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《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

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（ 1 ） Gauss’ law in electrostatic field

qSDs

d

（ 2 ） The loop theorem of the electrostatic field

0d lEl

1. Basic principles of time-dependent electric and magnetic fields

Conservative field ( 保守场 )

Active field

1) Displacement current

3

H 是涡旋场 , 因为它的旋度不为零

0d SBs

IlHl

d

Passive field

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 ） Gauss’ law for magnetic field

（ 4 ） The loop theorem of the magnetostatic field

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The relationship between the circulation ( 环量 ) of rotational field and the varying magnetic field is:

StB

tlE sL

m

dd

dd)2(

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?

tm

i dd

（ 5 ） Faraday’s law of electromagnetic induction

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2. Displacement current

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

I

1SC

2S

Sq

D

IldHC

I1S

C

2S

q q

E

A B

?C ldH

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Due to the current continuity equation

dt

dqSdJ

S

On the polar plate

SV

dSDdt

ddV

dt

d

S2 surface

S

D SdJ

where

t

DJ D

JD is a displacement current density

I1S

C

2S

q q

E

A B

7

21 S

dS

SdJSdJ

SC

Sdt

DJldH

Now Ampere’s law can be rewritten as:

H

t

D

The differential form of Ampere’s law can be expressed as

t

DJH

and H

t

D

in the loop C obey the right-hand screw rule

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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.

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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,

tEE m cosThe density of the displacement current is

tEt

DJ mrd sin0

The amplitude is given by

mmmrdm EEEJ 39

60 105.4

1094

181102

The density of the conduction current is

tEJ mc coswith the amplitude

mmcm EEJ 4

310125.1 cm

dm

J

JSo,

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1. Maxwell equation set in integral and differential forms

2) Maxwell equations

(1) The characteristic of electric field ( 电场特性 )

VS

dVSD

d Gauss’ law

The dielectric flux through a closed surface equals the total charges Q inside the closed surface.

Integral form

D

The source of a electric field is the free charge

Differential form

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S

SdB 0

(2) The characteristic of magnetic field

Continuity of magnetic flux

Magnetic field is passive field. There is no free magnetic charge in nature.

0 B

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(3) The relationship between the varying electric field and the magnetic field

SC

Sdt

DJldH

General Ampere law

The integral of magnetic field strength H along closed loop C equals the sum of conduction current and displacement current

t

DJH

The vorticity source of a magnetic field is the conduction current and displacement current.

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(4) The relationship between the varying magnetic field and the electric field

SC

Sdt

BldE

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

t

BE

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Maxwell equation set(( 需熟记需熟记 MaxwellMaxwell 方程组，并明确各个方程的物理含义方程组，并明确各个方程的物理含义 ))Maxwell equation set(( 需熟记需熟记 MaxwellMaxwell 方程组，并明确各个方程的物理含义方程组，并明确各个方程的物理含义 ))

VS

dVSD

d

S

SdB 0

SC

Sdt

DJldH

SC

Sdt

BldE

Integral form

D

t

BE

0 B

t

DJH

Differential form

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2. Constitutive equation 材料的本构关系（方程）

EED r

0

HHB r

0

EJ

)

/

/ / .

J E Ohm

I U R

U R E

是微观欧姆( 定律.

下面用宏观欧姆定律 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=( )/ S＝(EL（（1/ ）L/ S）) / S＝

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3. The relationship between electric field and magnetic field

charge current

Magnetic field

Electric field

motion

varying

Agitation（电流能激发磁场）

Agitation（电荷能激发电场）

varying

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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

lht

DlJlHlHldH STtt

C

21

1, 1

2, 2

h

Rectangle loop in the interface

JST is the component of J vertical to l. When h0, the second term in the right equation is 0. Then we have

When JS ＝ 0

021 tt HH or 021 HHn

STtt JHH 21 SJHHn 21ortangential:切向的normal: 法向的

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2. Boundary condition of electric field strength E

lht

BlElEldE tt

C

21

When h0, the right term in the above equation is 0

1 2 0 n E Ett EE 21 or

1, 1

2, 2

h

Rectangle loop in the interface

tangential:切向的normal: 法向的

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3. Boundary condition of magnetic field strength B

nn BB 21 021 BBnor

The normal component of the magnetic flux density B in the interface is continuous

