A. Marconi

Radiation from Moving Charges

Lecture 2Radiative Processes in AstrophysicsCourse on Relativistic Astrophysics AA 2016/2017

Ll EMARD - WIE CHART POTENTIALS

In the previous lecture we found the

retarded potentials of a distributions

of moving charges and currents:

KEH . f ftp.t?#d3i'

HAH .

t.fi?r?ihId3&with E

'

,retarded time

, given byt

'

a t -

KIC

Let's now consider a mooing pantlike charge e

withegudwn of matron

given by to G) ; we have

rift ) - dR¥tEYD . d¥oA÷-2

We can then compute the chargeand current densities

manythe

three - dimensional delta function

scat ) - @ See . too ]

jfi ,+ ) . erect ) See - za ]

The retarded potentials then

become

HAH - e .|stTf!rY?Td3t'

ATFH - ±cµAtsE-£AD_ dose

t'

.t

.

key

t.tl

.

C

by taking advantage of e changeof integration variable it is easyto demonstrate the followinghonesty of the delta function

{YfHS[g4]dx - §!fCxD ¥ , ,

with x ;,

i - 1,2 . ..

M, being the solutions

of the equation gcx ) a o

andg' Coe ) a dgcoydse

In the case of a three dimensional

delta - function the above blown

can be generalised to

ffcinslgci 'D oh'

. {" bciihsfe,with F.

,

i - 1,2,

..

.M, being the M

solutions of vectored equationGCI ) - o

.

] is the Jacobianof the Fhonsfrm

Raglin )

that

;ne( I ) - t8ak⇒

see

We now need to find the

solutionsof gfe ) - o that is the

points khere the argument ofDirac 's delta is zero

I'

- £Ct'

) - 0

I'

- 8 ( t - t¥D - o

Since che hostile ( change ) velocityis < C

,

the above equation

has only one solution forgiven

I,

t khech is explained

by the following figure

for simplicity let's denote with

to'

and t'

the solutions.

The Jecolian

matrixne

. ÷ { xh - [Kt% } -

p +derive car

.

t.

k componentintegration vouoble of I

'

- Ur '

) - °

= Sue - ÷ae[&oADnbut

taking into account that

t'

- t .

theyC

we olrtoim

teakettle. { s÷EHDu}Tk÷

= - r÷÷di . it

with K - }p[ RAY ]u ,

the hamp .

of the velocity at time t'

.

In

order to compute the derivative

We set

B - I -I

' E. Ict '

)then We have

Tye e - Sie £'

- Cx 's ,xi , xD

hencez.it -

2µe( ERIRD --2 Re

finallyR - },eF2 -- Rg e

- me

with methe l - th component

of versos m→ of R ( os - RHRDThe Jacobian matrix is finallyFee - Sue - { - mzfne ) }z

a Sue - Vu÷We need to amputee the

Jacobian that as

I -

Tense -

Tend- try

st÷÷÷t÷i¥÷tit is easy

to show that

] a I-

repC

Finally ,we can brute down

the retarded potentials for a

point - likemoving charge as

tfeit ) a ÷R

A (a) . endC K R

K e L - T.me-

R→- I

- Loft'

)

all these quantities are computedat the time

t'

- t - layoff fmnhutefd . )

Be I- BE

'

)

K - I - T(t').n(t=I

- v→Ct'

)

m→ - R→¥

r:these potentials are known as the

Liburd - Wiechert potentials .

They include the The factor chvh

is fundamental of the radiation

parents of a

moony change .

The physical meaning of the Yk

factor can be intuitively understood

by noting that the signal emitted

by a

moonysource

Lemore frequent (

"

closer"

) m the

direction of mdhm while theyare less frequent (

"

maser"

) m the

opposite direction

(see Igpbr effect ) .

Finally We note that, if the change

is notmoving , 10 becomes the

electrostatic potential while A is

ten,

as We expect .

ELECTROMAGNETIC FIELD OF A MOVING

CHARGE

he retarded potentials of a

many change with equation of

motion I e BCD oregiven

be at ) . exRATR ,t) - ed

C K R

K a L -

T.mgC

R→.

L. to Ct ) M - Ee

I

and all qualities ( re,R

,er

,

To ) one

amnuted at the retarded time t'

t'

. t .

