Download Lecture Notes 14: Electromagnetic Radiation from An Arbitrary Source, Radiation Reaction on a Moving Point Charge

UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede
LECTURE NOTES 14
EM RADIATION FROM AN ARBITRARY SOURCE:
We now apply the formalism/methodology that we have developed in the previous lectures
on low-order multipole EM radiation {E(1), M(1), E(2), M(2)} to an arbitrary configuration of
electric charges and currents, only restricting these to be localized charge and current
distributions, contained within a finite volume v near the origin:
  
r  r  r   tr 
 
r  r  r   tr 
 
r  r 2  r 2  t r   2r  r   t r 
t  t  tr  r c
tr  t  r c

For arbitrary, localized {total} electric charge and current density distributions tot  r , tr 
 
and J tot  r , tr  , the retarded scalar and vector potentials, respectively are:
1

Vr  r , t  
4 o
 

Ar  r , t   o
4


tot  r , tr 
r
v

 
J tot  r , tr 
r
v
d  
1
4 o

d   o
4

tot  r , t  r c 

r
v

 
J tot  r , t  r c 
v
r
d  with tr  t  r c
 
d  and r  r 2  r 2  tr   2r r   tr 

For EM radiation, we assume that the observation / field point r is far away from the localized
  r or: rmax
 r  1.
source charge / current distribution, such that: rmax
2
 
 
 r   tr   2r  r   tr 
2r  r   tr 
 r 
 r 1
Then keeping only up to terms linear in   : r  r 1  
 
2
r
r
r2
r


But:
And:
1   1   for  1
1
2


 

 r  r   tr  
 rˆr   tr  
r  r 1 
  r 1 

r2 
r 


 

1
1
1
1  r r   tr   1  rˆr   tr  

 1  for:  1
 1 
 
  1 
 using:
2
r r  r r   tr   r 
1 
r
r
r 

1 

r2 

1
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
1
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede
 



r
r  r r   tr   
r rˆr   tr  




Now: tot  r , tr   tot  r , t    tot  r , t  1 
   tot  r , t  

2
c
c
r
c
c 







Expand tot  r , tr  as a Taylor series in the present time t about the retarded time, at the origin { r   0 }:
Defining the retarded time at the origin: to  t  r c {valid in the “far-zone limit}
Then:

 2
 3
rˆr   1
rˆr   1
rˆr  








tot  r , tr   tot  r , to    tot  r , to  
tot  r , to  
  tot  r , to  
  
  ...
 c  2!
 c  3!
 c 

d tot  r , to 

Where:  tot  r , to  
etc.
dtr
We can drop / neglect all higher-order terms beyond the  tot term, provided that:
 
rmax
c
c
,
  
 
1
,
2
c

 
1
3
, ... is satisfied…
For a harmonically oscillating system (i.e. one with angular frequency ω), each of these ratios
c
c
c
c
 d 
 
e.g.
, etc. is =
and thus we have: rmax
if rmax  d  , then 
 1,



 
 c 
 r 
or equivalently{here}:  max   1 .
 c 
r   


rmax
 rmax
 1 etc…
 1 , or more generally: max
 1 and
c
r
c
amount to keeping only the first-order {the lowest-order, non-negligible} terms in r  .

The retarded scalar potential Vr  r , t  then becomes:
The two approximations
1

Vr  r , t  
4 o

v

tot  r , tr 
r
d  
1
4 o

v
1
r

tot  r , tr  d 



 rˆr   to  
1  rˆr   to   








r
t
r
t
1
,
,
...








 d 




o
tot
o
 tot
r
4 o v r 

 c 


1 



 rˆ  
 rˆ  
tot  r , to  d       r   to  tot  r , t0  d       r   to  tot  r , to  d   ...




v
v
v
4 o r 
c
r

1


 Qtot  to 
 ptot  to 
 ptot  to 
 




1 
rˆ  
rˆ  d 






Vr  r , t  
tot  r , to  d       r   to  tot  r , to  d       r   to  tot  r , to  d   ...



v
4 o r
r v
 c  dt v


2
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
UIUC Physics 436 EM Fields & Sources II

Or: Vr  r , t  
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede



1  Qtot  to  rˆ ptot  to  rˆ p tot  to 


 ... 

2
r
cr
4 o  r

In the static limit: monopole dipole
term
term
vanishes in the
static limit
The retarded vector potential, to first order in r   r  r  {with to  t  r c } then becomes:
 

Ar  r , t   o
4

v
 
J tot  r , tr 
r

d   o
4

 
J tot  r , t  r c 
v
r
d  
 
o


J
tot  r , to  d
4 r v
Griffiths Problem 5.7 (p. 214) showed that for localized electric charge / current distributions
contained in the source volume v , that:
 
 
dptot  r , t   


v J tot  r , tr  d  dt  ptot  r , t 
Thus:

 
 o  p tot  to 
Ar  r , t   

 4  r

Note that ptot  to  is already first order in r   any additional refinements are therefore
second order in r  ; thus, the higher-order terms can be neglected/ignored (here).


Next, we calculate the retarded E and B fields. Since we are only interested in the EM
radiation fields (in the “far-zone” limit), we drop / neglect 1 r 2 , 1 r 3 , 1 r 4 , etc. terms, and keep
only the 1 r radiation-field terms.
Note that the radiation terms come entirely from those terms in the Taylor series expansions for
 
 


tot  r , to  and J tot  r , to  in which we differentiate the argument to of tot  r , to  , J tot  r , to  .
Since retarded time: to  t  r c



1
1
then: to   r but: r  rˆ  to   rˆ
c
c




  1 rˆ p tot  to  
ptot  to   
1  rˆ 
1  rˆ ptot  to  

rˆ
Thus: Vr  r , t    


 to  
cr
cr
r
4 o c 2
 4 o
 4 o 

 

Ar  r , t   o  
ptot  to 

And:

t
 4  r
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
3
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede
The retarded electric field for EM radiation in the “far-zone” limit is:
 


 

Ar  r , t 
o ptot  to 
1  rˆ ptot  to  


Er  r , t   Vr  r , t  
rˆ 
t
4 o c 2
4
r
r
but:
1
  o o
c2
 
o  



rˆ ptot  to  rˆ  
ptot  to    o  rˆ  rˆ  
ptot  to   {using the BAC-CAB rule}
 Er  r , t  




4 r
4 r





where the second time-derivative of the total electric dipole moment 
ptot  to  is evaluated at the



retarded time to  t  r c and computed from the origin,  { r   0 }: 
ptot  to   
ptot  0, t  r c  .
The retarded magnetic field for EM radiation in the “far-zone” limit is:

 
  
 o   p tot  to   o    
Br  r , t     Ar  r , t   


   p  to  
r
 4 
 4 r  



  
ptot  to     o  rˆ  
ptot  to  
  o  to  
4 rc
 4  
 


Where in first step we have used the relation   v  tr   a  tr   tr {see “term (3)” P436 Lect.
Notes 12 p. 11 and/or Griffiths Equation 10.55, p. 436} and in the last step on the RHS we have

1
{again} used the relation to   rˆ .
c
 
o

 rˆ  
ptot  to  
 Br  r , t   
4 rc

where the second time-derivative of the total electric dipole moment 
ptot  to  is evaluated at the



retarded time to  t  r c and computed from the origin,  { r   0 }: 
ptot  to   
ptot  0, t  r c  .

