Download Lecture Notes 12: Lienard-Wiechert Retarded Potentials for Moving Point Charge, Retarded Electric and Magnetic Fields Associated with Moving Point Charge

UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 12
Prof. Steven Errede
LECTURE NOTES 12
 

THE LIÉNARD-WIECHERT RETARDED POTENTIALS Vr  r , t  AND Ar  r , t 
FOR A MOVING POINT CHARGE
Suppose a point electric charge q is moving along a specified trajectory = locus of points of

w  tr  = retarded position vector (SI units = meters) of the point charge q at retarded time tr .
The retarded time (in free space/vacuum) from point charge q to observer is determined by:

ct  r  r
where the time interval t  t  tr
 



and: r  r  t   r   tr   r  t   w  tr 
Observation / field point
at time t (doesn’t move)
position of the point charge q at the retarded time tr

w  tr  = retarded position of charge at retarded time tr
 

r  r  r  t   w  tr   ct  c  t  tr 

r = separation distance of point charge q at the retarded position w  tr  at the retarded time tr

to the observer’s position at the field point P  r  t   , which is at the present time, t  tr  r c .

r = vector separation distance between the two points, as shown in the figure below:
Position of pt. charge
q at retarded time, tr
Position of pt. charge
q at present time, t
ẑ
Trajectory of pt.
charge, w(tr)

Source Pt.

S  w  tr  
 

r  r  t   w  tr 

w  tr 

r t 
ŷ
Field Pt.

P  r t 
x̂

n.b. At most one point (one and only one point) on the trajectory w  tr  of the charged

particle can be “in communication” with the (stationary) observer at field point P  r  t   at the
present time t, because it takes a finite/causal amount of time t  t  tr for EM “news” to


propagate from w  tr  at the retarded time tr to the observation/field point P  r  t   at position

r at the present time t  tr  r c .
© 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 12
Prof. Steven Errede
In order to make this point conceptually clear, imagine replacing the point charge q moving

along the retarded trajectory w  tr  with a moving point light source. The stationary observer at

the field point P  r , t  at the present time t will see the point light source move along the retarded

trajectory w  tr  ; but it takes a finite time interval for the light (EM “news”) to propagate from


where the light source was at the retarded source point location S  r   tr   w  tr   at the retarded

time tr to the observer’s location at the field point P  r , t  at the present time t.
This situation is precisely what an observer sees when looking at stars, planets, etc. in the night sky!

Suppose that {somehow} there were e.g. two such source points along the trajectory w  tr 

“in communication” with the observer at the field point P  r  t   at the present time t with
retarded times tr1 and tr2 respectively.

Then: r 1  c t  tr1


and: r 2  c t  tr2


 
 
thus: r 1  r 2  c t  tr1  c t  tr2  c tr2  tr1



 The average velocity of this charged particle in the direction of the observer at r is c !!!
{n.b. the velocity component(s) of this particle in other directions are not counted here}.
However, we know that nothing can move faster than the speed of light c !!!
 


 Only one retarded point w  tr  can contribute to the potentials Vr  r , t  and Ar  r , t  at the

field point P  r  t   at any given moment in the present time, t for v < c !

 For v < c, an observer at the field point P  r  t   at a given present time t “sees” the moving
charged particle q in only one place.
{Note that a massless particle, such as a photon (which in free space/vacuum does move at the speed of light, c)
could/can be “seen” by a stationary observer as being at more than one place at a given {present} time, t !!!
Note further that it is also possible that no points along the trajectory of the photon are accessible to an observer….}
A “naïve” / cursory reading of the formula for the retarded scalar potential

tot  r , tr 
1

Vr  r , t  
d 
4 o v
r
might suggest that the retarded scalar potential for a moving point charge is {also}
1 q
4 o r
(as in the static case), except that r = the separation distance is from observer position to the
retarded position of the charge q.
However, this would be wrong – for a subtle conceptual reason!
It is true that for a moving point charge q, the denominator factor 1 r can be taken outside of

the integral, but note that {even} for a moving point charge, the integral:   tot  r , tr  d   q !!!
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
Fall Semester, 2015
Lect. Notes 12
Prof. Steven Errede

In order to calculate the total charge of a configuration, one must integrate tot  r , tr  over the
entire charge distribution at one instant of time, but {here} the retardation tr  t  r c forces

us to evaluate tot  r , tr  at different times for different parts of the charge configuration!!!
Thus, if the source is moving, we will obtain a distorted picture of the total charge!


r  r  t   r   tr  is a function of r  t  and r  tr 
Before integration:
 


r   w  tr 
is fixed after integration: tot  r , tr   q 3  w  tr  



r  r  t   w  tr  is a function of r and t because tr  t  r c .
After integration:
One might think that this problem would be understandable e.g. for a moving extended charge
distribution, but that it would disappear/go away/vanish for point charges. However it doesn’t !!!
In Maxwell’s equations of electrodynamics, formulated in terms of electric charge and current

densities tot and J tot , a point charge = limit of extended charge when the size → zero.


For an extended charge distribution, the retardation effect in
1
1
1




 1  rˆv  tr  c
1  rˆ   tr 

 

v

  r , tr  d  throws in a factor of:
 tot

v  tr 
where:   tr  
c



 

