04/06/2015PHY 712 Spring 2015 -- Lecture 281 PHY 712 Electrodynamics 9-9:50 AM MWF Olin 103 Plan for Lecture 28: Continue reading Chap. 14 – Radiation

PHY 712 Spring 2015 -- Lecture 28 104/06/2015

PHY 712 Electrodynamics9-9:50 AM MWF Olin 103

Plan for Lecture 28:

Continue reading Chap. 14 –

Radiation by moving charges

1. Motion in a line

2. Motion in a circle

3. Spectral analysis of radiation



Radiation from a moving charged particle

x

y

z

Rq(t)r

r-Rq(t)q

RRrR

vR

R

rqr

r

rqrq

tt

dt

tdt

:(notation) Variables


Liénard-Wiechert fields (cgs Gaussian units):

2

2

r

rqrqr

r

rqrq

dt

tdtt

dt

tdt

RvRRrRv

RR


Electric field far from source:

R

tt

cc

R

cR

qt

,,

,23

rERrB

vvRR

RvrE

tt

cR

qt

ccR

,ˆ,

ˆˆˆ1

,

ˆLet

3

rERrB

ββRRRβ

rE

vβ

vβ

RR


Poynting vector:

6

2

2

22

3

ˆ1

ˆˆˆ

4,ˆ

4,

,ˆ,

ˆˆˆ1

,

4,

Rβ

ββRRRrERrS

rERrB

ββRRRβ

rE

BErS

cR

qt

ct

tt

cR

qt

ct

Note: We have assumed that

ˆ , 0t R E r


Power radiated

22

3

222

6

22

2

6

2

2

22

sin4

ˆˆ4

1 :limit icrelativist-non In the

ˆ1

ˆˆ

4ˆ

ˆ1

ˆˆˆ

4,ˆ

4,

vβRR

Rβ

ββRRRS

Rβ

ββRRRrERrS

c

q

c

q

d

dP

c

qR

d

dP

cR

qt

ct


Radiation from a moving charged particle

x

y

z

Rq(t)r

r-Rq(tr)=Rq

RRrR

vR

R

rqr

r

rqrq

tt

dt

tdt

:(notation) Variables

Q

22

3

2

sin4

vc

q

d

dP

v.


2

3

2

22

3

2

3

2

sin4

v

v

c

q

d

dPdP

c

q

d

dP

Radiation power in non-relativistic case -- continued


Radiation distribution in the relativistic case

cRttr

c

qR

d

dP

/

6

22

2

ˆ1

ˆˆ

4ˆ

Rβ

ββRRRS

This expression gives us the energy per unit field time t. We are often interested in the power per unit retarded time tr=t-R/c:

Rβ ˆ1

r

trr

dt

dt

dt

dt

d

tdP

d

tdP

cRtt

rr

r

c

q

d

tdP

/

5

22

ˆ1

ˆˆ

4

Rβ

ββRR


Radiation distribution in the relativistic case -- continued

cRtt

rr

r

c

q

d

tdP

/

5

22

ˆ1

ˆˆ

4

Rβ

ββRR

For linear acceleration: 0ββ

5

22

3

2

/

5

22

cos1

sin

4ˆ1

ˆˆ

4

vRβ

βRR

c

q

c

q

d

tdP

cRtt

rr

r


5

22

3

2

/

5

22

cos1

sin

4ˆ1

ˆˆ

4

vRβ

βRR

c

q

c

q

d

tdP

cRtt

rr

r

Power from linearly accelerating particle

b=0b=0.5

b=0.7


Power from linearly accelerating particle

2

62

3

2

5

22

3

2

/

5

22

1

1 where

3

2

cos1

sin

4ˆ1

ˆˆ

4

v

vRβ

βRR

c

qd

d

tdPtP

c

q

c

q

d

tdP

rrrr

cRtt

rr

r

1

r

r

P

P

2/E mc


Power distribution for linear acceleration -- continued

y

z

x

b

b

.

r

q

2

62

3

2

5

22

3

2

/

5

22

1

1 where

3

2

cos1

sin

4ˆ1

ˆˆ

4

v

vRβ

βRR

c

qd

d

tdPtP

c

q

c

q

d

tdP

rrrr

cRtt

rr

r


Power distribution for circular acceleration

y

z

x

b

b

.r

q

42

3

2

/

5

22222

/

5

22

3

2

ˆ1

1ˆˆ1

4

ˆ1

ˆˆ

4

v

Rβ

βRRββ

Rβ

ββRR

c

q

d

tdPdtP

c

q

c

q

d

tdP

rrrr

cRtt

cRtt

rr

r

r


y

z

x

b

b

.r

q

Power distribution for circular acceleration

f

2 2 2

3 23 2

2 22 22

5

/

2

ˆ ˆ1 1

4 ˆ1

cos

sin

co = 1

4 s( ) cos(1 )1

r

r r

t t R c

dP t q

d c

q

c

β β R R β

β R

v


Spectral composition of electromagnetic radiation Previously we determined the power distribution from a charged particle:

22

/

3

2

2

/

6

22

2

~

:angle solidper power integrated Time

ˆ1

ˆˆ

4 re whe

ˆ1

ˆˆ

4ˆ

AA

A

A

dtdtd

tdPdt

d

dW

c

qt

t

c

qR

d

tdP

cRtt

cRtt

r

r

Rβ

ββRR

Rβ

ββRRRS


Spectral composition of electromagnetic radiation -- continued

titi edtetdt

dtdtd

tdPdt

d

dW

~

2

1

2

1~

:amplitudeFourier

~

:angle solidper power integrated Time

22

AAAA

AA

Parseval’s theorem Marc-Antoine Parseval des Chênes 1755-1836 http://www-history.mcs.st-andrews.ac.uk/Biographies/Parseval.html

http://www-history.mcs.st-andrews.ac.uk/Biographies/Parseval.html



22

2

0

22

0

2

22

~2

~~~

~~ : thatNote

~

:analysis sParseval' of esConsequenc

A

AAA

AA

AA

*

I

Iddd

d

dW

dtdtd

tdPdt

d

dW



cRttr

c

qt

/

3

2

ˆ1

ˆˆ

4 :caseour For

Rβ

ββRR

A

ti

cRtt

ti

edtc

q

etdt

r

/

32

2

ˆ1

ˆˆ

8

2

1~

:amplitudeFourier

Rβ

ββRR

AA



ctRti

cRtt

r

ctRti

cRttr

r

ti

cRtt

ti

rr

r

rr

r

r

edtc

q

edt

dtdt

c

q

edtc

q

etdt

/

/

22

2

/

/

32

2

/

32

2

ˆ1

ˆˆ

8

ˆ1

ˆˆ

8

ˆ1

ˆˆ

8

2

1~

:amplitudeFourier

Rβ

ββRR

Rβ

ββRR

Rβ

ββRR

AA



ctRti

cRtt

rrr

r

edtc

q /

/

22

2

ˆ1

ˆˆ

8

~

:expressionExact

Rβ

ββRR A

rR

rrRr

RRrRvR

R

ˆˆ :ionapproximat of level same At the

ˆ whereˆ For

:Recall

rtrtRtRr

ttdt

tdt

rqrrq

rqrr

rqrq



2/

22

/

2ˆ //

22

/

Exact expression:

ˆ ˆ

8 ˆ1

Approximate expression:

ˆ ˆ

8 ˆ1

r r

r

r q r

r

i t R t c

r

t t R c

i t t ci r cr

t t R c

qdt e

c

qe dt ec

r R

R R β β

β R

r r β β

β r

A

A

2

2 2ˆ /

22

/

Resulting spectral intensity expression:

ˆ ˆ

4 ˆ1r q r

r

i t t c

r

t t R c

I qdt e

c

r Rr r β β

β r


Example – radiation from a collinear acceleration burst

2

2 2ˆ /

22

/

ˆ ˆ

4 ˆ1r q r

r

i t t c

r

t t R c

I qdt e

c

r Rr r β β

β r

22 2

ˆ

220

2 2

2

2

3

2

23

cos

sin sin( (1 cos ) / 2)

( (1 cos

ˆ0

Suppose that

0 otherwise

ˆˆ ˆˆ Let

4 ˆ1

) / 2)c4 1 os

r ri tr

r

t

v

c

v

v

t

I qdt e

c

I q

c

r β

ββ

r r ββ r

β r


2

23

2 2

2

ˆ0

Suppose that

0 other

sin sin( (1 cos ) / 2)

( (1 cos ) / 2)

wi

cos

se

4 1

rt

I q

v

v

c

c

ββ

Example:

r

qDv



2

2 2 2ˆ /

2

Integration by parts and assumptions about the integration

limit behavior shows that the spectral intensity depends on

the following integral:

ˆ ˆ4

r q ri t t c

r r

I qdt t e

c

r R

r r β

2

Alternative expression --

It can be shown that:

ˆ ˆ ˆ ˆ

ˆ1ˆ1 r

d

dt

r r β β r r β

β rβ r

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04/06/2015PHY 712 Spring 2015 -- Lecture 281 PHY 712 Electrodynamics 9-9:50 AM MWF Olin 103 Plan for Lecture 28: Continue reading Chap. 14 – Radiation