If we use spherical-polar coordinates, with the ẑ -axis  
pTot  to  , then noting that:

rˆ  
ptot  to   
ptot  to   rˆ  zˆ 

but:
zˆ  cos  rˆ  sin ˆ

ptot  to   rˆ  cos  rˆ  sin ˆ 
= 


ptot  to  sin ˆ
=  
rˆ  rˆ  0
r̂  ˆ  ˆ
ˆ  ˆ  r̂
ˆ  r̂  ˆ
4
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Thus:

 p  t   sin 
Er  r ,  , t   o tot o 
4
 r
And:

 p  t   sin  
Br  r ,  , t   o tot o 
 ˆ
4 c  r 
ˆ


Lect. Notes 14
Prof. Steven Errede
← r̂   ˆ   ˆ




1
and we also see that {again} Br  r , , t   rˆ  Er  r , , t  , Br  Er  rˆ  kˆ
c
 
The instantaneous retarded EM radiation energy density u  r ,  , t  in the “far-zone” limit is:

1
1 2
urrad  r ,  , t     o Er2  r ,  , t  
Br  r ,  , t  
2
0

2 2
p  to   sin 2   1 o2 
p 2  to   sin 2 
1   o o 
 
 2 
2 2 
2
2  16 2
 r  o 16 c  r
p 2  t0  o 
p 2  to    sin 2 
1  o 
 


2  16 2 c 2
16 2 c 2   r 2
rad
Thus: ur  r ,  , t  

1
  but:  o 
o c 2

p 2  to   sin 2  
 o 


2 2 
2
 16 c  r 
o p 2  to   sin 2  

 (Joules/m3)
16 2 c 2  r 2 
The instantaneous retarded Poynting’s vector in the “far-zone” limit is:

1 
 Watts 
S rrad  r ,  , t  
Er  r ,  , t   Br  r ,  , t  
2 
o
 m 
 rad
o p 2  to   sin 2 
Sr  r , , t  

16 2 c  r 2
o p 2  to   sin 2 
 ˆ
ˆ
   

16 2 c  r 2
 


 rˆ

c  crˆ

 rad
 rˆ  cur  r ,  , t  with
rˆ  kˆ

The instantaneous retarded EM power radiated per unit solid angle in the “far-zone” limit is:
dPrrad  r , , t   rad
 p 2  t 
 S  r , , t r 2 rˆ  o 2 o sin 2 
16 c
d
 Watts 


 steradian 
The instantaneous retarded total EM power radiated into 4 steradians, with vector area element

da  r 2 sin  d d rˆ  r 2 d  rˆ in the “far-zone” limit is:

 p 2  t    2   2
o p 2  to 

sin
sin

Prrad  t    Srrad  r , , t da  o 2 o 


d

d

(Watts)
S
16 c  0  0
6 2 c
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
5
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede
The instantaneous retarded EM radiation linear momentum density in the “far-zone” limit is:
o p 2  to   sin 2 
1  rad
  r , , t   2 Sr  r , , t  

16 2 c 3  r 2
c
rad
r
  kg 
 rˆ  2

  m sec 
The instantaneous retarded EM radiation angular momentum density in the “far-zone” limit is:
 rad
 
r  r , , t   r rrad  r , , t   0
The scalar EM wave characteristic radiation impedance of the antenna associated with this
lowest-order EM radiation is:


Er
Er
o
 o c 
 Z o  120   377 
Z rad   

1

Hr
o
Br
o
The scalar EM wave radiation resistance of the antenna associated with this lowest-order EM
radiation is:
Rrad
p 2  to 
 rad o 
 2 
 Ohms 
Io
6 cI o2
Note that in the above, we deliberately/consciously neglected the electric monopole {E(0)} term
 r:
in the retarded scalar potential for “far-zone” limit, rmax

VrE(0)  r , t  
1 1
1 Qtot  t0 

  r , to  d  


v
r
4 o r
4 o
As mentioned previously (P436 Lect. Notes 13, p. 4), that because of electric charge
conservation, a spherically-symmetric electric monopole moment cannot radiate transverselypolarized EM waves – spherical symmetry of the monopole moment restricts oscillations only to
the radial direction – thus one could get radiation of one polarization from a certain dΩ solid
angle element, but then radiation from other dΩ’s on the sphere also contribute, such that the net
EM radiation from the entire sphere = 0 – total destructive interference. (Gauss’ Law  
encl
 E da  Qtot  o independent of the size of the spherically symmetric charge distribution
S
enclosed by the surface S´.




Note also that for free-space EM radiation, B must be  to E , and with both E and B  to kˆ ,
the propagation direction. How do you do this for a spherically-symmetric source, where kˆ  rˆ ?
Note also that if electric charge were not conserved, then we would get a retarded electric

1 Q  to 

rˆ  n.b. this says nothing
monopole field proportional to 1 r : ErE(0)  r , t  
4 o c r
about the physical size of the spherically-symmetric charge distribution.
6
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede
Contrast the behavior of transverse waves associated with EM radiation from a sphericallysymmetric source (an oscillating electric monopole moment) (  no net EM radiation) to that of
longitudinal sound waves / acoustic waves radiated from a spherically symmetric oscillating
acoustic monopole sound source – e.g. a radially inward / outward oscillating sphere (a breathing
bubble) – the latter of which very definitely can propagate / create sound precisely because
sound waves are longitudinal, not transverse waves!!
Now think about the electron – for EM radiation fields, electric dipole / quadrupole / etc.
higher EM moments break the rotational invariance / rotational symmetry associated with the
spherical monopole electric charge distribution of the source – thus transverse EM waves
(EM radiation) can couple to such electric monopole {E(0)} sources – and also ones that
lack rotational invariance!!!
 

In the above Taylor series expansions for tot  r , tr  and J tot  r , tr  , we only kept terms to
first-order in r´ in these expansions and then demonstrated that the first-order “far-zone” limit
radiation terms were associated with the electric dipole moment {E(1)} .
  r the instantaneous retarded
For E(1) electric dipole EM radiation to first-order in r´ for rmax
scalar and vector potentials, electric and magnetic fields are:

1  rˆ p  to  
Vr


4 o  cr 

  (1) 
o  p  to  
Ar  r , t  


4  r 
 (1)

r,t  

  (1) 
o  rˆ  rˆ  p  to 

Er  r , t  
r
4 


  (1) 
o  rˆ  p  to  
Br  r , t   


r
4 c 



n.b. proportional to p
 t0  (first time

derivative of p  to  - “velocity”)
 



n.b. proportional to 
p  t0  (second time

derivative of p  to  - “acceleration”)
Suppose the (localized) charge / current distributions are such that there is no (time-varying)
 
 
 
p  r , tr   0 .
E(1) electric dipole moment, p  r , tr   0 and/or: p  r , tr   0 , 
 

Then the Taylor series expansion of tot  r , tr  and J tot  r , tr  to first order in r  would give
nothing for potentials and fields associated with “far-zone” EM radiation. However, higherorder terms in these expansions might give rise to non-vanishing potentials and fields.
The second order terms in r  correspond to M(1) magnetic dipole and E(2) electric
quadrupole EM radiation terms – in order to see/verify this, the second-order contribution needs
to be / can be separated out into M(1) and E(2) terms.
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
7
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede
Indeed, if we compare e.g. the ratio of EM power radiated for M(1) magnetic dipole vs. E(2)
electric quadrupole radiation (in the “far-zone” limit):
mo   b 2 I o   b 2 q
 o mo2 4 


12 c3 
I  q
 
where: o
2
6
e
 oQzz  
Qzze  qdd   2b 2 q

3 

d  b
 60 c 
 rad
M (1)
 rad(2)
Thus:
 rad
M (1)
 rad(2)
  o  2b 4  6 q 2 


3

 5 2b 4
12

c
5 1


 2    1
4
d

2

2 4
6 
 o q d  
 60  c3 


5


 

Similarly, the third order terms in r  in the Taylor series expansion of tot  r , tr  and J tot  r , tr 
correspond to M(2) magnetic quadrupole and E(3) electric octupole radiation terms –
i.e. the third-order contribution needs to be / can be separated out into M(2) and E(3) terms!
 