We define the retardation factor   1  rˆv  tr  c , where v  tr  {more precisely v  r   tr   }

is the velocity of the moving charged particle at the source position r   tr  at the retarded time tr .
This is a purely geometrical effect, one which is analogous/similar to the Doppler effect.
{However, it is not due to special / general relativity (yet)!!}
Consider a long train moving towards a stationary observer. Due to the finite propagation
time of EM signals, the train actually appears (a little) longer than it really is! (If c ≈ 10 m/s
rather than 3 × 108 m/s, this motional effect would be readily apparent in the everyday world!!)
As shown in the figure below, light emitted from the caboose (end of the train) arriving at the
observer at time t must leave the caboose earlier trend than light emitted from the front of the
 
 
train trfront , both arrive simultaneously at the observer at the same present time t. The train is
further away from the observer when light from the end of the train is emitted at the earlier time
trend , compared to the train’s location for the light emitted from the front of the train at the later
 
 
time trfront . The observer thus sees a distorted picture of the moving train at the present time t.
© 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 12
Prof. Steven Errede
In the time interval tc that the light from the caboose takes to travel the distance L (see figure
above) the train moves a distance L   L  L  . Then since ctc  L , then: tc  L c .
But during the same time interval tc , the train moves a distance L  vtc   L  L  , or:
tc 
L L  L  L 
L
1
L  L  L 


L
but: tc 
thus: tc  
 L 
c
v
v
c
v
v
1 v c
 Trains moving towards / approaching an observer appear longer, by a factor of 1 1  v c  .
 Conversely, it can similarly be shown that trains moving away / receding from an observer
appear shorter by a factor of 1 1  v c  .

 In general, if the train’s velocity vector v makes an angle θ with the observer’s line of sight r̂
(n.b. assuming that the train is far enough away from the observer that the solid angle subtended by
the train is such that rays of light emitted from both ends of train are parallel) the extra distance that
light from the caboose must cover is L cos  (see figure below). The corresponding time interval is
tc  L cos  / c . Note that the train also moves a distance L  L  L in this same time interval.
L cos 
tc  L cos  / c but: tc 
 1 cos 
Or: L  
c
v
L L  L
L cos  L L  L
L cos  L  L




 tc 
i.e.
v
v
c
v
v
c
v
1  v cos 
 L
or: L  1 

v 
c
 v
1
 L
  or: L 
 v cos 
 v
1 
c




L
1
v
L with  
c
1   cos  
 
From the above figure, the angle   cos 1 rˆvˆ = opening angle between r̂ and v̂ .
 v

1
1
L
Thus:  cos   rˆ where:   . Hence: L 
 L.
c
1   cos  
1  rˆ 


Again, this retardation effect is due solely to the finite propagation time of the speed of light – it
has nothing to do with special / general relativity – e.g. Lorentz contraction and/or time dilation
and simultaneity.
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
Lect. Notes 12
Prof. Steven Errede
The apparent volume   of the train is related to the actual volume of train   by:
 

v t 
1
1
1


  
     where   tr   r and   1  rˆv  tr  c  1  rˆ   tr 


c
1  rˆv  tr  c
1  rˆ   tr 

 
and where rˆ  r r  r r = unit vector associated with the separation distance between the

position of a {stationary} observer r  t  at the present time t to a position somewhere on the

  




train r   tr  at the retarded time tr . Explicitly: rˆ  r r  r r   r  t   r   tr   r  t   r   tr 

The stationary observer’s position vector r  t  is constant in time, whereas the retarded position

vector of the moving train r   tr  changes in time.
 

Hence, whenever we carry out integrals of the type  tot  r , tr  d  {or  J tot  r , tr  d  }
v
v
 

where the integrand(s) tot  r , tr  {or J tot  r , tr  } are associated with {some kind of} moving
charge {current} distribution(s), evaluated at the retarded time tr , the apparent volume   of these


v  tr 
1
1
1


integrals is modified by the factor 
where:   tr  
, and the

c
 1  rˆv  tr  c 1  rˆ   tr 


1
1
1

d  
d 
retardation factor   1  rˆv  tr  c  1  rˆ   tr  , i.e. d   d  


1  rˆv  tr  c
1  rˆ   tr 
The figure shown below graphically depicts this effect, for a snapshot-in-time t  tr  r c :
True volume at the
present time t
Observer position
P(r,t) at present
time t (fixed)

v t 
Apparent volume at
the retarded time tr

v  tr 
 See animated demo of this effect: https://en.wikipedia.org/wiki/Relativistic_Doppler_effect
Note that because the motional correction factor makes no reference to the actual physical
size of the “particle”, it is also relevant/important for point charged particles.
© 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 12
Prof. Steven Errede
The retarded scalar potential associated with a point electric charge q moving along a



retarded trajectory w  tr  , with:  q  r , tr   q 3  w  tr   is:

1

Vr  r , t  
4 o

 q  r , tr 

r
v
d  
1 1
4 o r v

q 3  w  tr  
r

q 3  w  tr  
d  
d 

4 o r v r 1  rˆv  tr  c
1 1




1 
1 
q
q

 1 q


 

4 o  1  rˆv  w  tr   c r  4 o  1  rˆ   w  tr   r  4 o  r









   



Where: rˆ  r r  r r , r  r  t   r   tr   r  t   w  tr 
 
 
 
v  w  tr  
 
  1  rˆv  w  tr   c  1  rˆ   w  tr   with:   w  tr   
,
And:
c
 
Where: v  w  tr   = velocity vector of charged particle evaluated at the retarded time tr  t  r c .
 
The retarded current density J tot  r , tr  for a rigid object is related to its retarded charge

 
density tot  r , tr  and its retarded velocity v  r , tr  by the relation:
 

 
J tot  r , tr   tot  r , tr  v  r , tr  .
The retarded vector potential associated with a point electric charge q moving with retarded velocity
 

 
 

 

v  w  tr   along a retarded trajectory w  tr  , with: J q  r , tr    q  r , tr  v  r , tr   qv  r , tr   3  w  tr   is:
 

Ar  r , t   o
4
 
J q  r , tr 

v
r

d   o
4




 
 q  r , tr  v  r , tr 
r
v

d   o
4

 

qv  r , tr   3  w  tr  
v
 

 

qv  r , tr   3  w  tr  
qv  w  tr  
o 



d


 
v r 1  rˆv  t  c

ˆ
4


v
t
c

r
r
1
w





r
r






  qv  w  t    q  
qv  w  tr  
 
r
 o
 o
 o
v  w  tr  
 
r
4  r
4  1  rˆ   w  tr   r  4



 o
4


r
d 


Thus, we have obtained the so-called Liénard-Wiechert retarded potentials for a point electric
 

charge q moving with retarded velocity v  w  tr   along a retarded trajectory w  tr  :

1 
q

 1 q
Vr  r , t  
 
4 o  1  rˆ   w  tr   r  4 o  r


 

 
qv  w  tr  
o 

  o q v  w
Ar  r , t  
 
 tr  
4  1  rˆ   w  tr   r 
4  r


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
Where:
Fall Semester, 2015
 
 
  1  rˆv  w  tr   c  1  rˆ   w  tr  
Lect. Notes 12

with:   w  tr   
Prof. Steven Errede
 
v  w  tr  
c
.
 
