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
PHY 712 Spring 2015 -- Lecture 28 304/06/2015
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
PHY 712 Spring 2015 -- Lecture 28 404/06/2015
Liénard-Wiechert fields (cgs Gaussian units):
2
2
r
rqrqr
r
rqrq
dt
tdtt
dt
tdt
RvRRrRv
RR
PHY 712 Spring 2015 -- Lecture 28 504/06/2015
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
PHY 712 Spring 2015 -- Lecture 28 604/06/2015
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
PHY 712 Spring 2015 -- Lecture 28 704/06/2015
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
PHY 712 Spring 2015 -- Lecture 28 804/06/2015
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.
PHY 712 Spring 2015 -- Lecture 28 904/06/2015
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
PHY 712 Spring 2015 -- Lecture 28 1004/06/2015
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
PHY 712 Spring 2015 -- Lecture 28 1104/06/2015
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
PHY 712 Spring 2015 -- Lecture 28 1204/06/2015
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
PHY 712 Spring 2015 -- Lecture 28 1304/06/2015
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
PHY 712 Spring 2015 -- Lecture 28 1404/06/2015
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
PHY 712 Spring 2015 -- Lecture 28 1504/06/2015
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
PHY 712 Spring 2015 -- Lecture 28 1604/06/2015
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
PHY 712 Spring 2015 -- Lecture 28 1704/06/2015
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
PHY 712 Spring 2015 -- Lecture 28 1804/06/2015
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
PHY 712 Spring 2015 -- Lecture 28 1904/06/2015
Spectral composition of electromagnetic radiation -- continued
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
PHY 712 Spring 2015 -- Lecture 28 2004/06/2015
Spectral composition of electromagnetic radiation -- continued
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
PHY 712 Spring 2015 -- Lecture 28 2104/06/2015
Spectral composition of electromagnetic radiation -- continued
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
PHY 712 Spring 2015 -- Lecture 28 2204/06/2015
Spectral composition of electromagnetic radiation -- continued
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
PHY 712 Spring 2015 -- Lecture 28 2304/06/2015
Spectral composition of electromagnetic radiation -- continued
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
PHY 712 Spring 2015 -- Lecture 28 2404/06/2015
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
PHY 712 Spring 2015 -- Lecture 28 2504/06/2015
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
PHY 712 Spring 2015 -- Lecture 28 2604/06/2015
Spectral composition of electromagnetic radiation -- continued
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