Similarly, the fourth order terms in r  in the Taylor series expansion of tot  r , tr  and J tot  r , tr 
correspond to M(3) magnetic octupole and E(4) electric sextupole radiation terms –
i.e. the fourth-order contribution can be separated out into M(3) and E(4) terms!

And so on, for each successive higher-order term r  in the Taylor series expansion of tot  r , tr 
 
and/or J tot  r , tr  !!!
Griffiths Example 11.2:
a.) An oscillating (i.e. harmonically varying) electric dipole has time-dependent dipole moment:

p  tr   po cos tr  where: p  tr   p  tr  zˆ  po cos tr  zˆ
p  tr  
dp  tr 
  po sin tr 
dtr

p  tr  
dp  tr  d 2 p  tr 

  2 po cos tr 
dtr
dtr2
zˆ  cos  rˆ  sin ˆ with: to  t  r c
Then:
Vr
8
 (1)

r,t  

 po
1  rˆ p  to    po  rˆ zˆ 
sin to  
cos  sin to 




4 o  cr  4 o  cr 
4 o cr
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
UIUC Physics 436 EM Fields & Sources II
And:
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede

  (1) 
o  p  to  
o po
zˆ  cos  rˆ  sin ˆ
sin to  zˆ
Ar  r , t  


4  r 
4 r

  (1) 
o  rˆ  rˆ  p  to   o  rˆ   rˆ  zˆ  
2


r  r , t  

   po  cos to 
4 
r
r
 4 




  (1) 
o  rˆ  p  to  
o  rˆ  zˆ 
Br  r , t   
 2 po  cos to 





4 c 
4 c  r 
r



But:
 rˆ  zˆ   rˆ   cos  rˆ  sin ˆ    sin   rˆ  ˆ    sin ˆ
And:
rˆ   rˆ  zˆ    rˆ  ˆ     sin    ˆ   sin     sin ˆ
Thus:
p   cos 

Vr (1)  r , t   o 
4 o c  r
   r 
 sin   t   
   c 

 p   1    r 

Ar (1)  r , t    o o   sin   t    zˆ
4  r    c  
with:
to  t  r c
where:
zˆ  cos  rˆ  sin ˆ
  (1) 
o po 2  sin  
  r  ˆ
Er  r , t   

 cos   t    
4  r 
  c 

 p  2  sin  
  r 

Br (1)  r , t    o o 
 cos   t    ˆ
4 c  r 
  c 
Compare these results for the E(1) electric dipole EM radiation “far-zone” limit case with
those we obtained P436 Lecture Notes 13 {see pages 8-11}, and/or P436 Lecture Notes 13.5
{the E(1)/M(1) summary / comparison page 11} – they are (of course) identical!
b.) A single, point electric charge q can have (by definition) an electric dipole moment



p  tr   qd  tr  where d  tr  is the position vector of the point electric charge q at the retarded
time tr with respect to the (local) origin  . (n.b. subject to all the caveats r.e. choice of origin
for an EDM having a net charge – see P435 Lecture Notes. . . )


d d  tr 
dp  tr 


p  tr  
q
 qv  tr 
However:
n.b. these two quantities do not
dtr
dtr
depend
on the choice of origin !!!


dp  tr 
dv  tr 


p  tr  
q
 qa  tr 
And:
dtr
dtr



v  tr  = velocity vector of point electric charge q at the retarded time tr

a  tr  = acceleration vector of point electric charge q at the retarded time tr
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
9
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede
Everything goes through as before – get the same retarded scalar and vector potentials, same



retarded E and B fields, same u, S , P, etc.
In particular, the radiated EM E(1) power associated with a moving point charge q is:
o p 2  to 
Pq 
(Watts) But:
6 c
 Pq 

p  to   qa  to 
 o q 2 a 2  to 
 Famous Larmor formula (EM power radiated from a point charge q)
6 c
Note that the E(1) EM power radiated by a point charge q is proportional to the square of the
acceleration a and also is proportional to the square of the electric charge q.
This is the origin of statement: “Whenever one accelerates an electric charge q, it radiates
away EM energy in the form of (real) photons”. It is the E(1) electric dipole term which
dominates this radiation process.
n.b. This is also true for decelerating charged particles – the time-reversed situation!!!

Pq ~ a2 ← doesn’t care about sign of a {The EM interaction is time-reversal invariant}!!!
Radiation from accelerated / decelerated +q vs. –q charges is the same if  q  q .
(Pq doesn’t care about the sign of q!)
But:
Pq ~ q2 → so if double q → then Pq increases by factor of 4!
 For the same acceleration/deceleration, high-Z nuclei radiate EM energy {in the form of
photons} much more than e.g. a proton (= hydrogen nucleus) – process is known as
bremsstrahlung {= “braking radiation”, auf Deutsch}.
e.g. A fully-stripped uranium nucleus (Zu = 92) gives 922 = 8464 more EM radiation than a
proton for the same acceleration, a.
EM Power Radiated by a Moving Point Electric Charge:
The retarded electric field of an electric charge q in arbitrary motion is:
 
q
r  2 2    
Er  r , t  
 c  v  u  r  u  a 

4 o  r u 3 
The associated retarded magnetic field is:
 
 
1
Br  r , t   rˆ  Er  r , t 
c
 
As mentioned before, the first term in Er  r , t  ,
where:


u  crˆ  v  tr 
  
 
r  r  r   tr   r  w  tr 
 
r  r  r   tr   ct  c  t  tr 
or: tr  t  r c .
r  2 2 
c  v  u 

4 o  r u 3 
q
is known as the generalized Coulomb field, or velocity field.
10
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
UIUC Physics 436 EM Fields & Sources II
 
The second term in Er  r , t  ,
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede
r    
r   u  a   is known as the acceleration field

4 o  r u 3 
q
(a.k.a. the radiation field).
 
 
 
 
1  
1
Er  r , t   Br  r , t  where: Br  r , t   rˆ  Er  r , t 
The retarded Poynting’s vector is: S r  r , t  
c
o


  
  
  
Use the A  B  C  B AC  C A B rule:



 

 
 
  
1  
1  2 
Sr  r , t  
Er  r , t   rˆ  Er  r , t   
Er  r , t  rˆ  rˆ Er Er  r , t  
  c

o c 
o




However, note that not all of this EM energy flux constitutes EM radiation (real photons) –
 




some of it is still in the form of virtual photons, S r  r , t   S rvirt  r , t   S rrad  r , t 
The metaphor Griffiths uses, that of flies “attached” to a moving garbage truck, is a
reasonable picture to imagine here….

n.b. – In order to “detect” the total EM power radiated by a moving point charge q, we draw

a huge sphere of radius r centered on the position of the charged particle w  tr  at the
retarded time tr  t  r c and wait the appropriate time interval t  t  tr  r c for the EM
radiation radiated at the retarded time tr to arrive at the surface of the imaginary sphere.
Note that in the “far-zone” limit, the retarded time tr is the correct retarded time for all points
on the surface of the sphere S  .