 

 
 
v  w  tr  
Vr  r , t 
1  v  w  tr   


1
Vr  r , t   
 Vr  r , t     w  tr  
Note that: Ar  r , t  
using: c 2 
2
c
c 
c
c
 o o

 

 
 
 
v  w  tr  
Vr  r , t 
Ar  r , t     w  tr  
with:   w  tr   
.
Or:
c
c

Where Vr is in Volts, Ar is in Newtons/Ampere (= momentum per Coulomb); they are related to

each other by a factor of 1 c and  for the case of a moving point electric charge q.

Recall that the relativistic four-potential is: A  V c , A , hence {here} the retarded relativistic

 
four-potential for a moving point charge is: Ar  Vr c , Ar  Vr c ,   w  tr   Vr c .



 

Griffith’s Example 10.3:
Find/determine the Liénard-Wiechert retarded potentials associated with point charge q

moving with constant velocity v .
For convenience sake, define tr  0  retarded time the charged particle passes through the origin.





Then: w  tr   v  tr  tr  vtr because v  tr   v is a constant vector.
The retarded time is: tr  t  r c
 


or: r  c  t  tr   ctr
with: tr  t  tr
 
r  r  r   tr   r  w  tr   r  vtr but we also have: r  c  t  tr   ctr
But:
 
Solve for the retarded time tr by relating: r  r  vtr  c  t  tr 
Square both sides:
But:
Thus:
Solve this quadratic equation for tr :
  2
2
r  vtr  c 2  t  tr   c 2  t 2  2ttr  tr2 
  2
 
 
 
r  vtr   r  vtr  r  vtr   r 2  2r vtr  v 2tr2
 
r 2  2r vtr  v 2tr2  c 2  t 2  2ttr  tr2 
 
 v 2  tr2  2  c 2t  r v  tr   c 2t 2  r 2   0
c2



a
 c2t  rv  
b  b 2  4ac
tr 
 tr 
2a
b
c
 c t  rv    c
2
2
c2  v2
2
 v 2  r 2  c 2t 2 
**
Which sign do we choose? + or  ?? Must make physical sense!
© 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 12
Prof. Steven Errede
Consider the limit as v → 0: tr  t  r c ← what we want!

And, if v = 0, the point charge q is at rest at the origin ( r   0 ), because it is there at time tr  0 .
 

Then: r  r  r   r  r
Thus, its retarded time should be: tr  t  r c  tr  t  r c when v → 0.
 We must choose the – sign on physical grounds, i.e. we must choose:
 c t  rv    c t  rv    c
2
tr 
Now: r  ctr  c  t  tr 
2
2
2
 v 2  r 2  c 2t 2 
c2  v2
  
 
r  vtr 
ˆr  r  r  vtr  
.
r ctr c  t  tr 
and:
Therefore, the quantity:
 

 

 
 
c
c


r  vtr v 
r  vtr v 
r v v v

r 1  rˆv c   ctr 1 
tr

   c  t  tr  1 
   c  t  tr  
c  t  tr  c 
 c  t  tr  c 
 
1
r v v 2
 
 c  t  tr  
 tr   c 2t  r v    c 2  v 2  tr 
c
c
c

Then, insert the retarded time tr from the expression (**) {above} with the minus sign, i.e.:
 c t  rv    c t  rv    c
2
tr 
2
2
2
 v 2  r 2  c 2t 2 
c2  v2


1

Thus: r 1  rˆv c   r 
c

 c t  rv    c
2
2
2
into the above formula & carry out the algebra:


 v 2  r 2  c 2t 2  with:   1  rˆv c

{here}

The general form of the Liénard-Wiechert retarded scalar and vector potentials associated with a
 
point electric charge q moving with a time-dependent velocity v  w  tr   are:


1 
q
1 
q

 1


 

4 o  1  rˆv  w  tr   c r  4 o  1  rˆ   w  tr   r  4 o




 
 



 
qv  w  tr  
qv  w  tr  
o 

o


  o

Ar  r , t  
 
 
4  1  rˆv  w  tr   c r 
4  1  rˆ   w  tr   r 
4





Vr  r , t  



Where:
8

 
 
  1  rˆv  w  tr   c  1  rˆ   w  tr  


 





with:   w  tr   
 
v  w  tr  
c
q
r
q  
v  w  tr  
r
.
© 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 12
Prof. Steven Errede
The Liénard-Wiechert retarded scalar and vector potentials associated with a point charge q

moving with constant velocity v are:

Vr  r , t  

1 
q
 1 q  1

4 o  1  rˆv c r  4 o  r 4 o


 
Ar  r , t  

  qv
o 

qv
 o
 o

4  1  rˆv c r  4  r
4






qc
 c t  rv    c
2
2
2
 v 2  r 2  c 2t 2 

qcv
 c t  rv    c
2
2
2
 v 2  r 2  c 2t 2 


where:   1  rˆv c = retardation factor {here} and: r  rrˆ .
 
 


Note again that {here}: Ar  r , t    Vr  r , t  c  where:   v c = constant vector.


