8.07 Leture 37: Deember 12
2012
(THE LAST��
RADIATION
Radiation: Eletromagneti felds that arry energy of to
infnity.
- and B
- fall of only as 1/r, so the Poynting
At large distanes, E
vetor falls of as 1/r2. If the Poynting vetor is then integrated
over a large sphere, of area 4 r2, the ontribution approahes
a onstant as r
.
Alan Guth
Massahusetts Institute of Tehnology
8.07 Leture 37, Deember 12, 2012
-1-
Reall the Lieenard-Wiehert potentials:
V (~r ; t) =
q
1
40 j~r
~r p j(1
q~vp
~A(~r ; t) = 0
4 j~r ~r p j(1
~vp
c
~vp
c
^)
;
~vp
= 2 V (~r ; t) ;
^) c
(1)
(2)
where -r p and -vp are the position and veloity of the partile at tT ,
tT = t and
r
r
p
i=
-r - -r p
-r - -r p
= -r - -r p
(3)
(4)
Alan Guth
Massahusetts Institute of Tehnology
8.07 Leture 37, Deember 12, 2012
-2-
Eletri Dipole Radiation
Simplest dipole: two tiny metal spheres
separated by a distane d along the
z -axis, onneted by a wire, with
harges
q (t) = qo os(wt)
+
+q
r
θ
(5)
on the top sphere, and q (t) =
-qo os(wt) on the bottom sphere.
Then
z
-
y
d
-q
Image by MIT OpenCourseWare.
qo os[w (t - + /)] qo os[w (t - - /)]
V (-r t) =
40o
+
1
All diagrams are from D.J. GriÆths, Introdution to Eletrodynamis, 3rd Edition
(6)
-3-
Approximation 1:
±=
�
=>
d
« r.
r2 = rd os e (d/2)2
1 r 1 1 ± . os e
2T
± T
�
�
.
os[w (t - ± /)] r os w (t - r/) ± 2 os e
(7)
d « r is ALWAYS valid for radiation, whih is defned in the
r ! 1 limit.
.
Approximation 2: d
�
Sine A = 2 /w , this is equivalent to d « A. This is the IDEAL
DIPOLE APPROXIMATION. This is really the frst term in a
power expansion in d/A, the multipole expansion for radiation,
but we will go no further than the dipole. Implies
«
os[w (t -
wd
± /)] r os[w (t - r/)] = 2 os e sin[w (t - r/)]
(�)
- -
Then, defning Po = qo d,
p0 cos 1
V (r; ; t) =
cos[! (t r=c)]
40r r
»
!
sin[! (t r=c)] : (9)
c
Approximation 3: r
o.
The region r » A is alled the radiation zone. This approximation
is ALWAYS valid for disussing radiation. Implies that the frst
term in urly brakets an be dropped:
p0 ! cos V (r; ; t) =
sin[! (t r=c)] :
4o0 c
r
(10)
Summary of approximations: d « A « r.
Alan Guth
Massahusetts Institute of Tehnology
8.07 Leture 37, Deember 12, 2012
-5-
- , whih is due to the urrent in the wire. Reall
Need also A
- (-r t) =
A
Here
1
4o0
'
0
~
1
(
r
J
~
; tTr )
3
'
dx
:
'
0
j-~r - -~r j
dq
3
'
d x 1 = d = dz zi
dt
./2 -q w sin[w (t - /)]
J
o
- (-r t) = o
A
dz
4 -./2
so
Jo Po w
= sin[w (t - r/)] iz
4
r
(11)
(12)
(13)
Alan Guth
Massahusetts Institute of Tehnology
8.07 Leture 37, Deember 12, 2012
-6-
Diferentiating,
~= r
~V
E
2
~
@A
0 p0 ! sin =
cos[! (t r=c)] ^ :
@t
4
r
(14)
-B
~ (~r ; t) = 1 ^r E
~ (~r ; t) :
c
(15)
Alan Guth
Massahusetts Institute of Tehnology
8.07 Leture 37, Deember 12, 2012
-7-
Poynting Vetor:
2
~S = 1 (E
~ B
~ ) = 0 p0 ! sin cos[! (t
0
c 4
r
2
r=c)]
^r :
(16)
Intensity:
�
2
Average the Poynting vetor over a omplete yle: os = 1/2
2
2
4
p
!
sin
~S = 0 0
^r :
2
2
32 c
r
D E
�
(17)
Image by MIT OpenCourseWare.