 
Again, since the area of the sphere Asphere   r 2 ( ~ r 2 ), then any term in S r  r , t  that varies
 

as 1 r 2 will yield a finite answer for radiated EM power, Prad   S r  r , t da .
S
 
However, note that terms in S r  r , t  that vary as 1 r 3 , 1 r 4 , 1 r 5 … etc. will contribute
nothing to Prad in the limit r → ∞.
For this reason, only the acceleration fields represent true EM radiation (real photons) –
hence their other name, that of radiation fields:

  
q
r

 r   u  a  
Erad  r , t  

3 

4 o  r u 
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
11
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede
The EM velocity fields do indeed carry EM energy – as the charged particle moves through
space-time, this EM energy is dragged along with it – but it is not in the form of EM radiation.

  

Note that Erad  r , t  is  rˆ (due to the  r   u  a   term).


 The second term in S rad  r , t  vanishes:





1  2 
1 2 



S rad  r , t  
Erad  r , t  rˆ  rˆ Erad  r , t  Erad  r , t   
E  r , t  rˆ
 o c rad
o c 

Now if the point charge q happened to be {instantaneously} at rest ( v  tr   0 ) at the retarded
0


time tr, then: u  tr   crˆ  v  tr   crˆ {here}. Then in this case:

  
q
r
q
r 







Erad  r , t  
u
a
r
r  crˆ  a 




3 
3 


4 o  r u 
4 o r crˆ


1 
q

r  rˆ  a  
2 
4 o c r
4 o c 2

q
2



1

rˆ   rˆ  a  

r

o q  ˆ ˆ  
r  r a 
4 r 
1


since 2   o o 
c


 q
q 



 o  rˆa rˆ  rˆ rˆ a   o  rˆa rˆ  a 

4 r 
  4 r 


1



   
 

Then {here} in this case { v  tr   0 }:

 q2
1 2 

 2
S rad  r , t  
Erad  r , t  rˆ  o 2  a 2  rˆa  rˆ

o c
4 crˆ 
 


But: rˆa  a cos  where θ = opening angle between r̂ and acceleration a .

o q 2 a 2
o q 2 a 2  sin 2 

2
ˆ


1  cos   r 
 S rad  r , t  

4 cr 2 
4 c  r 2

 rˆ

Here again, we see that no power is radiated in the forward/backward directions (θ = 0 and

θ = π) – radiated power is maximum when    2  90o , i.e. when rˆ  a - get a donut-shaped

intensity pattern about the instantaneous acceleration vector a  tr  :
12
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede
The power radiated by this point charge (instantaneously at rest at time tr) is:

o q 2 a 2  tr  sin 2  2


r sin  d d
Prad  t    Srad  r , t da 
S
16 2 c  r 2


o q 2 a 2  tr 
 o q 2 a 2  t r  4 o q 2 a 2  tr 
3
d

2

sin

 



0
3
6 c
8

16  2 c
8
Prad  t  
4
2
3
o q 2 a 2  tr 
 Larmor power formula {again} !!!
6 c

This formula was derived assuming v  tr   0 , but in fact, we get the same formula as long as
v  tr   c (i.e. non-relativistic motion).






An exact treatment of v  tr   0 is (much) more difficult / tedious.
Note that in special relativity {inertial (non-accelerated) reference frames}, the choice

v  tr   0 merely represents a judicious choice of an (inertial) reference frame, with no
loss of generality.
If we can determine how Prad  t  transforms from one reference frame to another, then we


can deduce the more general v  tr   0 result (Liénard) from the (Larmor) v  tr   0
result. (See e.g. Griffiths problem 12.69, p. 545).




For the v  tr   0 case, Erad  r , t  is more complicated (than the v  tr   0 case).



For the v  tr   0 case, Srad  r , t  = the rate of energy passing through the (imaginary)


large-radius surface S  of the sphere, Srad  r , t  is NOT the same as the rate of energy
when it left the charged particle at the retarded time tr.
Consider the example of a person firing a stream of bullets (photons) out the window of a
moving car, parallel to the direction of motion of the car:
The rate at which the bullets strike a target, Rtgt (#/sec) is not the same as the rate of bullets
leaving the gun, Rgun (#/sec) because of the relative motion of the car with respect to the target.
This is again analogous to the Doppler effect. It is purely due to a motional geometrical factor
(i.e. it is not due to special relativity). For bullets moving parallel to the car’s velocity vector:
Rgun  1    tr   Rtgt
or:
Rtgt 
1
1    tr 
Rgun
where:   tr  
v  tr 
c
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
13
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede
Whereas for bullets moving anti-parallel to the car’s velocity vector:
Rgun  1    tr   Rtgt
or:
Rtgt 
1
1    tr 
Rgun

v  tr 
where:   tr  
c

For arbitrary directions, with rˆ  unit vector from car to target:



Rgun  1  rˆ   tr  Rtgt or:


v  tr 
1
Rtgt 
Rgun where:   tr  

c
1  rˆ   tr 


dW
= rate of energy passing through sphere of radius r then the rate at which energy leaves the
dt

dW dW dt dW tr  rˆu  dW
tr
c
rc






     with u  crˆ  v  tr  .
since:
charge q is:

t  c  dt
dt
dtr
dt dtr
t rˆu r u
(see P436 Lect. Notes 12, p. 14-15, and/or Griffiths problem 10.17, p. 441)
So if

rˆu
But:
c


rˆ crˆ  v  tr  
c


 1  rˆv  tr  c  1  rˆ   tr    = retardation factor




v  tr 
dW  rˆu  dW
dW
dW

 1  rˆ   tr 

Then:
where:   tr  

c
dtr  c  dt
dt
dt


Thus, the power radiated into a patch of area da  r 2 sin  d d  r 2 d  on the sphere S  , where
d   sin  d d = solid angle into which the EM power is radiated into area element da on the

q
r    

r   u  a   is given by:
surface of the sphere S  , with Erad  r , t  

4 o  r u 3 


 rˆu  tr   1 2 2
dPrad  tr   rˆu  tr  

2

S
r,t r  
E r
 c  rad  
 c   c rad
d



 o

2
q2
1  rˆu  tr   r 2  r 2  


r   u  t r   a  tr  

  
6
2 2
16  o o c  c   r u  t  
r

 
2
q2
r4
1  rˆu  tr  




u
t
a
t
r









r
r
16 2 o2 o c  c   r u  t  6
r

Thus:
14
q2
16 2 o


rˆ   u  tr   a  tr  
 rˆu  t  
2
5
r
2


2
ˆ
r

u
t

a
t






dPrad  t 
r
r
q

5

16 2 o
d
rˆu  tr 


© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede
dPrad  t 
S  d  d  over the sphere S  (i.e. over θ and φ angles) is a pain….
However, the result of this integration {again!} yields the famous Liénard formula:
Integrating
2



2
v  tr   a  tr   o q 2 6 2
o q 2 6  2


Prad  t  
  a  tr  
 a  tr     t r   a  tr 
 6 c
c
6 c 





Where:   tr   v  tr  c and:   tr  
0   1
1
1   2  tr 

= Lorentz factor.
1   
Note that the Liénard formula reduces to the Larmor formula for Prad  t  when v  c .
Note also that when v → c, the γ6 factor in the Liénard formula goes “berserk” – as the charged
particle travels closer and closer to the speed of light c, the more one tries to accelerate it (in
order to make it travel even closer to the speed of light, c), it radiates away more and more of the
(absorbed) energy as v → c!!!
 very high energy electron accelerators are problematic in this regard, because the electron is
so light, mass-wise, e.g. relative to the proton: me  0.511 MeV c 2 whereas m p  938.28 MeV c 2 .
Griffiths Example 11.3:


Suppose v  tr  and a  tr  are instantaneously collinear (i.e. parallel to each other). Find the angular
distribution of radiated power
dPrad  t 




when v  tr a  tr   v  tr  a  tr  (i.e. when v  tr   a  tr  )
d



Then in this case: u  tr   crˆ  v  tr 



because v  tr   a  tr 
0











Thus: u  tr   a  tr   crˆ  v  tr   a  tr   crˆ  a  tr   v  tr   a  tr   crˆ  a  tr 