The Electromagnetic Fields Associated with a Moving Point Charge
We are now in a position to derive the retarded electric and magnetic fields associated with a moving
point charge using the Liénard-Wiechert retarded potentials associated with a moving point charge:

1 
q

 1 q
Vr  r , t  
 
4 o  1  rˆ   w  tr   r  4 o  r


 

 
qv  w  tr  
o 

  o q v  w
Ar  r , t  
 
 tr  
4  1  rˆ   w  tr   r 
4  r


with:




 
 
 

 
Ar  r , t     w  tr   Vr  r , t  c with:   w  tr    v  w  tr   c and: c  1
 o o {in free space}
 
where: r  r  w  tr 
and:
tr  t  r c = retarded time


 
and:   1  rˆv  w  tr   c = retardation factor.


The equations for the retarded E and B -fields in terms of their retarded potentials are:
 
 

 
  
Ar  r , t 

Er  r , t   Vr  r , t  
and: Br  r , t     Ar  r , t 
t
Again, the differentiation has various subtleties associated with it because:
 



r  r  t   r   tr   r  t   w  tr 
and:

w  tr  

v  tr  
 w  tr 
t
both quantities are evaluated at
the retarded time tr  t  r c
© 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 12
Prof. Steven Errede
  
 

Note that: r  r  r  r   tr   r  w  tr   ctr  c  t  tr   r is a function of both r and t.


Now: Vr  r , t  

  1 
1
 
 

4 o  r  1  rˆv  tr  c  

 
q

 


1
q
1





r  r v  t r  c 



2

4 o  r  r v  tr  c  4 o  r  r v  t  c 
r





r  c  t  tr   r  c  t  tr   ctr


Thus: Vr  r , t  
But:


with: rrˆ  r
q
  
  
     
  
Then, using Griffiths “Product Rule # 4”  A B  A B  B A  A    B  B    A
  
on the second term   r v  tr  c  we obtain:

(1)
 
 
(2)


(3)



(4)




  
  1 

 
 

1   
1 
1

  r v  tr  c   r  v  tr   v  tr  r  r    v  tr   v  tr     r
c
c
c
c








For term (1):



  
dv  tr  tr
dv  tr  tr
dv  tr  tr
 

 
 r   v  tr    r x x  r y y  r z z  v  tr   r x dt x  r y dt y  r z dt z

Again:


d
d

dtr dt
r
r
r
since: tr  t  r c

 
 
dv  tr   tr
t
t  
 r y r  r z r   a  tr   r tr 
 r   v  tr  
r x
dt
x

y
z 


dv  tr  dv  tr 


where: a  tr  
= acceleration of charged particle at the retarded time tr  t  r c .
dtr
dt
  
 
r  r  r   tr   r  w  tr 
  
  
  
 


Thus:  v  tr   r   v  tr    r  w  tr     v  tr   r   v  tr   w  tr 
Second term (2):
 
 (2 a )

Term (2a):
10
 v  t   r   v  t  x  v
r
 (2 b )

 
 vz  t r    xxˆ  yyˆ  zzˆ 
y
z 


 vx  tr  xˆ  v y  tr  yˆ  vz  tr  zˆ  v  tr  !!!
x
r
y
 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 12
Prof. Steven Errede
Term (2b):
 



 

 v  tr  w  tr     vx  tr   v y  tr   vz  tr   w  tr 
x
y
z 




dw  tr  tr
dw  tr  tr
dw  tr  tr
  vx  t r 
 v y  tr 
 vz  t r 
dtr x
dtr y
dtr z


t
t
t  dw  tr 
   vx  tr  r  v y  tr  r  vz  tr  r 
x
y
z  dtr




dw  tr 

v  tr  
But:
dtr
 


t
t
t 




Thus:  v  tr  w  tr   v  tr  vx  tr  r  v y  tr  r  vz  tr  r   v  tr  v  tr tr
x
y
z 

 

  



  
n.b.  v  tr  w  tr   v  tr  v  tr tr is analogous / similar to: r  v  tr   a r tr





  
Third term (3): r    v  tr 









 
 
First, work out:   v  tr  Since: A  B   Ay Bz  Az By  xˆ   Az Bx  Ax Bz  yˆ   Ax By  Ay Bx  zˆ
Then:
 
 v  t  v  t  
 v  t  v  t  
 v  t  v  t  
  v  tr    z r  y r  xˆ   x r  z r  yˆ   y r  x r  zˆ
z 
x 
y 
 z
 y
 x
 dv  t  tr dv y  tr  tr 
 dv y  tr  tr dvx  tr  tr 
 dvx  tr  tr dvz  tr  tr 
 z r



 xˆ  
 zˆ
 yˆ  
dtr z 
dtr x 
dtr y 
 dtr z
 dtr y
 dtr x

tr
t 
 a y  tr  r  xˆ 
 az  tr 
y
z 


t
t 
   a y  tr  r  az  tr  r  xˆ 
z
y 



  a  t r   t r


tr
t 
t
t 

 az  tr  r  yˆ   a y  tr  r  ax  tr  r  zˆ
 ax  tr 
z
x 
x
y 


tr
t

 ax  tr  r
 az  tr 
x
z



  

 

 r    v  tr   r  a  tr tr   r  a  tr tr






tr
t  

 a y  tr  r  zˆ 
 yˆ   ax  tr 
y
x  



© 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
 

Fourth Term (4): v  tr     r
Lect. Notes 12
Prof. Steven Errede

  with: r  r  r  t   r  w  t 
 
  
   
First, work out:    r      r  w  t      r    w  t 
r
r
r
r

Of course!!! Because r is
0
0 
0
0 
0
0 



 
position vector of field
z y 
x z 
y x 

zˆ  0 
xˆ  
yˆ  



But:   r  
point/observer P ( r )
 y z 
 z x 
 x y 







r = constant vector !!!

r  xxˆ  yyˆ  zzˆ
 

 



