have
 H  J 
D
t
 E  
B
t
Combine the above equations
H    E   E    H    H 
B
D
 EJ  E
t
t
If we assume that the medium is linear, we can
obtain
H
B
 1


 H 2 
t
t  2

E
D
 1


 E 2 
t
t  2

24
For linear media
J   E  E i 
Ei is impressed electric field, JEi is the power of impressing (external)
sources per unit volume. 外电源也产生了一个电场
E  J   Ei
If we substitute the above expression into the
equation
H    E   E    H    H 
B
D
 EJ  E
t
t
we have
Ei  J 
J2


 1 2 1

 E  H 2     E  H 
t  2
2

25
Let us multiply this equation by a volume element d and integrate over
an arbitrary volume  of the field, we have

v
E i  Jdv 
The power of all the
sources inside v

v
J2

dv 

t
1 2 1

 E  H 2 dv 
v 2
2


Transformed inside v
into heat （焦耳热）
 E  H  dS
change rate of the
energy localized in the
electric and magnetic
field inside v
S
Power transferred
through S to a region
outside S
How the power is
classified
26
2. Poynting’s vector
S  EH
W/m2
Energy flow density
（能流密度）
J2

的物理意义是：焦耳热（Joul e heat ）密度（体积密度），
即单位体积内产生的电功（焦耳热的微观表达式）。
2
下面由宏观焦耳热I R表达式导出微观焦耳热表达式
J2

：
一段金属导体, 截面S , 长L,电导率 ，电流密度J，I  JS，
电阻R＝L /( S )，
I R   JS  L /( S )＝
2
2
J2

SL
SL是金属导体体积，那么
J2

（焦耳热的微观表达式）。
就是单位体积内产生的电功
27
Example: in passive free space, the time-dependent electromagnetic
field is
E  ey E0 cos(t  kz) (V / m)
Determine: (1) magnetic field strength;（2）instantaneous Poynting’s
vector；（3） average Poynting’s vector
(1)
 E  
B
t
E y
E y
B

 ez
 ex
 ex kE0 sin( t  kz )
t
x
z
kE0
B
H 
dt  ex
cos( t  kz )

0 t
0
1
28
(2)
S (t )  E (t )  H (t )
 ey E0 cos( t  kz )  ex
 ez
(3)
kE0
2
0
kE0
 0
cos( t  kz )
cos 2 ( t  kz )
1 T
S av   E (t )  H (t )dt
T 0
kE0 2
 ez
0T
kE0 2
 ez
 0T

T
0

cos 2 ( t  kz )dt
T
0
cos(2 t  2kz )  1
dt
2
kE0 2
 ez
(W / m2 )
20
29
5) The generalized definition of conductors and insulators
(导体与绝缘体的推广定义)
光学上判断绝缘体还是良导体，可以看传导电流( conduct i on cur r ent ) J
与位移电流( di spl acement cur r ent )
D
哪个大。
t
For linear media and time-harmonic （时谐）fields
D
 H  J 
D E
t
  H    j  E
E  exp( jt )

σ  ωε Good conductor (良导体)
【理想导体，电导率conduct i vi t y σ为无穷大】
σ  ωε Good insulator
【电磁波频率升高，良导体也成为了绝缘体】
30
6) The Lorentz potential
Ｍagnetic vector potential and electric field strength in terms of
retarded potentials (延迟势)


B   A
magnetic vector potential （Wb /m）
Therefore




 E  
 A
t




A 
0
   E 
t 


 A
E
 V
t


A
E
 V
t
Scalar potential (V)
Electric field strength in terms of retarded potential
31
If
V
  A   
t


 A
 

 E  
 V  
 t
 
 V  
2
 2V
t 2






 1

E

A 


 H   A  J  
 J     V 

t
t 
t 

 A


V
2
 A   J  
  2     A
t
t
2
If
We get

Helmholtz theorem

V
  A   
t

 A  
2

 A
2
t 2

  J
32
D’ Alembert’s equation
(达兰伯方程)
Lorentz condition:
 2V  
 2V



t 2

2


 A
2
 A   2   J
t

V
  A   
t
For sinusoidal electromagnetic field





2
2
 A  k A   J
 2V  k 2V  
Lorentz condition:

 A
V 
 j
33
说明：为什么要引入一个Lorentz条件？
在经典电动力学中，我们可以用Ｅ，Ｂ，Ｈ，Ｄ来描述电磁场．与之
等价的方案是，可以用上面提到的四维电磁势来代替Ｅ，Ｂ，Ｈ，
Ｄ．但是我们发现，同一个电场Ｅ与磁场Ｂ可以对应无穷多套电磁
势．也就是说，对于描述电磁场，电磁势是“超定的”，不是“欠定
的”．为了把电磁势定下来，我们需要额外的约束．这个约束条件就
是Lorentz条件．当然，约束条件是可以随意选的．我们不一定必须
选Lorentz条件．有时我们可以选择库仑规范．
无论选择什么约束条件，都是等价的．选择什么约束条件，主要看问
题方便而定．如为了照顾到狭义相对论不变性，我们就使用Lorentz
条件．
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Lorentz potential and gauge transformation (规范变换)
1) Vector and scalar potential



Since   B  0, we always have B    A. Thus, Faraday' s law
 E  
can be written as
  (E 
B
t
A
)0
t
This implies that the term in ( ) can be written as the gradient of a scalar potential V, i
A
E  V 
t
At this stage it is convenient to consider only the vacuum case. Then the Maxwell
equation
 E   /  0
1 E
 B  2
 0 J
c t
35
can be expressed in terms of the potentials as