4. Boundary condition of dielectric flux density D

When S ＝ 0

Snn DD 21 SDDn 21

or

021 nn DD 021 DDn

or

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5. Summary of the boundary conditions

Stt JHH 21

tt EE 21

nn BB 21

Snn DD 21

SJHHn 21

1 2 0 n E E

021 BBn

SDDn 21or

21

021 tt HH

021 tt EE

021 nn BB

021 nn DD

021 HHn

1 2 0 n E E

021 BBn

021 DDn

or

Interface of two passive media无源，交界面上的边界条件

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St JH 1

021 tt EE

021 nn BB

SnD 1

Ideal medium 1 and ideal conductor 2理想介质１与理想导体２

0 ,0 ,0 ,0 2222 BHDE

or

SJHn

1

01 Bn

SDn 1

01 En

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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。

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1. Poynting’s theorem

t t

B D

HH EE E H E J

Combine the above equations

From Maxwell equation set, we have

t

D

H Jt

B

E

If we assume that the medium is linear, we can obtain

2

2

1H

tt

BH

2

2

1E

tt

DE

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iEEJ

For linear media

Ei is impressed electric field, JEi is the power of impressing (external) sources per unit volume. 外电源也产生了一个电场

iE J E

If we substitute the above expression into the equation

t t

B D

HH EE E H E J

we have

HEHEt

JJE i

222

2

1

2

1

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Let us multiply this equation by a volume element d and integrate over an arbitrary volume of the field, we have

dSHEdvHEt

dvJ

dvJESvvv

i

222

2

1

2

1

The power of all the sources inside v

Transformed inside v into heat （焦耳热）

change rate of the energy localized in the electric and magnetic field inside v

Power transferred through S to a region outside S

How the power is classified

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2. Poynting’s vector

W/m2

Energy flow density （能流密度）

S E H

2

2

222

2

, , ,

/( )

/( )

J

JI R

S L J I JS

R L S

JI R JS L S SL

JSL

2

的物理意义是：焦耳热（J oul 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）。

下面由宏观焦耳热 表达式导出微观焦耳热表达式 ：

一段金属导体 截面 长 电导率 ，电流密度 ， ，电阻 ＝ ，

＝

是金属导体体积，那么 就是单位体积内产生的电功

（焦耳热的微观表达式）。

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Example: in passive free space, the time-dependent electromagnetic field is

0 cos( ) ( / )yE e E t kz V m

Determine: (1) magnetic field strength; （ 2 ） instantaneous Poynting’s vector ；（ 3 ） average Poynting’s vector

(1)B

Et

0 sin( )y yz x x

E EBe e e kE

t x zt kz

0

0 0

( )1

x

kEBH dt e c s t kz

to

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00

0

cos( ) ( )y xe E t kzkE

cos t kze

0 22

0

cos ( )z t zkE

e k

(2) ( ) ( ) ( )S t E t H t

(3)0

1( ) ( )

T

avS E t H t dtT

2

0

20

0

cos ( )z

Te t kz

kEdt

T

20

00

cos(2 2 ) 1

2

T

z

t kze

kEdt

T

220

0

( /2

)z

kEme W

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5) The generalized definition of conductors and insulators( 导体与绝缘体的推广定义 )

For linear media and time-harmonic （时谐） fields

exp( )D

H J D E E j tt

H j E

Good conductor ( )σ ωε

σ

良导体【理想导体，电导率conducti vi ty 为无穷大】

Good insulator σ ωε

【电磁波频率升高，良导体也成为了绝缘体】

J

D

t

光学上判断绝缘体还是良导体，可以看传导电流(conducti on current)

与位移电流(di spl acement current) 哪个大。

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Therefore

At

E

0

t

AE

6) The Lorentz potential

Ｍ agnetic vector potential and electric field strength in terms of retarded potentials ( 延迟势 )

magnetic vector potential （ Wb /m ）AB

Scalar potential (V)

Vt

AE

Electric field strength in terms of retarded potential

Vt

AE

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If

Vt

AE

2

22

t

VV

If

t

VA

We getJ

t

AA

2

22

t

AV

tJ

t

EJAH

1

At

A

t

VJA

2

22

Helmholtz theorem

VA

t

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t

VA

Lorentz condition:

D’ Alembert’s equation

( 达兰伯方程 )

Jt

AA

2

22

2

22

t

VV

For sinusoidal electromagnetic field

VkV 22

JAkA

22

j

AV

Lorentz condition:

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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 )

law sFaraday' Thus, .AB have always we,0 Since

B

BE

t

can be written as

( ) 0A

Et

This implies that the term in ( ) can be written as the gradient of a scalar potential V, i.e.,

AE V

t

At this stage it is convenient to consider only the vacuum case. Then the Maxwell equation

0/E

02

1 EB J

c t

1) Vector and scalar potential

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can be expressed in terms of the potentials as

20( ) /V A

t

22

02 2 2

1 1( )

A VA A J

c t c t

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.

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2) Gauge Transformations, Lorentz Gauge, Coulomb Gauge

As ' , is unchanged. In order that be

unchanged as well

A A A B E

' 'A A

E V E Vt t t

We choose 'V V

t

The transformations '

'

A A

V Vt

V).,A( ofset a choose tofreedom

have weobviously, ations, transformgauge called are

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* Lorentz Gauge2

10

VA

c t

20( ) /V A E

t

22

02 2

1/ (L1)

VV

c t

Also,

0 2

1( ) ( )

AA J V

c t t

2

202 2 2

1 1( )

A VA A J

c t c t

22

02 2

1 (L2)

AA J

c t

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These two equations are equivalent to 4 Maxwell equations.Under the Lorentz condition:

22

2 2 2 2

1 ' 1 1' 0

V VA A

c t c t c t

01

2

2

22

tc

In other words, as long as satisfies the above equation, the Lorentz condition preserves under the gauge transformation.

** Coulomb Gauge

0 A

20/V

The solution is ''

),'(),(

drr

trtrV

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The scalar potential is just the instantaneous Coulomb potential due to the charge density . This is the origin of the name “Coulomb gauge”.

The vector potential satisfies

),( tr

22

02 2 2

1 1A VA J

c t c t

The term involving the scalar potential V can, in principle, be calculated from the previous integral.

tl JJJ

where 0 and 0. One can check thatl tJ J

Note that can be written as the sum of the longitudinal ( or irrotational)

and transverse currents

J

1 ' ( ', )'

4 'l

J r tJ d

r r

1 ( ', )'

4 't

J r tJ d

r r

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With the help of the continuity equation Jt

lJ

of expression thefrom findcan we

0

lJV

t

22

02 2

Therefore the source for the wave function for A can be expressed entirely in

terms of the transverse current

1

This is the origin of the name "transvers

t

AA J

c t

e gauge".

t

A

cE

1

AB

The Coulomb gauge is often used when no sources are present. Then V=0

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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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0

0.

0

d dxv

dt t dt x t x

vt

Jt

Jt

DH J

t

DH J

t

DJ D

t

电荷守恒：

＝

电荷守恒定律数学表达式(微分形式)：

麦克斯韦方程的魅力之一是它自动包含了电荷守恒定律：

对 两边求散度，

。利用旋度的散度必为零，

。再利用 ，得到

0Jt

电荷守恒定律

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(B) 能量守恒 (energy conservation)

2 2

0.

0.

0

1 1+

2 2

.

Jt

wS

t

w S

wS

t

w E H

Poynting S E H

我们已经知道电荷守恒定律为

所以，凡是守恒定律，必然有这样的结构形式：

为守恒量的密度， 为守恒量的流密度（fl ow densi ty）.

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：

，

电磁能量密度 ＝ ，

矢量（能流密度） ＝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。