From the result of term (3) above:   v  tr    a  tr   tr we see that:   w  tr   v  tr     tr 
  
 

   0   





 v  t r     r  v  t r      r    w  t r     v  t r     w  t r    v  t r   v  t r   t r








Collecting all of the above individual results (1) – (4) for the term:



1  
  r  r  v  t r  c   r    r  v  t r  
c


    
  
1   


 r   r  v  tr   v  tr  r  r    v  tr   v  tr     r 
c    
 

(1)
(2)
(3)
(4)










We see that:




 
1 



  r  r v  tr  c   ctr  a  tr  r tr  v  tr   v  tr  v  tr tr
c


 


 r  a  t r   t r  v  t r   v  t r    t r





But:

And:
  
  
  
A  B  C  B AC  C A B



 





  
  
  
  
  
 r  a  tr   a r tr  tr  r a   a r tr  tr  r a 

 




  
  
  
v  v  tr  v v tr  tr  v v 

 

Then, some truly amazing/fortuitous cancellations occur:




1 
  r  r v  tr  c   ctr  a  tr 
c

 a  tr 
 

 r t   v  t   v  t   v  t t 
r
r
r
12
r
 r t   t  r a  t    v  t   v  t t   t  v  t v  t  
 


r
r

r


 
1 
 ctr  v  tr    r a  tr    v 2  tr   tr
c

r
r
r

r
r
r

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
r
UIUC Physics 436 EM Fields & Sources II
Thus:


Vr  r , t  
Fall Semester, 2015
Lect. Notes 12
Prof. Steven Errede


1
v  t r  c 


r
r


4 o  r  r v  t  c  2
r


q
 
 
1
1 
 
ctr  v  tr    r a  tr    v 2  tr  tr  

2 

4 o  r  r v  t  c  
c

r



 
 
 v  tr  1 2 2
1
q



c
v
t
a
t
r











r
r  tr 
4 o  r  r v  t  c  2  c
c

r



q

Now, what is tr {here}??
tr  t 
r
c
t
 
r  r   tr 
c
 
r  w  tr 
t
c



But we already found that: r  ctr , and noting that: r  r  r

   1 1   
1   
Then: r   r  r 
    r r     r r 
2 r r
2r
  
  
  
  
Now:  A B  A    B  B   A  A B 
        
  
  r  r   r    r  r    r  r  r 


But:







  
    B  A {Griffiths Product Rule # 4}
     
  
    r   r  2r     r   2  r   r

   1   
1   
1    
r    r  r  
2r    r  2 r  r    r    r 
 r
2r
2r 
  

   
  r  v  tr   tr {from term (4) above}  r    r  r 


 




    


 v  t   t 
r
r
 
 r   r  r   r  r  t    r   r  w  t  
     
  r   r   r   w  t 
r
r
from (2) above:
r
And:
  
  r   r 
 
 x
y
z  
xˆ  r y
yˆ  r z zˆ   v  tr  r tr
 r x
y
z 
 x
 
 
 r  v  tr  r tr



 
 
 r   w  tr   v  tr   r tr 



  
1   
ctr  r   r    r  r  r 

r


 
 
1  
  r  v  tr   tr  r  v  tr  r tr 

r 

 



1  
Thus: ctr   r  v  tr 
r


BAC  CAB rule

  
 
 

r tr  v  tr  r tr  tr  r v  tr    more cancellations occur!!




© 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 12
Prof. Steven Errede


1   
ctr   r   r v  tr   tr 
Or:
r





1  
r
r
1
r


  
Now solve for tr :  c   r v  tr    tr   rˆ  tr 
r
r
r  1 r v t  c   r v  tr   rc 


 r  
 r 





r
r
r
1
1
rˆ
rˆ
t r   







c 1  rˆv  tr  c 
c
rc 1  rˆv  tr  c 
 r v  tr   rc   rc  r v  tr  





1
rˆ
rˆ
r

Thus: tr  
, rˆ 

c 1  rˆv  tr  c 
c
r




r  rrˆ and   1  rˆv  tr  c = retardation factor


 
 
1  v  tr  1 2 2
q




c
v
t
a
t
r
Vr  r , t  









r
r  tr 
4 o  r 2  c
c

Then:

 

1  v  tr 
1
q

 2 c 2  v 2  tr   r a  tr   rˆ 
2 
4 o  r   c
c



Thus: Vr  r , t  
q
1
4 o  r 2


  c 2  v 2  tr   r a  tr  
v  tr  


 rˆ 
c 
 c2

 
 
2

 1   v  tr   r a  tr   ˆ v  tr  
r 
 1  

 
c2 
c 
    c 


Or:

1
q

Vr  r , t  
4 o  r 2
Now:


 
qv  tr 
o
o qv  tr 

Ar  r , t  
4 1  rˆv  tr  c r 4  r



v  tr 

 
n.b.  2 Vr  r , t  
c


 


 

v  tr 
Ar  r , t  o   v  tr 
1

o






q
q


t
4 t  r 1  rˆv  tr  c  4 t   r  r v  tr  c  




Then:

 
v  tr  
o 

1
1


 v  tr  
q


t   r  r v  tr  c   
4   r  r v  tr  c  t





Now:



v  tr  dv  tr  tr 
dv  tr 
tr


 a  tr 
where: a  tr  
t
dtr t
t
dtr
What is
tr
{here} ??
t
14
© 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
Now: ctr  c  t  tr   r

c 2 tr2  c 2  t  tr   r 2  r  r
2
Or:
Fall Semester, 2015
Lect. Notes 12
Prof. Steven Errede
 differentiate this expression with respect to t.
 2 2

  
2
c tr   c 2  t  tr    r  r 

t
t
t

 r
 tr 
2
 2c  t  tr   1 
  2r 
t 
t

But:
 
 




r  r  r  t   r   tr   r  t   w  tr   ctr  c  t  tr  and also: r  r  t   w  tr 
Thus:
 r
  