 V  (  A)    /  0
t
2
1 2 A
1 V
 A  2 2  (  A  2
)   0 J
c t
c t
2
We have now reduced the set of four Maxwell equations to two equations. But
they are still coupled. The uncoupling can be achieved by using the arbitrariness
in the definition of the potentials.
36
2) Gauge Transformations, Lorentz Gauge, Coulomb Gauge
As A  A '  A   , B is unchanged. In order that E be
unchanged as well
A

A
E '  V '

 E  V 
t
t
t
We choose
V ' V 

t
The transformations
 A '  A  



V '  V 
t

are called gauge transform ations, obviously, we have

freedom to choose a set of (A, V).
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* Lorentz Gauge
1 V
 A 2
0
c t


 V  (  A)    E    /  0
t
2

1  2V
2
 V  2 2   / 0
c t
(L1)
Also,
 ( A)  0 J 
1 
A
(

V

)
2
c t
t

1  A
1 V
 A  2 2  (  A  2
)   0 J
c t
c t
2
2
2
1

A
 2 A  2 2   0 J
c t
(L2)
38
These two equations are equivalent to 4 Maxwell equations.
Under the Lorentz condition:
2
1 V '
1 V
1


  A ' 2
 0   A 2
  2  2 2
c t
c t
c t

1  2
  2 2 0
c t
2
In other words, as long as  satisfies the above equation,
the Lorentz condition preserves under the gauge transformation.
** Coulomb Gauge

 A  0
The solution is

 2V    /  0


 (r ' , t )
V (r , t )     d '
r r'
39
The scalar potential is just the instantaneous Coulomb potential due to the charge
density  ( r, t )
. This is the origin of the name “Coulomb gauge”.
The vector potential satisfies
2
1

A
1 V
2
 A  2 2   0 J  2 
c t
c
t
The term involving the scalar potential V can, in principle, be calculated from the
previous integral.
Note that J can be written as the sum of the longitudinal ( or irrotational)
and transverse currents
  
J  Jl  Jt
where  Jl  0 and  Jt  0. One can check that
1
 ' J (r ', t )
Jl  

d '
4
r r '
Jt 
1
J (r ', t )
 
d '
4
r r '
40
With the help of the continuity equation


   J
t

we can find from the expression of J l
V J l


t  0
Therefore the source for the wave function for A can be expressed entirely in
terms of the transverse current
1 2 A
 A  2 2   0 J t
c t
This is the origin of the name "transverse gauge".
2
The Coulomb gauge is often used when no sources are present. Then V=0


1 A
E
c t


B   A
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谈谈电荷守恒和能量守恒
(A)电荷守恒（charge conservation）
什么是电荷守恒？
电荷守恒的数学表达式是什么？
答：古典的“电荷守恒”定律，是指电荷不能凭空创生，也不能凭空消
失。自从量子电动力学(quantum electrodynamics)诞生（1940年代）
以来，电荷可以创生，也可以湮灭，如一个高能光子（gamma射线
光子）可以变成正负电子对，正电子（带正电）与普通电子（带负电）
相遇可以变成光子。
正电子质量与普通电子一样，只是带正电。反质子（带负电）与正
电子可以构成反氢原子，即反物质。反物质世界好比正物质世界的
“镜像世界”（如反物质世界的左、右定义与我们正物质世界的左、
右定义相反）。来自反物质世界的友好人士与你握手，你俩瞬间变为
光子和各种射线, 真正的“灰飞烟灭”。所以要谨慎交友。
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电荷守恒：
0＝
d   dx  





v
dt
t dt x t
x

    v 
t


 J
t

电荷守恒定律数学表达式( 微分形式) ：

   J  0.
t
麦克斯韦方程的魅力之一是它自动包含了电荷守恒定律：
D
两边求散度，
t

D 
  H   J 
。利用旋度的散度必为零，

t


对  H  J 


0   J 

 D

 J  0
t
t
 。再利用  D  ，得到电荷守恒定律
43
(B)能量守恒(energy conservation)

   J  0.
t
所以，凡是守恒定律，必然有这样的结构形式：
w
   S  0.
t
w为守恒量的密度，S 为守恒量的流密度（f l ow densi t y）.
我们已经知道电荷守恒定律为
对于没有损耗（焦耳热）也无外功的情况，电磁场能量守恒便是：
w
   S  0，
t
1
1
电磁能量密度w＝  E 2 +  H 2，
2
2
Poynting 矢量（能流密度）S＝E  H .
有关数学推导可见任何一本电磁学或电动力学教材，如谢处方、饶克谨
《电磁场与电磁波》( 第四版) pp. 175- 177.
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动量守恒与Lorentz力公式的推导
• 值得一提的是, Poynting 定律（电磁学能量守恒定
律）是一个功率方程（其微分形式为功率密度方
程）。能量在时间上的变化率即为功率；能量在空
间上的变化率即为力。除了上述功率方程，由
Maxwell 方程组亦可以得到力方程，即含有Lorentz
力 公 式 的 电 磁 学 动 量 守 恒 定 律 。
由于其比较复杂，一般电动力学和电磁波理论教材都
不讲。在文件夹“供学有余力或课时有多时讲授”
内有一个“Lorentz力公式的推导.pdf”(文题是：
由Maxwell方程推导Lorentz力公式)可供参考。
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