2



dPrad  t 
q 2 rˆ   u  tr   a  tr  
q 2 c 2 rˆ  rˆ  a  tr 


Then:
5
5


16 2 o
16 2 o
d
rˆu  tr 
rˆu  tr 





2

v  tr 
  tr  
c






Work on denominator term: rˆu  tr   rˆ crˆ  v  tr   c  rˆv  tr   c 1  rˆ   tr    c   1  rˆ   tr 

Work on numerator term:
Thus:



1






rˆ  rˆ  a  tr   rˆa  tr  rˆ  rˆ rˆ a  tr   rˆa  tr  rˆ  a  tr 

 
  


rˆ   rˆ  a  tr    a 2  tr    rˆa  tr  
2


2
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
15
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015

Lect. Notes 14


Prof. Steven Errede

 a 2  t   rˆa  t  2 
 a 2  t   rˆa  t  2 
2
r
r
r
r
dPrad  t 
q c 
q
 



Then:


5
5
2
2
3
5
d
16  o c 1  rˆ   t 
16  o c
1  rˆ   tr 
r
2 2








If we let the ẑ -axis point along v  tr  – along   tr   v  tr  c {and hence also along a  tr  }:


and: rˆv  v cos  or: rˆ    cos  where θ = opening angle

between r̂ and acceleration a , as shown on page 12 above.

Then: rˆa  a cos 
1  cos 2  

dPrad  t  q 2 a 2  tr 

Thus:
16 2 o c 3 1    tr  cos  5
d

but:
1
  o o
c2
v  tr 
dPrad  t  o q 2 a 2  tr 
sin 2 

with:   tr  
5
2
c
d
16 c 1    tr  cos  
When β → 0:
 v 0 
dPrad  t  o q 2 a 2  tr  2

sin   Srad
 r , t  r 2 rˆ
2
16 c
d
When β → 1: The donut of EM radiation intensity is folded forward by the factor 1 1   cos   :
5
Note that there is still no radiation precisely in the forward direction, rather it is in a cone which
becomes increasingly narrow as β → 1, of half-angle:
 max 
1   
2
{see Griffiths problem 11.15, p. 465}
 
The total EM power radiated into 4 steradians by the point charge for v  a is:
dPrad  t 
o q 2 a 2  tr 
sin 2 
sin  d d
Prad  t   
d 
16 2 c  1    tr  cos  5
d

Let:
16
o q 2 a 2  tr   
sin 2 
 0 1    t  cos  5 sin  d
8 c
r
u = cosθ
θ = 0 → u = +1
du = sinθdθ θ = π → u = 1
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
UIUC Physics 436 EM Fields & Sources II
Then: Prad  t  
Prad  t  
Fall Semester, 2015
Lect. Notes 14
o q 2 a 2  tr  1 1  u 
1 1   u 5 du Integrate by parts:  vdu  uv   udv
8 c
2

3 
o q 2 a 2  tr   4
o q 2 a 2  tr  
1
2


t


1
 r  
 3 
 1   2  t  3 
8 c
6 c

r

But:

  tr  
Prof. Steven Errede
1
1 
2
with:
 tr 



  tr   v  tr  c
o q 2 a 2  tr  6
  tr 
Prad  t  
6 c
This is the same/identical result as obtained directly from the Liénard formula when


v  tr   a  tr  . It is also known as the classical formula for bremsstrahlung (“braking radiation”
in german).
Again, note that because Prad  t  ~ a 2  tr  , the EM power radiated doesn’t depend on the sign

of a  tr  – i.e. whether the charged particle is accelerating or decelerating.
2
2
Now it can also be shown that the Lorentz factor   E mc , where E   pc    mc 


= total relativistic energy associated with a charged particle moving with   tr   v  tr  c .
2
2
Thus, when v → c, for a given {high} total energy E, then  ~ 1 mc 2 and thus: Prad  t  ~ 1 m6 .
Comparing EM bremsstrahlung radiation from an accelerated electron { me  0.511 MeV c 2 }
vs. that of e.g. an accelerated muon { m  105.66 MeV c 2 }, for the same total energy E, an
electron will radiate  m me    206.8   7.8  1013 times more EM energy than a muon.
6
6
This explains why muons have such high penetrating power in traversing matter – they lose
relatively little energy via bremsstrahlung, whereas high-energy electrons radiate EM energy
“like crazy” in matter.
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
17
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede
The Radiation Reaction on a Radiating Charged Particle
According to the laws of classical electrodynamics, an accelerating electric charge radiates
electromagnetic energy in the form of real photons ( = quanta of the EM radiation field).
By conservation of energy, the EM radiation carries off / carries away energy – which must
come at the expense of the charged particle’s kinetic energy {since its rest mass cannot change}.
In other words, one puts in energy to accelerate the charged particle, but the charged particle
winds up being accelerated less than e.g. an electrically neutral particle {of the same rest mass of
the charged particle}, for the same amount of input energy!
The devil is in the microscopic details of precisely how this is accomplished in both cases.
At the microscopic level, an electrically charged particle of mass m is accelerated/increases its
2
{kinetic} energy T     1 mc by absorbing EM energy (either in the form of virtual or real
photons) from a source of EM field(s). In order to accelerate/increase the {kinetic} energy
T     1 mc 2 of an electrically neutral particle, it too must interact, at the microscopic level,
via one of the four fundamental forces of nature, with a source {of fields} associated with that
fundamental force.
Feynman/Space-Time Diagram for
Electron-Positron Scattering {QED}
ct {time}
e
e

e
e
x {space}
In the electromagnetic case, if an electrically charged particle is decelerated and radiates EM
energy away in the form of {real} photons, by energy conservation, the change in the kinetic
energy of the charged particle must equal the sum of the energies associated with each of the
n individual {real} photons radiated by the charged particle:
n
n
i 1
i 1
KEq   E i   hf i
18
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede

This implies that the radiation must {somehow!} exert a force, Frad back on the electrically
charged particle – i.e. a recoil force, analogous to that associated with firing a bullet from a gun.
Thus, linear momentum p must also conserved in this process. In the emission of EM radiation
{real photons}, linear momentum p i  h  i  hf i c is also carried away by each of the {real}

photons. This comes at the expense of the charged particle’s momentum pq and
(non-relativistically, for vq  c ): KEq  pq2 2m

E i
n
n
n
h


pqrecoil c   p i c   ckˆ i   hf i kˆ i
i 1
i 1

i
i 1
kˆ i
kˆ i = wave vector for the ith photon

B i
Thus, if a similarly accelerated/decelerated neutral particle doesn’t radiate force quanta {of
some kind} because it is accelerated/decelerated, then because the electrically-charged particle
does radiate EM quanta {real photons} in the acceleration/deceleration process, then we can see
that the final-state KEq  KEo for similarly accelerated/decelerated neutral particle of the same
mass m and initial/original kinetic energy as that of the electrically-charged particle.
The Radiation Reaction Force on a Charged Particle
For a non-relativistic particle ( vq  c ) the Larmor formula for the total instantaneous
EM radiated power is:
Prad  t  
o q 2 a 2  tr 
(Watts)
6 c
Conservation of energy would then imply that this radiated EM power = the instantaneous
rate at which the charged particle loses energy, due to the effect of the EM radiation back
reaction / recoil force Frad  tr  :
dW  tr  
o q 2 a 2  tr 