 2 2

 t 
 t 
c tr   2c  c  t  tr   1  r   2cr 1  r   2r   2r   r  t   w  tr  

t
t 
t 
t
t



 t
 cr 1  r
t


  0


r
   r w  tr  

0
n.b.
  r 
t
t
t 




is a constant vector – i.e. stationary observer!
 w  tr   dw  tr  tr
  t
 tr 
 tr 
 r 
  r v  t r  r
Thus: cr 1 
  rc 1 
  r 
t 
t 
t
dtr t
t


  t
t
 t 
 rc  1  r    r v  tr  r  Solve for r
t 
t
t

rc  rc
Thus:


tr  
t
t
 r v  tr  r   rc  r v  tr   r
t
t
t

Because r = vector to field point P  r 


dw  tr 

v  tr  
dtr
tr
rc


t rc  r v  tr 
 rc 
tr
rc
1
1
1








t rc  r v  tr   rc  1  rˆv  tr  c
1  rˆv  tr  c 

 

tr
1
1



where:   1  rˆv  tr  c = retardation factor

t 1  rˆv  tr  c 



v  tr  
a  tr 
tr a  tr 
 a  tr 


Then:

t
t

1  rˆv  tr  c


© 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
Using:
 1 u  x  
x

Fall Semester, 2015
Lect. Notes 12
Prof. Steven Errede
u 1  x 
u  x  1 u  x 
 1 u 2
 2
x
x
u x
Then:



1

1


v  t r  c 

r
r



2
t   r  r v  tr  c    r  r v  t  c  t
r



1  r 1   

r v  tr  

2 

 r   t c t


1  r 1  
r  v  tr   

  v  t r   r 

2 
t
t  
 r   t c 
r  ctr  c  t  tr 
  t  tr 
r 
But:

t

t
c  t  tr   c
t
tr 1
1
 t t 
 t 
 c   r   c 1  r  where:
 

t  1  rˆv  tr  c
 t t 
 t 



1  rˆv  tr  c  1
rˆv  tr 
r v  t r 
r
 1
  1 
 c 1    c 


Thus:
c
t


r
 
  



0

 





dw  tr  tr
v  tr 
v  tr  dv  tr  tr 
1
r   r  w  tr   r w  tr 

 a  tr 





and:
t
dtr t


t
dtr t
t
t
t
But:
Thus:


 r 1  
r  v  tr   
   v  t r   r 

t
t  
 t c 

 
2
1  r v  tr  1  v  tr  r a  tr   


 


2 

  
c
 r    r


1
1



2
t   r  r v  tr  c  
 r 



1
  r 
2
 
1
ˆ 

2
 r v  tr    v  tr   r a  tr   
c


 

 
Ar  r , t  o  1 v  tr  

1

q

 v  tr  

4   r  t
t
t   r  r v  tr  c   



Then:


 
  1 a  tr  v  tr   ˆ 
1
 
2




v
t
v
t
a
t
r
r
 o q










r
r
r  
2 
c
4   r  
 
  r  
Or:
16
 

 
Ar  r , t 
v  tr   
1  q  1  
1
 
1
2

 a  tr  
 2
 rˆv  tr    v  tr   r a  tr     using o  2
4 o  c    r  
c
t
 r  
c o
 
© 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 12
Prof. Steven Errede
 
Ar  r , t 
Does this expression = Griffiths result for
, eqn 10.63 bottom of page 437 ?? YES, it does!
t
 
 


Ar  r , t 
v 2  tr  v  tr   r a  tr   v  tr  
q  
1




 ra  tr   rˆv  tr  v  tr  

t
c
c
4 o  3 r 2 c 2 



 
Thus:


 



v 2  tr  v  tr   r a  tr   v  tr  



c 1    v  tr    ra  tr  

c
c






v 2  tr  v  tr   r a  tr   v  tr  
q 




 cv  tr    ra  tr   cv  tr  

 3 r 2c 2 
c
c


 
r 2 2
q 



2
r
r


r
 a  t r   v  tr  
v
t





cv
t
a
t
c








r
r
r
3 3 2
 rc 
c

 
r 2 2
q 




r
r
r
 a  tr   v  t r  






c
v
t
a
t
c
c
v
t










r
r
r
3 3 2
 rc 
c

 
Ar  r , t 
1
q

3 2 2
4 o  r c
t










c 1    v  tr   c 1  1  rˆv  tr  c v  tr   rˆv  tr  v  tr 
But:
But:


1
4 o
1
4 o
1
4 o








 rc  1  rˆv  tr  c rc   r  r v  tr  c  c   rc  r v  tr  
 

 
Ar  r , t 
r
q 
1



rc  r v  tr    v  tr   ra  tr  c    c 2  v 2  tr   r a  tr   v  tr  

3 3 2 
c
t
4 o  r c 

Or:
 

 
Ar  r , t 
qc
1
r





rc  r v  tr    v  tr   ra  tr  / c    c 2  v 2  tr   r a  tr   v  tr  

3 

t
c
4 o  rc  r v  t   

r
 Griffiths Equation 10.63 on page 437
Then {“finally”!} the retarded electric field for a moving point electric charge q:
 
 

Ar  r , t 

Er  r , t   Vr  r , t  
t

 1   v  t  2 r a  t  
v  tr  
r
r
 rˆ 
 1  

 
c2 
c 
4 o  r 2     c 

 


 
v  tr   
q
1  1 
1

2

 2   a  tr  
 rˆv  tr    v  tr   r a  tr    
c
4 0   r   c  
r 

 
Er  r , t  
q
1
© 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 12
Prof. Steven Errede


With some more algebra, more vector identities, and defining a retarded vector u  tr   crˆ  v  tr  ,
the retarded electric field for a moving point charge can equivalently be written as:
 

q
r
 c 2  v 2  tr   u  tr   r   u  tr   a  tr   
Er  r , t  

3 


4 o  r u  t  
r
 


 
 
