7.1 、 potentials of electromagnetic field, gauge invariance 7.2 、 d’Alembert equation and retarded potential

7.3 、 electric dipole radiation

7.4 、 EM radiation from arbitrary motion charge

Ch 7 Radiation of Electromagnetic Waves

1. What is EM radiation

EM field is excited by time-dependent charge and currents. It may propagate in form of waves. The problem is usually solved in terms of potentials.

2 ． It is a boundary value problem

Source (charge and current) excites EMF, EMF in turn affects source distribution --- boundary value problem!

For convenience, our discussions are limited to a simple case –Distribution of source is known.

特征：与 1/r 正比的电磁场！

§7.1 vector potential and scalar potential

potentials are slightly different from the static cases

since ， we can introduce vector

potential as the static field,

0 B

A

AB

1.a ） vector potential

Since ， scalar potential can not

be defined as before

0

t

BE

1.b ） scalar potentialAB

t

AA

tE

6

0)(

t

AE

t

AE

Define scalar func

t

AE

22)) ．． Gauge invarianceGauge invariance

t

AAA

),( A

Potentials are not uniquely determined,

they differ by a gauge transformation.

Gauge: Given a set of

give identical electric and magnetic fields' '( , )A

Prove: since and ， and can not change E

and B, so

A

A

B A A A

AA

t

A

ttt

A

t

AE

)(

A

t

t

  规范不变性规范不变性：在规范变换下物理规律满足的动力学方程保持不变的性质（在微观世界是一条物理学基本原理）。

  Coulomb gauge condition 0 A

3) ． Two typical gauges

transverse ( 横场 ) ， longitudinal ( 纵场 ) 。 is determined by instantaneous

distribution of charge density (similar to

static coulomb field)

A

To reduce arbitrariness of potential, we give

some constraint --- Gauge fixing 。

Symmetry or explicit physical interpretation

02 Function satisfies

  Lorenz gauge

condition 012

tc

A

satisfy manifest relativistic covariant equations

,A

Function satisfies

01

2

2

22

tc

0)1

()1

(2

2

22

2

tctc

A

0 AA

02 Prove

01

2

2

22

tc

2

2

222

111

tctcA

tcA

prove ：

Ludwig Lorenz

22

02 2 2

2

0

1 1( )

( )

AA A J

tc t c

At

Prove ： substitute ， into

Maxwell eqs

And using

AB

t

AE

0000 ,

EJt

EB

AAA

2)()(

4) ． D’Alembert equation

22

02 2 2

2

0

1 1AA J

tc t c

So satisfies Poisson equation as in static case. instantaneous interaction?

4.a) Under coulomb gauge

4.b) Under Lorenz gauge

2 22 2

02 2 2 20

1 1AA J

c t c t

洛仑兹规范下的达朗贝尔方程是两个波动方程，因此由它们求出的 及 均为波动形式，反映了电磁场的波动性。

),( A

),( BE

wave properties

highly symmetric and independent to each other

Get one, get 2nd for free.

Solution of d’alembert eq under Lorenz gauge indicates that EM interaction takes time.

To study radiation, we use Lorenz gauge.

§7.2§7.2 Retarded potentialRetarded potential

AAssumessume is known. We first solve is known. We first solve

point charge problempoint charge problem ，， then use then use

superposition to get general solutionsuperposition to get general solution

),( tx

1. Solve d’Alembert equation1. Solve d’Alembert equation

Assume point charge at Assume point charge at originorigin ， ，， ， symmetry indicates symmetry indicates is independent of is independent of

, so d’Alembert eq for scalar potential is , so d’Alembert eq for scalar potential is

)()(),( xtQtx

),( tr

,

asas 0r2

22 2

1 1( ) 0r

r r r c t

22

2 2 20

1 1 ( ) ( )( )

Q t rr

r rr c t

＊＊

r

trutr

),(),(

2 2

2 2 2

10

u u

r c t

letlet 　　

rcrtf )(

rcrtg )(

Outward spherical waveOutward spherical wave

Inward waveInward wave

The general solution for 1D wave equation isThe general solution for 1D wave equation is