18 - tC

The fields E and BE can be

obtained"

simply"

byamwtmy

Be ( I ,t ) -Px Ifft )

Efi ,t ) - - THE ,t ) - ± ¥ FKTD

Gladiolas are quite Anyhty but

straightforward .

For instance one

should recall hew We

amputed 2%¢,24yd ends

on to get on idea of what to

expect .

The full computationsare

shownm Egidio Landi 's

book.

Here We present the find results

straightaway

Ehr ) . €4 - na . E) ]+

+ ¥17 . ¥6 . i ) - na ]with the wlokvistc ndatrm

p→. re p . Ec

We can finally write

€44 . # ( s . pyfi . F) +

+ # Fxkefyxjy

Bfe,t ) - - Fp ( s - P ) se xp

. ¥ { xxittxx Exam }

Let's focus on the electric field( similar considerations ofhly for BYWe can cerate

Ea Econ ( e ,t ) + Ehn ( I ,t )

that is,

the electric field is the

sum of two contributions :

the Coulomb not ,Whose module

→

IE . .nl art

and the"

heohdeve"

notI Errol x R

- 1

The expressions for E and B

are general and, given

the

relativistic invariance of the well 's

equations ,ore valid in on odrtoy

menthol reference frame .

If We tbke the non zelokvstc

limit to the first order in PWe find the Well known

laws for Aohmfy phenomena.

Finally ,We on verify that

Be ,+ ) e ne x Etfe ,t )

RADIATION FROM A MOVING CHARGE

We just found that the Clectuc

field from emooing charge with

velocity P ( p - In ) and aeule.

ration @ ( p . I ) can be expressedas

Eta ) - lands . pyfi . F) +

tea m→×[ Can xp ]

with B a m→xE

All quantities are commuted at

the "

retarded"

time

t'

. t . lazed . t.ir#R→

-T

. EE'

) n→ .IR

K a 1 - T.LI a 1 - of . m→

we note that the first term ofE. depends only on P ,

while

the second one depends also

on P ,i.e. from acceleration

.

Therefore ,

the "

rodohm"

field ,

as We called it,

exists only ifthe change is accelerated

.

Let's try to understand its

meaning .

Let 's first cmader a chargewith a constant velocity ; Eras

is the and there is only the

first term Eau→

•Pct

,+ )

rR

•xp

e piety

o zip Hey se

velocity is constant, ebony as se

to ( + ) isthe monthly charge posture

attune +;

to compute Es one needs the

change month at time +'

- t - REILet 's cell £ the rector of the

direction pointingto the true hostile

portion at the t;

since the velocity is constant we

can wwe

Rok ) - Ici ) - I( + . t'

)hit t

'

- t - REI hence

to Ct ) -I ( t

'

) a F r±£ is related to the other quantities

;EA) - ear . ) ] + I - RA '

)that

:→- RCI ) - v→r¥

Ia Rct

'

) [ ni - F)

smee with a amtont velocityE- Fault ,

+ ) - offs . py ( a . g)We Can conclude the Ethos the

dhechm cmneetmy p to the True

paddle month at time t.

This results is obtained regardlessof the fact that We argued

old qualities at the worded

time t'

.

Also we showed that m→(+'

) - op ( t'

)is the directors between the

point P ( there we Want the fieldat time t ) and the rated

partum at time &.

Let's now assume that in a

time interval At,

the velocityChanges to a different value

to which it remains constant

again.

In the case shown m the figurethe pottle moves with constant

velocity then stops at x - ° forthM a short time It

.

At time tie ,

The for away observer ( so forthat the travel time from se - °

is > 1) doesn't know yetthat the noble has domed

so it sees the nalthle as it

wore m see 1 ( i.e. cohere it

Wohld be if it contemned with

Constant velocity ).

Hower the" close

"

observers sees the north

as it is,

i.e. stormed m neo

In both cases we have whole

field lines centered onthe

expected pottle forth .