 
 
v  w  tr  
v  tr 
Vr  r , t 

Since: Ar  r , t     w  tr  
with:   w  tr   
then: Ar  r , t   2 Vr  r , t 
c
c
c
The retarded magnetic field of a moving point charge q can be written as:
 
  
 

1  
1




B  r , t     Ar  r , t   2    v  tr  Vr  r , t    2 Vr  r , t    v  tr   v  tr   Vr  r , t  

c
c 

 

 
n.b. The relation on the RHS used Griffiths Product Rule # 7:   f A  f   A  A  f


 


 


We already calculated   v  tr    a  tr   tr {term (3) above, page 11 of these lecture notes}

1
rˆ
rˆ


{see page 14 of these lecture notes}:
and we found that: tr  

c 1  rˆv  tr  c 
c




 

 a  tr   r 
 rˆ 



Hence:   v  tr   a  tr   tr   a  tr        
 where:   1  rˆv  tr  c
 c 
  cr 

Vr  r , t  
q
4 o r


{from page 9} and Vr  r , t  is as given above {on page 14}.


n.b. v  tr   v  tr   0
Thus:


1 1  a  tr   r 


4 o c 2  r   cr 
 
2


v  tr  
q v  tr   1   v  tr   r a  tr  
ˆ
r 

  1  

 
4 o  r 2     c 
c 
c 

 



After some more algebra, and again using the retarded vector: u  tr   crˆ  v  tr 
 
  
Br  r , t     Ar  r , t  
q
the retarded magnetic field of a moving point electric charge q is:
 
 

1 q
r



Br  r , t   
rˆ   c 2  v 2  tr   v  tr    r a  tr   v  tr    r u  tr  a  tr   

3

c 4 o  r u  t  
r
Compare this expression to that for the retarded electric field of a moving point electric charge:
 

q
r
 c 2  v 2  tr   u  tr   r   u  tr   a  tr   
Er  r , t  

3

4 o  r u  t   
r
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 12
Prof. Steven Errede
 
 
The terms in the square brackets have some striking similarities between Br  r , t  vs. Er  r , t  .

Because of the cross product of terms such as rˆ  v  tr  contained in the square brackets of the
 




expression for Br  r , t  , notice that since: u  tr   crˆ  v  tr  or: v  tr   crˆ  u  tr 








rˆ  rˆ  rˆ  u  tr    rˆ  u  tr  , i.e. rˆ  v  tr    rˆ  u  tr  .
Then: rˆ  v  tr   rˆ  crˆ  u  tr   c
0
  
  
  


Then, using the A  B  C  B AC  C A B rule, and rˆ  v  tr    rˆ  u  tr  we see that:



 

 




rˆ    c 2  v 2  tr   v  tr    r a  tr   v  tr    r u  tr  a  tr   


 






2
2
 rˆ     c  v  tr   u  tr    r a  tr   u  tr    r u  tr  a  tr   

 



 r  u  tr a  tr  
 


 rˆ     c 2  v 2  tr   u  tr   r   u  tr   a  tr   
Thus:
 

1 q
r
ˆ
 c 2  v 2  tr   u  tr   r   u  tr   a  tr   
r

Br  r , t  

3


c 4 o  r u  t  
r
But:
 
Er  r , t  
 
r


2
2

r


  u  tr   a  tr   
c
v
t
u
t







r
r
3 

4 o  r u  t  
r
q
 
 




 We again see that: Br  r , t   1c rˆ  Er  r , t  i.e. Br is  Er ; note also that Br is  r where
  
 
r  r  r   tr   r  w  tr  = separation distance vector from retarded source point to field point.

 

The first term in Er  r , t  involving  c 2  v 2  tr   u  tr  falls off / decreases as r 2 . If both the
 
1 q
rˆ
velocity and acceleration are zero, then we obtain the static limit result: E  r  
4 o r 2
 
The first term in Er  r , t  is known as the “Generalized” Coulomb field, a.k.a. velocity field

which describes the macroscopic, collective behavior of virtual photons!!
 
  
The second term in Er  r , t  , involving the triple product r   u  a  falls off / decreases as

1 r  i.e. this term dominates at large separation distances!

 
The second term in Er  r , t  is responsible for radiation – hence the 2nd term is known as the

radiation field. Since it is also proportional to a  tr  it is also known as the acceleration field
which describes the macroscopic, collective behavior of real photons!!

 
The same terminology obviously also applies to the retarded magnetic field Br  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 12
Prof. Steven Errede

The EM force exerted on a point test charge qT at the observation/field point r at the present
time, t due to the retarded electric and magnetic fields associated with a moving point charge q
 
 
 
 
is given by the retarded Lorentz force law: Fr  r , t   qT Er  r , t   qT vT  r , t   Br  r , t  where

 
vT  r , t  = the velocity of the point test charge qT at the observation/field point r at the present
time, t.
 
 
 
 
Fr  r , t   qT Er  r , t   qT vT  r , t   Br  r , t 

 
qqT
r


 c 2  v 2  t r   u  t r   r   u  t r   a  t r   

3

4 o  r u  t   
r

 
vT  t 



  rˆ   c 2  v 2  tr   u  tr   r   u  tr   a  tr    


c


  


 
where: r  r  r   tr   r  w  tr  and: u  tr   crˆ  v  tr 
 
The Lorentz force Fr  r , t  is the net force acting on a point test charge qT moving with
 
velocity vT  r , t  {n.b. which is evaluated at the present time {i.e. the non-retarded} time t }.
 
 
The Lorentz Force Fr  r , t  is due to the retarded electric and magnetic fields Er  r , t  and
 

Br  r , t  associated with the point charge q moving with {its own} retarded velocity v  tr  and

acceleration a  tr  .