All diagrams are from D.J. GriÆths, Introdution to Eletrodynamis, 3rd Edition
-8-
Polarized Blue Skies
Blue is the highest frequeny in the visible spetrum. The sky
appears blue largely beause the w
! 4 fator in dipole radiation
implies that blue light is sattered more strongly than other
frequenies.
Sunsets are red beause the blue light has been sattered out
of the path of the sunlight.
Alan Guth
Massahusetts Institute of Tehnology
8.07 Leture 37, Deember 12, 2012
-9-
Sun’s rays
This dipole does
not radiate to the
observer
Image by MIT OpenCourseWare.
When the line of sight is perpendiular to the Sun's rays, the
light is polarized horizontally.
All diagrams are from D.J. GriÆths, Introdution to Eletrodynamis, 3rd Edition
Alan Guth
Massahusetts Institute of Tehnology
8.07 Leture 37, Deember 12, 2012
-10-
Total Power:
Integrate over a sphere at large r.
hP i =
Z
2
2
4
p
!
sin
2
~S d~a = 0 0
r sin d d
2
2
32 c
r
Z D E
0 p20 ! 4
=
:
12c
(18)
Alan Guth
Massahusetts Institute of Tehnology
8.07 Leture 37, Deember 12, 2012
-11-
Magneti Dipole Radiation
Consider a wire loop of radius
b, with alternating urrent
I (t) = I0 cos(!t) ;
z
(19)
r
with magnetic dipole moment
θ
m
~ (t) = b2 I (t) z^
(20)
= m0 cos(!t) z^ ;
ψ
b
x
where
φ’
dl’
m0 b2 I0 :
(21)
y
Image by MIT OpenCourseWare.
All diagrams are from D.J. GriÆths, Introdution to Eletrodynamis, 3rd Edition
Alan Guth
Massahusetts Institute of Tehnology
8.07 Leture 37, Deember 12, 2012
-12-
Make the same approximations as before, with b replaing d as the
size of the system: b « A « r. Find
2
~E = 0 m0 ! sin cos[! (t
4c
r
r=c)] ^ :
(22)
Compared to the eletri dipole radiation,
m0
p0 !
;
c
^ ! ^ :
(23)
~B (~r ; t) = 1 ^r E
~ (~r ; t) :
c
(24)
As always for radiation,
Alan Guth
Massahusetts Institute of Tehnology
8.07 Leture 37, Deember 12, 2012
-13-
(Eletri Dipole� Radiation
From an Arbitrary Soure
Consider an arbitrary time-dependent
harge distribution p(-r t). Then
(~r 0 ; tr ) 0
V (~r ; t) =
d : (25)
0
40 j~r ~r j
Expand 1=j~r ~r 0 j and tr in powers of ~r 0 ,
1
r
z
Z
r’
dτ'
y
x
using similar approximations as before.
Minor dierence: here we have no ! . Previously we assumed that
d or equivalently ! c=d. Here we need to assume that
 _ j c=d, with similar bounds on higher time derivatives.
j=
Image by MIT OpenCourseWare.
All diagrams are from D.J. GriÆths, Introdution to Eletrodynamis, 3rd Edition
Alan Guth
Massahusetts Institute of Tehnology
8.07 Leture 37, Deember 12, 2012
-1 -
We fnd
"
#
Q ^r ~p(t0 ) ^r ~p_ (t0 )
V (~r ; t) '
+
+
;
2
40 r
r
rc
1
(26)
where to is the retarded time at the origin. Final result:
~E (~r ; t) ' 0 [(^r ~p)^r ~p ]
4r
~B (~r ; t) ' 0 [^r ~p ] :
4rc
(27)
Although this looks diferent, it is really the same as what we
had for the simple eletri dipole, hanging to vetor notation and
replaing -w 2-Po by -Pp.
Alan Guth
Massahusetts Institute of Tehnology
8.07 Leture 37, Deember 12, 2012
-15-
The electric dipole radiation formula is really the rst term in a
doubly innite series. There is electric dipole, quadrupole, : : :
radiation, and also magnetic dipole, quadrupole, : : : radiation.