The thnntum tokes whaleinauywn

at distancecft- a)

from the msthm Cohere the

north stoned of t.to ;the

thickness of the transition worm

is also C Dt,

khere At is the

time m khnch the hotels

stoned . If stat . to,

the

electric field mthe tzanathm

region is1- to the redid

dheathm ; mourner this"

Fhonnthn"

field honeyedalong the bold direction

Corth velocity C : it is as

electromagnetic Wove !

Clearly this e.mn. cook

originatedfrom the deceleration

of the hated : the emnmm

of redrawn is thereforeamounted net to the velocity

of the paddle but to its

acceleration.

If We go lock tothe expression

for the E field

Estrin - ja ( s . Pyfi . F) +

+ ⇐ axxnroyxpj

we clearly note that the

Coulomb term,

2 R- 2

,dominoes

at small R,

khile the zoduothm

term,

X R' '

,dominates at

layer .

Both terms are annular forR a Rc such that

* e

= e÷q3Rcthat is for Rex I

a

If Lis

the typical dimension

of theregion

where the chargemoves

,

and e is the time

scale fr velocity variationsthem

a = ± 2

and

Rate ¥As we will seem tie following ,

the change zealots at typicalfrequencies v = € so

We

can write

Rca CNI - { jt - ItThe

zeyum for R > > Rc is called

the redrawnregion

and is

the beganwhere the "

radiate"

Contribution dominates the Es field ,

i.e. the part which depends on

acceleration.

For R→Rc we then have

Eficthefafreit) - Irritated xDwith B→

a sex E.

Os explained already ,

allqualities

are computed at time

t'

- t - II - toC

However, if the observer is at

large distance from the change ,

such that R > > L,

the differenceE

'

- trs Constanta .e.

outcrop

m the

direction from the change motion

mt and The hour under

examination.

We now cash to compute the

bohemian flux emitted fromthe halide ( obviously m the

radiationregion ) .

As We reviewed,

The module ofthe Poynty vector representsthe amount of energy her

nut surface herunit time

,

perpendicular to the propagationdirection

5 - Et Exb- f- Ex ( sex E) -

a act E2m→

that is 5is directed form the

charge to the mainunderexamination

.

Let's select the non - relativistic

Casefor a start :

la as

he a e -

P.sx X 1

n→ - of a a

the fheohdum ) field is

Ek, Dx Ekmfe , D a£rm→×Ee×£]

- - free - Canopy

for the poverty of the double

Cross product

n→×(T×w→ ) - ( we .u→)v→ . 4.1 )w→EKTH - - eg[ or - ( a . myx ]

E. is directed 1- tom and lays m

the plane identified by R and oh

AT g

*

, 'd>£ .>D• > PaEs:- s

if g isthe angle between m→ and of

( this is at time t'

,

but,

as We

have seen E'

= t + At where At is

Constant,

i.e. We can neglect about

it ) We obtain

E . - at [a→ -aasori ]

E2- Ie [ oitoeaio

-ceased . If

a cod [ alt olaio -2dg ?e]

z *reQ2Hn2&

Finally→

- ITc¥pa2xn2em→

§ -

e2a2xiis-

(3 RZ

giventhe rhymed meaning of $ )

( amount of energy her unit time

perunit surface beyond ailor

to the panayotwn direction )The power emitted in the solid

angle dr is

dp - 151 pidr

R2dR is the surface perpendicularTo the honeydew of em

. fieldcrossed by the energy .

Then

off - 151 R2- Get said

The radiation power of themoving

charge depends of the duecthm

as sale.

Wecan draw the zodathm pattern as

follows

@€I¥¥'-

rFeaFm*o

0

In the diagram We drew a aeroe

whose distance from the policeportion 0 is a 154 ,

i.e.,proportionalhe said

.

This is the antenna diagram of a

man ulotvstc hatch .

We note that there is no emission

mFhe acceleration direction

,able

emission is maximal in the

direction perpendicular to the

acceleration.

To find the total tower of the

zoohdhm emitted by the hailedwe need to integrate on a

Zphere of zodms R

P - § ddfsdr - { 5.I d E

'

ee .

the total mower isthe flux of

the Poyang vector across a

Closed surface Celuch we can

Choose as the sphere of bodies R

Centered on the hostile.In this

case

g.ms. 1ST

and

P -

{ lstrdr -

{.d£rdra

- Pdt ftdo[Eff*xio]r*sne- HI 2T fFen3o de

since [m3o do - ub we finally got

Pz xeI⇒n2*43- 252¥

P - za2e÷(3

This is the Well known Larmore 's

Formula.

a few things to zemak in Larner 's

formulap . ze2a÷3

D power is mgnrthnd to thesquare

of change and deceleration

2) off - eIg÷ neo

emission is not isotopic and has

the characteristic ndthn ofthe dipole ( x senior ) ; there is no

emitted Zodrothm m the acceleration

direction,

and maximum emission

is nertplndrahr to it.

3)the rotontoneous direction of Erasis determined by a and X

;

if the pottle is accelerated

bneohy ( i.e. 8 has a constant

direction ) bdotwn will be

Loo % polarised in The dire phone

RADIATION SPECTRUM FROM A MOVING

CHARGE

Conadevng the non - uldtostc case

We found that the electric field( module ) is :

E - e# send

Let's now find The zediothmapectmmWe wall from the first lecture that

Ecw ) - t.IE#eiutdwand

Ect ) - €¥(w) e iutdw - eatenCZR

let's define the dude momentum

as

F - e I - FG )

them its Former Transform is

pat :{Fcayeiutdu

then taking the second time derive

twee+ ) . each - -6+52 pyujeiatdwthem

EA ) - JFEC w ) eiutdw- ea{÷oo=

- - rate .§E2§cw ) eiutdw

that is

E ( w ) .- w2F ( w ) are2 R

we have then connected the spectrum

of E to trot of the dipole .

The total flux of energy hermit

pnfoie is

F . ddt÷ - § Fat . ¥§¥CDdtwe found

j . d¥a . c FTECWHDW

the amount of radiated energyher unit surface and unit

frequency bond isthen

Fu - ddadeun e a IEESP

Integrating over a sphericalsurface A of bolus R

dd±w - fc IECWIPIA .

A

a fca"

lee)P¥tr÷*dR

:w4g¥dPf←dt§xn3ode

izu 43

a 8Fw÷ IFEDP

that is,

mthe non - zelotvshe approx .

the one ctwm of embed radiation

is monotone to w"

/F(w)/ ?

MULTI POLE EXPANSION IM THE

RADIAIUM 20 ME

Up to now we have seen the

Zaohdhem Gam one non - relatevista accelerated change .

If we have e set of Mmoving

charges roe can take advantage

of the linearity of the Maxwell 's

equations and Wuk

EH,+ ) - EH Eice

, D

BGTD . Em Biter ) - §"m?×E,

with

Eifert ) .

cenpinr ,kix[CniiefxcifWe have considered only the

Zedeothm field and fr the E- th

particle We use the time

ti - t-

II - F.C*a

If L asthe typical she of the

regionkhere paths one confined

and wego

to distance R >>L

them we can take a

central reference point C

Luth To,

R,n→ and assume

MT a m→A

→

rite .÷÷;#¥÷given

the reference point Rc

ST - F - Iod

t'

; . t - 1¥ =t -

kettlet -

RR. I ) .CL .

8 ;D't-

Z

zE

-

[ pits ? -25 .R]÷

C

t t - Rees + £ .z Erard ]

'

2

khech,

to the first order in Fr ,

t.tt= t

- refs - Band ) -

e Et + SITC

Luth He - t - Ec"

contended"

time

m C

Togive

the some"

onhzinded"

time to all changes it isthen

necessarythat the time scales

of motions ore

Cmonom D IBEX = £C

the wovekmght typed of the

path emesoum is = Cc

hen

: ace DL

When I > > L it is ramble to

define the electric dipole

moment of the astern of

changes as

5- EM e ; E

finally obtainingEofe

,

t ) - see m→×( sex is )and all quantities are computed

at the anti noted time Ec.

This is the some expression

obtained for the single noble

( if we set pee E) hence we

can write the Lorne formulafor the distribution of patches

P a

2 lD±3 C

3

in the case in which the halidedistribution is symmetric and

lift a o We need to expandto the terms in (E)

2

, (E)3

and

20 on.

Going to higher order exponam

it will be homble to define higherorder moments such as the

magnetic dipole moment,

the electric quedmpole moment

and x on.