 Frad  tr v  tr   
Pq  tr  
(Watts)
dt
6 c
This relation / equation is actually wrong. Why???
The reason is, that we calculated the radiated EM power by integrating Poynting’s vector


S rad  r , t  for the EM radiation associated with the accelerating point charged particle over an
“infinite” sphere of radius r ; in this calculation the EM velocity fields played no role, since they
fall off too rapidly as a function of r to make any contribution to Prad  t  . However, the EM
velocity fields do carry energy – because the total retarded electric field associated with the
electrically charged particle is the sum of two terms – the EM velocity field and the EM
acceleration field terms:
 
 
 
Ertot  r , t   Erv  r , t   Era  r , t 
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
19
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede
The total retarded EM energy density associated with the total retarded electric field is:
 
  2
2


utot  r , t   12  o Ertot  r , t   12  o Erv  r , t   Era  r , t 
 
 
2
2


 12  o  Erv  r , t   2 Erv  r , t  Era  r , t   Era  r , t  



Energy stored in velocity
field only (virtual photons)
Generalized Coulomb
fields only

Cross term!!! Energy stored
in mixture of velocity and
acceleration field (both
virtual & real photons!!)
“Conversion” field
virtual → real photons
{and vice versa!}
Note that:
The Generalized Coulomb fields vary as ~ 1 r 4
The “Conversion” fields vary as ~ 1 r 3
The Radiation fields vary as ~ 1 r 2
Energy stored in
acceleration field only
(real photons)
Radiation fields only
Neither the Generalized Coulomb field
nor the “Conversion” field contribute to
EM radiation in the “far-zone” limit r   r
Clearly, the first two terms in the EM energy density formula associated with the electric field
have energy associated with them. However, this energy stays with the charged particle – it is
not radiated away.
As the charged particle accelerates / decelerates, energy is exchanged between the charged

particle and the velocity and acceleration fields. For the latter term (the last/ 3rd term in utot  r , t 
above), this energy is irretrievably carried away (by real photons) out to r = ∞.
dW  tr  
 q 2 a 2  tr 

 Frad  tr v  tr    o
only accounts for the last / 3rd term
6 c
dt
2


( Era  r , t  ) in utot  r , t  above.
Thus, Pq  tr  
If we want to know the total recoil force exerted by the EM velocity and the EM acceleration
fields on the point charge, then we need to know the total instantaneous power lost, not just the
radiation-only contribution.
Thus, in this sense, the term “radiation” (back)-reaction is a misnomer because it should more
appropriately be called an EM field (back)-reaction. Note further that this EM field (back)reaction is also intimately connected with the issue of the so-called “hidden” EM momentum.


Shortly, we’ll see that Frad  tr  is determined by the time derivative of the acceleration a  tr  ,

and can be non-zero even when the acceleration a  tr  is instantaneously zero!
(The charged particle is not radiating at that retarded instant in time!)
20
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede
By energy conservation, the energy lost by the electrically charged particle in a given
retarded time interval tr  tr2  tr1 tr2  tr1 must equal the energy carried away by the EM


radiation, plus whatever extra energy has been pumped into the EM velocity/generalized
Coulomb field.
If we consider time intervals tr  tr2  tr1 such that “the system” (consisting of the pointcharged particle q and the EM velocity field – see drawing on following page) returns to its
initial state, then (assuming that the energy in the EM velocity fields is the same at time tr2 as at
time tr1 , then the only net energy loss is in the form of EM radiation (due to the emission of
n real photons).
dW  tr  
 q 2 a 2  tr 

 Frad  tr v  tr    o
is incorrect,
dt
6 c
by suitably averaging this relation over a finite time interval, it is valid, with the restriction that
state of “the system” is identical at the retarded times tr1 and tr2 :
Thus, while instantaneously Pq  tr  
1
tr

tr  tr2
tr tr1

1 o q 2

Frad  tr v  tr  dtr  
tr 6 c

tr  tr2
tr tr1
a 2  tr  dtr
For the case of periodic/harmonic motion, this means that the above integrals must be carried
out over at least one (or more) complete / full cycles, tr  tr2  tr1  n  , n = 1, 2, 3, . . .
For non-periodic motion, the condition that “the system” be identical at times tr1 and tr2 is
more difficult to achieve – it is not enough that the instantaneous velocities and accelerations be
equal at tr1 and tr2 , since the (retarded) fields farther out (at the present time t  tr  r c ) depend


on v  tr  and a  tr  at the earlier retarded time tr !!!
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
21
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede
For non-periodic motion, the condition that “the system” be identical at times tr1 and tr2




technically requires that not only v tr1  v tr2 and a tr1  a tr2 , but all higher derivatives of

v  tr  must also likewise be equal at times tr1 and tr2 !!!
   
 
 
However, in practice, for non-periodic motion, since the EM velocity fields fall off rapidly with




r , it is sufficient that v tr1  v tr2 and a tr1  a tr2 , for a brief time interval, tr  tr2  tr1 .
   
 
 
The RHS of the above equation can be integrated by parts:

tr  tr2
a  tr  dtr  
2
tr tr1
tr  tr2
tr tr1
tr



2
tr  tr2 d v  t   
dv  tr   2
 dv  tr    dv  tr  

r
v  tr  dtr


 dtr   v  tr 
  tr tr
2
dtr  t
dtr
1
 dtr   dtr 




r1

 a  tr 




Because of the restriction on v tr1  v tr2 and a tr1  a tr2 at the time endpoints tr1 and tr2 ,
   
 
 
tr


dv  tr   2 
tr2

The term:  v  tr 
  v  tr a  tr  tr  0
1
dtr  t

r1
Thus:
Or:
 
tr  tr2
tr tr1

tr  tr2
tr tr1

o q 2




Frad  tr v  tr  dtr  
6 c


tr  tr2
tr  tr1
 a  t  v  t   dt 
r
r
r


o q 2 

a  tr  v  tr  dtr  0
 Frad  tr  
6 c


Mathematically, there are lots of ways this integral equation can be satisfied, but it will
certainly be satisfied if:

o q 2 

Frad  tr  
a  tr   Abraham-Lorentz formula
6 c
This relation is known as the Abraham-Lorentz formula for the EM “radiation reaction” force.

 q2 
Frad  tr   o a  tr  is the simplest possible form the EM radiation reaction force can take.
6 c
Physically, note that this formula tells us only about the time-averaged force {albeit} over a

very brief time interval tr  tr2  tr1 , of the force component parallel to v  tr  - because of the




original term Frad  tr v  tr  . As such, it tells us nothing about Frad  tr   to v  tr  .


n.b. These averages are also restricted to time intervals such that tr  tr2  tr1 is chosen to




ensure that v tr1  v tr2 and a tr1  a tr2 .
   