)()(),(c

rtg

c

rtftru

rcrtg

rcrtf

tr)()(

),(

)0( r

Compared with static potential , we have ：rcrtQ

tr04

)(),(

0)( c

rtg

For radiationFor radiation

If point charge is placed at x

rcrtxQ

tx04

),(),(

容易证明上述解的形式满足波动方程＊式

For contineous charge distribution

0

( , )( , )

4V

rx t cx t dVr

Vdr

crtxJ

txAV

),(

4),( 0

Since satisfies identical equation as ,

so the solution ：A

2.Show the solution 、 satisfies Lorenz conditionA

2

10A

tc

证：令 ( , , )rt t t t x xc

A

Vdr

txJV

),(

40

0 1 1[ ( , ) ( , ) ]

4J x t J x t dV

r r

( , ) 1 ( , ) 1 ( , )( , )

J x t J x t J x tJ x t t r r

t c t c t

r r ( , )

( , ) ( , )

1 ( , )( , )

t c

t c

J x tJ x t J x t t

t

J x tJ x t r

c t

),(),(),( txJtxJtxJct

A

VdtxJr ct

),(1

40

Vd

rtxJtxJ

r]

1),(),(

1[

0 1( , )

4 t c

J x t dVr

( , )J x t

dVr

( , )

0S

J x tdS

r

2 20

1 1 1 ( , )

4 V

x t tdV

t r t tc c

0 1 ( , )

4 V

x tdV

r t

tc

A

2

10 1 ( , )

[ ( , ) ] 04 t c

x tJ x t dV

r t

0 电荷守恒定律

The value of the retarded potential at , depends on charge/current distribution at .

xt

crt

physical excitation at reaches observation point by .

And the speed of signal traveling in vacuum is c.

crt

t

33 ．． Physical interpretationPhysical interpretation

Electromagnetic interaction takes time!

§7.3§7.3 Electric dipole Electric dipole radiationradiation

 we limit our discussion to charge distribution of periodic motion. Furthermore,size of charge distribution is much smaller than the distance between charge and the observation point.

电磁波是从变化的电荷、电流系统辐射出来的。电磁波是从变化的电荷、电流系统辐射出来的。Antenna with high frequency alternative electric Antenna with high frequency alternative electric current current

Non-uniform moving charged particlesNon-uniform moving charged particles

Substitute into retarded solution （ ）ck

0 0( , / ) ( )( , ) [ ]

4 4

ikri tJ x t r c J x e

A x t dV dV er r

let ， so0 ( )( )

4

ikrJ x eA x dV

r

( , ) ( ) i tA x t A x e

1. General formula for radiation field1. General formula for radiation field

Compared with static case, there is an additional phase factorikre

( , ) ( )

( , ) ( )

i t

i t

J x t J x e

x t x e

Charge/currentCharge/current ：： 随时间正弦随时间正弦或余弦变化或余弦变化

Similarly,

)()(2

xAic

x

Satisfy Lorenz condition

Electromagnetic fields are

tiexBtxAtxB )(),(),(

),(),( txBk

ictxE

（ ） 0J

0 0 0E

B Jt

0

( )( , ) [ ]

4

ikri tx e

x t dV er

( , ) ( ) i tx t x e

Assume source to field point distance Assume source to field point distance

(size of charge/current distribution), so(size of charge/current distribution), so

Perform power expansion around

x

lr

rR

xxr

110x

...1

...1

...111

23

R

xn

RR

xR

Rx

RRr

22 ．． Multiple expansionMultiple expansion

Where is unit vector along ,n

R

R x

1) ． Power expansion for small size of source

(R is distance (R is distance between center of between center of coordinate and field coordinate and field point)point)

xnRR

xnR

RxnR

xnR

Rr

)1()

/1

1(

2

VdxnR

exJxA

xnRik

)(0 )(

4)(

0 ( )( )

4

ikrJ x eA x dV

r

2 n x

Since , so in denominator can be neglected. But it maybe important in phase,

R x n x xn

because is not necessary small, compared with2

Keep first two terms, we have

VdxnikxJR

exA

ikR...)1)((

4)( 0

xl 2

2n x

kn x

if ,

( , ) ( )

( , ) ( )

i t

i t

J x t J x e

x t x e

The radiation field for is

The first term dominates

VdxJR

exA

ikR)(

4)( 0

Electric dipole radiation

Under ， ， we can further divide into three cases according to and

2) ． 与 的关系R Rl l

R

a ） （近区）R2

, 1ikRR ek

R

t Tc

, Time of propagation

EM field is similar to the static case.

c ） （远区，即辐射区）REM waves propagates away from the source. interested

Rb ） （感应区）Very complicate.

0( , )4

ikReA x t p

R

1) ． Re-express in terms of dipole momentp

3 ． Electric dipole dipole radiationradiation

0( ) ( )4

ikReA x J x dV

R

Vdtxxp ),(

( , )p J x t dV

2) ． Electric and magnetic fields

. . .0 0( ) ( )

4 4

ikR ikR ikRe e eB A p p p

R R R

0

3

1 1 1 1( ) ( )

ikRe R nikR ikR ikR ikR ikRe e e ike ik eR R R R R RR

ki

nkk

here

Rk

1

R

11

R

nR

ike

R

e ikRikR

Consider 远区 ， ，即 ，

so

0( , )4

ikRikeB x t n p

R

( , )p J x t dV

.. .p i p

it

. ..

/p i p

( )Magnetic induction

Rl

30

( , )4

ikReB x t p n

c R

/ ,k c 0 01/c

30

20

1( , ) sin

4

1( , ) sin

4

ikR

ikR

B x t p e ec R

E x t p e ec R

In spherical coordinates, Let along axisp

z

),(),( txBk

ictxE

20

( , ) ( )4

ikReE x t p n n

c R

using

电场线是经面上的闭合线

Discussion:（ 1 ） E oscillates along longitude and B along latitude lines. Direction of propagating, E and B are orthogonal to each other (right-hand).

（ 2 ） E,B are proportional to ， so they are propagating spherical waves. They are transverse （ TEM 波） and maybe regarded as plane wave as .

（ 3 ） without （ ） , it can be shown that E is no longer perpendicular to k, electric lines are not close, but magnetic lines are still close （ TM 波） .

1/ R

R

R1 1

R

4 ． Energy flux, angular distribution and power of

radiation

22 40

3012

pP SR d

c

Average power

1) 。与球半径无关，能量可以传播到无穷远。2) 。与电磁波的频率 4 次方成正比。

* 202 3 2

0

1Re( ) sin

2 32

pS E H n

c R

Average energy flux vector ( 平均能流密度矢量 )

角分布

Example: Short antenna

2,)

21(),( 0

lzez

lItzI ti

lIdzzIpl

l

0

2/

2/ 2

1)(

rRIl

Ic

lIP 2

0

220

0

0222

00

2

1

1248

2

197l

Rr 短天线辐射能力不强。通常天线长度与波长同数量级，不能用简单的偶极辐射公式。

例：半波天线（长度为半波长）0 ( )

( )4

ikrJ x eA x dV

r

必须直接用推迟势计算

天线电流要与外场联合作为边值问题求解，一般较复杂。对于细长直天线，电流分布应是驻波，两端是波节。如

kzIzlzlkI

lzzlkIzI l cos

02/),2/(sin

2/0),2/(sin)( 0

2/

0

0

z

ikR

ikzikR

zz

zRrikr

z

ekR

eI

dzkzeR

eIedzkzI

r

eeA

200

cos4/

4/

00cos0

4/

4/

0

sin

)cos2

cos(

2

cos4

cos4

ekR

eIcinBcE

ekR

eIiB

ikR

ikR

sin

)cos2

cos(

2

sin

)cos2

cos(

2

00

00

辐射电磁场

2

2

2

22

200* sin,

sin

)cos2

(cos

8)Re(

2

1tosimilarn

R

IcHES

2.73

444.2

844.2

0

2002

cR

IcdRnSP

r

The energy flux

要得到高度定向的辐射，可利用天线阵的干涉效应。张角

)/(sin 1 Nl

§7.4§7.4 Magnetic dipole radiation Magnetic dipole radiation and electric quadrupole radiation and electric quadrupole radiation

VdxnikxJR

exA

ikR...)1)((

4)( 0

)]()[(2

1)()( xJJxxJJxnJxnxnJ

Dnxxedt

dnxxne

dt

d

xdt

xdn

dt

xdxnexJJxnVd

D

6

1

2

1)(

2

1

])()[(2

1)[(

2

1

mnJxVdn

xJnJxnVdxJJxnVd

2

1

])()[(2

1)(

2

1

]6

1[

40)2( DnmnR

eikA

ikR D

nBcBk

icE

AnikAB radiation

电四极辐射电四极辐射

磁偶极辐射磁偶极辐射

0( , )4

ikReA x t p

R

§7.5§7.5 Radiation from a localized charge in Radiation from a localized charge in arbitrary motion (Bo-p124)arbitrary motion (Bo-p124)

xxrcrttwhere

r

xtjxdxtA

r

xtxdxt

V

V

,/

),(

4),(

),(

4

1),(

30

3

0

method1 ： use the retarded potential粒子看作小体积电荷分布，直接积分

, ( )et x t

( )v t

( , )P t xr

0~

,~4~

0

Ar

e

rc

ve

c

vA

r

e

~4~

~4~

20

2

0

method2 ： using Lorentz transformation

)(

)]()([

)~~(~

rr

xxttc

ttcr

)(4

)(4

0

0

rrc

eA

rr

e

Lienard-Wiechert potential

At rest frame

where

At Lab frame

1 ( ) 1 ( )1 1

11

1

t r t t r x t t

t c t t cr t tv r t t

cr t t n

,A

E B At

r

vr

t

xrrt

r

txxc

tc

rtt

)(1

))((1 2

Derivative was performed with respective to t ， x.However

( ), ( )et r x x t

2 3 3 20 0

2 3 3 3 2 30 0 0 0

|4 4

( ( ))4 4 4 4

t const

A ev r ev rB A t

t c r c r

ev er er eE t r r v

c r r t r c r

c

nt

t

tFor

,1,0

1 1( | )

( / )

t const

rt r r t

c c tr v r r

t tcr cr c r r v c

, ,Let p ex p ev identical to electric dipole

Enc

Bvrrrc

eE

1)),((

4 320

vandrbetweenangle

nrc

veHES

)(sin16

223

02

22

在与加速度垂直方向辐射最强！Q ：圆周运动和直线运动时的辐射方向？

Radiation field from a relativistic charged particle

nn

vnn

rc

encES

6230

2

22

0)1(

|))((|

16

当 v 趋于光速，辐射集中于朝前方向，张角为

1

drtd

dtnStP 2)(

Radiation power

Enc

B

n

vnn

rc

eE

1

)1(

))((

4 320

Radiation field for arbitrary moving charge particle

30

226

30

22

2

3

66)(

cm

Fe

c

vetP vmF

Acceleration is parallel to velocity( 轫致辐射）

Bremsstrahlung and Synchrotron radiation

2 2 22

2 3 50

( ) sin

16 (1 cos )

dP t dt e vr S n

d dt c

Angular distribution

32 3/ 22 (1 )1

d mv mvF mv

dt

where

Radiation power

Acceleration is perpendicular to velocity（同步辐射）

2

30

224

30

22

2

66)(

cm

Fe

c

vetP vmF

辐射功率与粒子能量平方正比 !

2 2 2 2 2 2

2 3 50

( ) (1 cos ) (1 )sin cos

16 (1 cos )

dP t e v

d c

Angular distribution

Power

where 21

mvF mv

e.g. BEPC E= 2.8GeV ， =5479

高能加速器设计与同步辐射光源