 


n.b. The lower-case quantities r  tr   r  w  tr  , v  tr  , u  tr   crˆ  v  tr  and a  tr  are all
evaluated at the retarded time tr  t  r c .
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 12
Prof. Steven Errede
Griffiths Example 10.4:
Calculate the electric and magnetic fields of a point charge q moving with constant velocity.

Constant velocity  a  tr   0 (no acceleration).
r
 c 2  v 2  tr   u  tr   with: u  tr   crˆ  v  tr 

3 


4 o  r u  t  
r
Then:
 
Er  r , t  
Trajectory:



w  tr   v  tr  tr  vtr for constant velocity.
q

 
 
The retarded time: tr  t  r c with: r  tr   r  w  tr   r  v  tr  tr and: r  ctr  c  t  tr 


Since: u  tr   crˆ  v  tr  , then:
ru  tr   r  crˆ  v  tr  
 
 cr  rv  tr 


 

 c  r  r   tr    rv  tr 



 cr  cw  tr   rv  tr 
 

 cr  cv  tr  tr  c  t  tr  v  tr 




 cr  ctr v  tr   ctv  tr   ctr v  tr 
 
 c  r  v  tr  t 
Note also:
 



r u  tr   r  crˆ  v  tr    cr  r v  tr 





It can be shown that, since {here} v  tr   v  constant velocity vector, then: v  tr   v  t   v ,
and thus:
 
r  u  tr  
 c t  rv    c
2
2
2
 v 2  r 2  c 2t 2 
 Rc 1  v 2 sin 2  / c 2
 Rc 1   2 sin 2 
(See p. 8 above, i.e. Griffiths
Example 10.3, p. 433)
(See Griffiths Problem 10.14, p. 434)
where   v c
  
where R  r  vt = vector from the present location of the point electric charged particle to the


observation/field point P  r  t   at the present time t moving with constant velocity v .
© 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 12
Prof. Steven Errede
 


The angle   cos 1 Rv is the opening angle between R and the constant velocity vector v ,


as shown in the figure below:
Present position
of point charge q
@ time t.

R

P R, t Observation/Field Point
 
 
  cos 1  R v 
q

v = constant velocity vector
1  2 
 
 Rˆ 

q
v
,
E
r
t
    = constant



Thus:
3 
2  with:
4 o 1   2 sin 2  2  R 
c


 
 
This expression for E  r , t  shows that E  r , t  points along the line from the present position of

the point charged particle to the observation/field point P R, t – which is strange, since the “EM
 
news” came from the retarded position of the point charge. We will see/learn that the explanation
for this is due to {special} relativity – peek ahead in Physics 436 Lect. Notes 18.5, p. 10-17.

Due to the  2 sin 2  term in the denominator of this expression, the E -field of a fast-moving
point charged particle is flattened/compressed into a “pancake”  to the direction of motion,

increasing the E -field strength in the  direction by a factor of   1 1   2 , whereas in the
forward/backward directions { i.e.  and/or anti-  to the direction of motion} the strength of the


E -field is reduced by a factor of 1   2 relative to that of the E -field strength when the point
electric charge is at rest, as shown in the figure:
 
E r ,t  
q
1   
2
4 o 1   2 sin 2 


3
2
 Rˆ 
 2
R 
v
 
    = constant
c
with:

Lines of E are compressed into a “pancake”  to the direction of motion as v → c
  
 
  
 
r  vtr  r  vt    t  tr  v R v
r

  since tr  t  r c or r  ctr
For B  r , t  we have: rˆ  
r
r
r
r c
Thus:
0
 



 




 




1
1 R v
1
1v   



ˆ
B  r , t   r  E  r , t        E  r , t   
R  E  r , t      E  r , t   n.b. R  E
c
c  r c 
cc
 cr

Or:
22


1  2 
 

q
1v   
1   
B r ,t     E r ,t     E r ,t  
cc
c
4 o c 1   2 sin 2  3 2



   Rˆ 
 2 
 R 
These expressions for E and
B were first obtained by
Oliver Heaviside in 1888.
© 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 12
Prof. Steven Errede

The lines of B circle around the point charge q as shown in the figure:


When v  c (i.e.   1 ) the Heaviside expressions for E and B reduce to:
 
E r ,t  
1
q ˆ
R
4 o R 2
 essentially Coulomb’s law for a point electric charge
 
  q 
B  r , t    o  2 v  Rˆ  essentially Biot-Savart law for a point electric charge
 4  R


The figure below shows a “snapshot-in-time” at time t2  sec  of the classical/macroscopic
electric field lines associated with a point electric charge q, initially at rest {at to  0  sec  , where
the green dot is located}, that undergoes an abrupt, momentary acceleration {i.e. a short impulse
lasting t  t1  to  t1  sec  , where the yellow dot is located} in the horizontal direction, to the
left in the figure. After the impulse has been applied, the charge continues to move to the left with
constant velocity v, at time t2  sec  the charge is where the pink dot is located.
The classical/macroscopic electric field lines associated with one “epoch” in time must
connect to their counterparts in another “epoch” of time. Here, in this situation, the spatial slopes
of the E-field lines are discontinuous due to the abrupt, momentary nature of the acceleration.
The spherical shell associated with the discontinuity(ies) in the electric field lines in the
“transition” region between the two “epochs” expands at the speed of light, as the EM “news”
propagates outward/away from the accelerated charge.
© 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 12
Prof. Steven Errede
Note that this picture also meshes in nicely (and naturally!) with the microscopic perspective –
namely that, when a point charge is accelerated it radiates real photons, which subsequently
propagate away from the electric charge at the speed of light. Real photons have a transverse
electric field relative to their propagation direction (whereas virtual photons associated with the
static/Coulomb field are longitudinally polarized). The spherical shell associated with the
discontinuity(ies) in the electric field lines of the “transition” region is precisely where the real
photons are located in this “snapshot-in-time” picture, having propagated that far out from the
charge after application of the abrupt, momentary impulse-type acceleration of the electric charge.
24
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.