22
 
 
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede

o q 2 

F
t
a  tr  also has disturbing,
The Abraham-Lorentz radiation reaction force rad  r  
6 c
seemingly unphysical implications that are still not fully understood today, despite the passage of
nearly a century!
Suppose a charged particle is subject to NO external forces. Then Newton’s 2nd law says that:

 q2 

Frad  tr   o a  tr   ma  tr  where m = (real) rest mass of the charged particle.
6 c
Then:

o q 2 
o q 2 







Frad  tr  
ma  tr   m a  tr   ma  tr  or:
a  tr    a  tr   a  tr 
6 mc
6 mc
The solution to this linear, first-order homogeneous differential equation is: a  tr   ao e tr 
  q2 
where ao = acceleration at the retarded zero of time, tr = 0, and    o  , which for the
 6 mc 
electron is a time constant of :  e  6 1024 sec .
If ao ≠ 0, the acceleration exponentially increases
(+ve, if ao > 0, ve, if ao < 0) as time progresses!
This is a runaway solution, which is CRAZY !!!
This can only be avoided if ao  0.
However, if the runaway solutions are excluded
on physical grounds, then the charged particle
develops an acausal behavior – e.g. if an external
force is applied, the charged particle responds before
the force acts!! This acausal “pre-acceleration”
“jumps the gun” by only a short time
 e  6 1024 sec, and since we know that quantum
mechanics and uncertainty principle are operative on short distance/short timescales, perhaps this
classical behavior shouldn’t be too unsettling to us. Nevertheless, to many it is…. (see Griffiths
Problem 11.19, p. 469 for more aspects/ramifications of the Abraham-Lorentz formula…)
Such difficulties also persist in the fully-relativistic version of the Abraham-Lorentz equation.
Griffiths Example 11.4 – EM Radiation Damping:
Calculate the EM radiation damping of an electrically charged particle attached to a spring of
natural angular frequency ωo with driving frequency = ω
The 1-dimensional equation of motion is:
mx tr   Fspring  tr  Frad  tr  Fdriving  tr 
  mo2 x  tr   m 
x  tr   Fdriving  tr 
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
23
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede
With the system oscillating at the driving frequency ω:
Instantaneous position:
x  tr   xo cos tr   
Instantaneous velocity:
x  tr    xo sin tr   
Instantaneous acceleration:

x  tr    2 xo cos tr   

x  tr    3 xo sin tr      2   xo sin tr    


Instantaneous jerk:
 x  tr 
Thus:

x  tr    x  tr 
2
Thus: mx tr    m 2 x  tr   mo2 x  tr   Fdriving  tr 
Define the damping constant:    2 (SI units: 1 sec )
Then: mx tr   m x  tr   mo2 x  tr   Fdriving  tr  ← 2nd-order linear inhomogeneous diff. eqn.
n.b. In this situation, the EM radiation damping is proportional to v tr  . Compare this to
e.g. “normal” mechanical damping, which is proportional to v  tr  (e.g. friction / dissipation).
The Physical Basis of the Radiation Reaction

 q2 
We derived the Abraham-Lorentz EM radiation reaction force Frad  tr   o a  tr  from
6 c
consideration of conservation of energy in the EM radiation process, from what was observable
in the far-field region, r → ∞.
Classically, if one tries to determine this radiation reaction force at the radiating point charge,
we run into mathematical difficulties due to the mathematical point-behavior of the electric
charge (e.g. at its origin) where the (static) electric field and corresponding scalar potential
become singular, this problem correspondingly has infinite energy density at the point charge.
This singular nature is also the present for the retarded EM fields associated with a moving point
charge:
 
q
r  2 2    
Er  r , t  
 c  v  u  r  u  a 

4 o  r u 3 
 
 
1


Br  r , t   rˆ  Er  r , t  with: u  crˆ  v
c
Today, we know that quantum mechanics is operative, e.g. from the Heisenberg uncertainty
principle on {e.g. 1-dimensional} distance scales of:
xpx   where:   h 2 = Planck’s Constant / 2π, h  6.626 1034 J-sec
Then:
24
x   px but:
px c  me c 2 {for electrons}
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
UIUC Physics 436 EM Fields & Sources II
Note that:
 x 
hc = 1240 eV-nm
and:
Fall Semester, 2015
(1 fm = 1015 m)
e
c

 386 fm = reduced Compton wavelength of the electron
2 me c 2
and: e 
Thus: x   e 
Prof. Steven Errede
1 nm = 109 m
c
1240 eV-nm / 2
 0.386 1012 m  386 fm

2
0.511 meV
me c
The quantity  e 
Lect. Notes 14
hc
 2427 fm = Compton wavelength of electron
me c 2
c
 386 fm  386 1015 m
2
me c
For short distance scales of order x   e  c me c 2  386 fm {and less} the behavior of an
electron will be manifestly quantum mechanical in nature. Thus, we should not be surprised that
when extrapolating classical EM theory into this short-distance regime, we obtain erroneous answers
– we have no reason to expect classical theory to {continue to} hold in the quantum domain !!!
Ve  r   
q 1
4 o r
Similarly, we have no business extrapolating quantum mechanics to distance scales less than:
reBH 
2GN me 2  6.673 1011 m3 kg 1s 1  9.109 1031 kg
 1.35 1057 m =

2
8
c2
 3 10 m / s 
Schwartzschild radius of
electron (event horizon)
Where GN = Newton’s gravitational constant. The electron is a black hole at this distance scale –
the Schwartzschild radius/event horizon of an electron is where space & time interchange roles!
However, long before this regime is reached, at distance scales corresponding to the Planck
energy/Planck mass m p c 2  c 3 GN  2.2 108 kg  1.2 1019 GeV  1.2 1028 eV
{1 GeV = 109 eV}, is the regime of quantum gravity, where space-time itself becomes
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
25
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede
“foam-like” (i.e. not continuous) – quantized/discretized {somehow…}. The distance scale
where quantum gravity is operative is known as the Planck length: LP  GN c 3  1.6 1035 m .
The Planck length corresponds to a time-scale {known as the Planck time} of
t P  LP c  GN c 5  5.4 1044 sec .
Nevertheless, back in the early 1900’s, ignorance of quantum gravity and quantum mechanics
did not stop Abraham, Lorentz, Poincaré {and many others} from applying classical EM theory electrodynamics to calculate the self-force / radiation back-reaction on a point electric charge.
These efforts by-and-large modeled the point electron as {some kind of} spatially-extended
electric charge distribution (of finite, but very small size), calculations could then be carried out
and then (at the end of the calculation) the limit of the size of the charge distribution → 0.
In general (as we have already encountered this before in electrodynamics), the retarded
classical/macroscopic EM force of one part (A) acting on another part (B) is not equal and
opposite to the force of B acting on A, Newton’s 3rd Law is seemingly violated:
  
  
Fr AB  rB , t    Fr BA  rA , t 
Adding up the imbalances of such force pairs, we obtain the net force (imbalance) of a charge
on itself – the “self-force” acting on the charge.
H.A. Lorentz originally calculated the classical self-force using
a spherical charge distribution – tedious – see J.D. Jackson’s
Classical Electrodynamics, 3rd ed., sec. 16.3 and beyond if
interested in these details….
A “less realistic” model of a charge is to use a rigid dumbbell
in which the total charge q is divided into 2 halves separated
by a fixed distance d (simplest possible charge arrangement
to elucidate the self-force mechanism):
Assume that the dumbbell moves in x̂ -direction and (for simplicity) assume that the dumbbell is
instantaneously at rest at the retarded time tr. Then the retarded electric field at (1) due to (2) is:
  
Er21  r1 , t  
26
r  2      
r  2      
q
c  r a  u   r u  a  
 c  r a  u   r u  a 


3 

4 o  r u 
8 o  r u 3 
1
2
q
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
UIUC Physics 436 EM Fields & Sources II
Here:
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede


u  crˆ {because v  tr   0 }

Note that: r  xˆ  dyˆ  rrˆ and thus: r   2  d 2 .