Alan Guth
Massahusetts Institute of Tehnology
8.07 Leture 37, Deember 12, 2012
-16-
Radiation by Point Charges
Reall the felds (found by diferentiating the Lieenard-Wiehert
potentials):
~ (~
E
r ; t) =
j~r
40 (~
u (~
r
q
~
rp
j
~
r p ))
2
3 (c
2
vp )~
u + (~
r
~
r p)
(~u ~ap )
;
����
where
~u = c ^
~vp ;
~ = ~r ~r p :
(29)
If -vp = 0 (at the retarded time), then
q
~Erad =
[^
2
0
40 c j~r ~r j
( ^ ~ap)] :
(30)
Alan Guth
Massahusetts Institute of Tehnology
8.07 Leture 37, Deember 12, 2012
-17-
Poynting Vetor (partile at rest):
~Srad = 1 jE
~ rad j2 ^ =
0 c
2 2
2
0 q a sin ^
:
2
2
16 c
(31)
Total Power (Larmor formula):
0 q 2 a2
P=
:
6c
(32)
Alan Guth
Massahusetts Institute of Tehnology
8.07 Leture 37, Deember 12, 2012
-18-
Total Power (Larmor formula):
0 q 2 a2
P=
:
6c
(32)
6
Lie
enard's Generalization if v p - 0:
0 q 2 6 2
P=
a
6c
~v
c
2 !
~a =
0 q 2 dp dp
6m20c d d
|
{z
}
:
(33)
For relativists only
P = rate at which energy that is destined to become
leaving the particle.
Alan Guth
Massahusetts Institute of Tehnology
8.07 Leture 37, Deember 12, 2012
-19-
Radiation Reaction
If an accelerating particle radiates energy, it must lose energy |
obviously! But it is not clear exactly how. Point charges and
conservation laws cannot be combined rigorously.
For a nonrelativistic particle, radiation power is given by the
Larmor formula:
0 q 2 a2
P=
:
6c
Maybe then,
~Frad ~v =?
0 q 2 a2
:
6c
(34)
But not necessarily, since PLarmor represents the power destined to
become radiation. There can be other energy exchanges with
the near elds. Since the RHS of Eq. (34) does not depend on
~v , there is no way to match the two sides.
-20-
For yli motion, the total energy loss over one yle should math
the energy loss desribed by the Larmor formula. If a yle extends
from t1 to t2 then
Z t2
t1
a2 dt =
Z
d~v d~v
dt = ~v dt dt
|
t2
d~v dt t1
{z
}
=0
Z t2
d2~v
~v dt : (35)
2
t1 dt
So, energy onservation holds if
Z t2 t1
whih will hold if
F~ rad
2
0 q _
~a ~v dt = 0 ;
6c
2
q
0
F~ rad =
~a_ :
6c
(36)
(37)
Alan Guth
Massahusetts Institute of Tehnology
8.07 Leture 37, Deember 12, 2012
-21-
2
q
0
F~ rad =
~a_
6c
(37)
is the Abraham-Lorentz formula for the radiation reation fore.
We have not proven it, but we have shown that it provides a fore
that is onsistent with onservation of energy, at least for periodi
motion.
The Abraham-Lorentz formula an be "derived" by modeling
the point partile as an extended objet, so that one an alulate
the fore that one part of the objet exerts on the other parts.
One then takes a limit in whih the size of the extended objet
approahes zero. But the result depends on the details of the
model, so the result remains inonlusive.
Alan Guth
Massahusetts Institute of Tehnology
8.07 Leture 37, Deember 12, 2012
-22-
The Abraham-Lorentz formula gives reasonable preditions for
the radiation reation fore under many irumstanes, but it also
has an important pathology: runaway solutions. Note that
2
q
0
F~ rad =
~a_ = m~a
6c
(3�)
a(t) = a0 et= ;
(39)
0 q 2
=
6 10 24 s
6mc
(40)
has the solution
where
for an eletron.
Alan Guth
Massahusetts Institute of Tehnology
8.07 Leture 37, Deember 12, 2012
-23-
In quantum electrodynamics the situation is at least under much
better control. Innities arise, but they can be removed from the
theory by a technique called renormalization. This process can
be proven to succeed order by order in perturbation theory (an
expansion in powers of the = e2 =(40 h c) 1=137), but the
perturbation expansion is not believed to converge. Nothing can
be proven rigorously, but there are reasons to believe that the
theory is not completely well-dened, but that it nonetheless gives
a consistent description up to extraordinarily high energies, well
beyond the Planck mass.
Alan Guth
Massahusetts Institute of Tehnology
8.07 Leture 37, Deember 12, 2012
-2 -
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