Note also that:   fcn  a  and: a  tr   a  tr  xˆ .

 r u  tr    xˆ  dyˆ crˆ  rrˆcrˆ  cr
 
and: r a  tr    xˆ  dyˆ a  tr  xˆ  a  tr  
  
We are in fact only interested in the x̂ -component of Er21  r , t  , since the ŷ -components of
  
  
Er21  r , t  and Er12  r , t  will cancel when we add forces on the two ends of the dumbbell.
Note further that since the two charges on the dumbbell are both moving in the same direction /
parallel to each other, the magnetic forces associated one charge acting on the other will also cancel,
thus Newton’s 3rd Law is manifestly obeyed {here}, in this particular situation / configuration.


cr
c
c
 

If u  crˆ , then: u x  u  xˆ   xˆ and since: r  xˆ  dyˆ then: u x   lxˆ  dyˆ  xˆ 
r
r
r



q
r

 c 2  r a  tr   u x   r u  tr   ax  tr  
Thus: Er21x  r1 , t  

3 


8 o  r u  t  
r

 

And:  r u  tr    cr and:  r a  tr     xˆ  dyˆ a  tr  xˆ  a  tr   since: a  tr   a  tr  xˆ  ax  tr  xˆ
Then:

q
r  2
c


Er21x  r1 , t  
c  a  tr     cra  tr  
3 3 
r
8 o c r 


1  2
q


c  a  tr     ra  tr  
2 2 
r
8 o c r 


2

1  c 2 a  t r  

 ra  tr  
2 2 
r
8 o c r  r

q
2
2
1  c 2 a  t r    r a  t r  



r
8 o c 2 r 2  r

q
1
c 2   r 2   2  a  tr   but: r 2   2  d 2 or: r 2   2  d 2

2 3 



8 o c r

q
 d2

q
1
c 2  d 2 a  tr  
8 o c r 
2 3
2
2
q  c  d a  t r  

Thus: E  r1 , t  
since: r   2  d 2
3
2
2
2
8 o c   d 2






By symmetry: Er21x  r1 , t   Er12x  r2 , t 

21
rx
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
27
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede
 The net retarded force on the rigid dumbbell is:

  
  
  
  

Frself  r , t   Fr21  r1 , t   Fr12  r2 , t   12 qEr21  r1 , t   12 qEr12  r2 , t  
8 o c
 c  d a  t   xˆ  Exact
  d 
2
q2
2
r
2
2
2
3
2


We now expand Frself  r , t  in powers of d. Then when the size d of the electrically-charged
dumbbell is taken to its limit of d → 0, all positive powers will disappear.
Taylor’s Theorem: x  t   x  tr   x  tr  t  tr  
1
1
2
3

x  tr  t  tr   
x  tr  t  tr   ...
2!
3!
Recall that:

x  tr   v  tr   0 and that:   fcn  a  tr  
Then:
   x  tr   x  tr   
1
1
a  tr  tr2  a  tr  tr3  ... where: tr   t  tr 
2
6
But:
ctr  r   2  d 2
 c 2 tr2  r 2   2  d 2
Or:
1
1

d  r    c t    c t   a  tr  tr2  a  tr  tr3  ... 
6
2

2
2
2
2
r
2
2
2
2
r
 a  tr  tr a  tr  tr2

 a  tr  tr a  tr  tr2

 c t  c t 

 ...   ctr 1  

 ... 
6c
6c
 2c

 2c

2
2
 ctr 
2
r
2
2
2
r
a 2  tr  3
tr    tr4  ...
8c
We want tr in terms of d. From above, it can be seen that we can solve for d in terms of tr .
But we can solve for tr in terms of d using the reversion of series technique, which is a
formal method that can be used to obtain an approximate value of tr by ignoring all higher
powers of tr . To first order in d, we have:
d  ctr  tr 
d
 use this as an approximation for obtaining a cubic correction term:
c
2
3
a 2  tr   d 
d a  tr  d
d  ctr 
   tr  
c
8c5
8c  c 
3
a 2  tr  3
1
d    d 4  ...
Keep going. . . tr  d 
5
c
8c
28
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede
Thus:    x  t   x  tr   
a  tr  2 a  tr  3
1
1
a  tr  tr2  a  tr  tr3  ... 
d 
d    d 4  ...
3
2
6
2c
6c


Then: Frself  r , t  
 c  d a  t   xˆ  q
4
  d 
2
q2
8 o c
2
r
2
2
2
3
2
2
o
 a  tr  a  tr 

   d  ... xˆ
 2 
3
 4c d 12c

{Note that a  tr  and a  tr  are evaluated at the retarded time tr.}
Using the Taylor series expansion of a  tr  , we can rewrite this result in terms of the present time:
a  tr   a  t   a  t  tr  t   ...  a  t   a  t  tr  ...  a  t   a  t 
Then:

q2

Frself  r , t  
4 o
d
 ...
c
 a  t  a  t 

  2  3    d  ... xˆ
 4c d 3c

The first term inside the brackets on the RHS is proportional to acceleration of the charge q.


If we put it on LHS, then by Newton’s 2nd Law F  ma , we see that it adds to the mass m of the
dumbbell – there is inertia associated with accelerating an electrically-charged particle.
The total inertial mass of the dumbbell is therefore:
mtot  mdumbbell
Or:
2
1  q2 
1   12 q  




  mdumbbell 
4 o  4dc 2 
4 o  dc 2 
2
1   12 q  


mtot c  mdumbbell c 
4 o  d 
2
 rest mass energy, E  mc 2
2
Note that the {repulsive} electrostatic potential energy associated with this dumbbell is:
UE r  d   
1
2
 12 q 
2
1   12 q  

 (Joules)

q V  r  d  
4 o d 4 o  d 
2
The fact that this works out “perfectly” is simply due to the fact that the initial choice of the
dumbbell’s orientation was deliberately/consciously chosen to be transverse to the direction of
motion. For a longitudinally oriented dumbbell, the EM mass correction is half this amount.
For a spherical charge distribution, the EM mass correction is a factor of ¾ !!
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved
29
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14
Prof. Steven Errede


The second term inside the brackets on the RHS of the Frself  r , t  relation is the EM radiation
reaction term:
 int 
q 2 a  t 
o q 2 a  t 
ˆ
Frad  r , t  
x
xˆ
12 o c 3
12 c
 int 
Note that Frad
 r , t  differs from Abraham-Lorentz result by a factor of 2:
 A-L 
o q 2 
Frad  r , t  
a t 
6 c
 int 
The reason for the factor of 2 difference is that physically, Frad
 r , t  is force of one end of the
dumbbell acting on the on other – i.e. an EM interaction between the two ends of the dumbbell.
There is also a force of each end of the dumbbell acting on itself – an EM self-interaction
 self 
Frad  r , t  for each end. When the EM self-interactions for each end are included (see Griffiths
Problem 11.20, p. 473), the total EM radiation-reaction is:
 tot 
 

 q 2 a  t   1 1 1 
o q 2 a  t 

ˆ



Frad
x
xˆ
 r , t   Fradint  r , t   2 Fradself  r , t   o
6 c  2 4 4 
6 c
which agrees perfectly with Abraham-Lorentz radiation-reaction force formula.
Thus, physically we see that the EM radiation reaction is due to the force of the charge acting on
itself – an {apparent} self-force!
 
 
Note also that Frad  r , t  does NOT depend on d ( Frad  r , t  is valid/well-behaved in limit of
the size of the dumbbell, d → 0).
2
1   12 q  

   when d → 0 !!!
However, note that: mtot c  mdumbbell c 
4 o  d 
2
2
The inertial mass of the classical electron becomes infinite when when d → 0, because:
UE r  d   
1
2
 12 q 
2
1   12 q  

   when d → 0 !!!

q V  r  d  
4 o d 4 o  d 
2
{But we already knew this, as we learned long ago, in P435/last semester…}
Note that this unpleasant/awkward problem also persists in the fully-relativistic, quantum
electrodynamical theory {QED}. Infinities/singularities there are dealt with/side-stepped by a
process known as mass renormalization, so as to avoid such infinities – look only at mass
differences / energy differences…
30
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved