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Transcript
```UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 11
Prof. Steven Errede
LECTURE NOTES 11
POTENTIALS & FIELDS
 

The Potential Formulation: Scalar and Vector Potentials V  r , t  and A  r , t 
 

How do the instantaneous sources tot  r , t  and J tot  r , t  (total electric charge density and
 
 
total electric current density) generate the electric and magnetic fields E  r , t  and B  r , t  ?
We seek general solutions to the full Maxwell equations:
(1) Gauss’ Law:
  
1




P
 E  r , t   tot  r , t  where: tot  r , t    free  r , t   bnd
r ,t 
(2) No Magnetic Charges:
  
 B  r , t   0
o
 
  
 

P 
M 
B  r , t 

  E r ,t   
where: J tot  r , t   J free  r , t   J bnd  r , t   J bnd  r , t 
t
 
  
 
E  r , t 
  B  r , t   o J tot  r , t   o o
(4) Ampere’s Law:
t
 


If time-dependent tot  r , t  and J tot  r , t  are given/specified at source position(s) r  , what
 
 

are the corresponding EM fields E  r , t  and B  r , t  at the observer position r ?

 
  
tot  r  
1
 For the static case {i.e.  , J , E , B  fcns  t  }: Coulomb’s law E  r  
rˆ d 
4 o v r 2
 
 
o J tot  r    rˆ
 
and the Biot-Savart law B  r  
d  provide the answers. {n.b. r  r  r  ,
2


v
4
r




where r  observer’s position at field point P  r  and r   source position at point S  r   }.
 We seek time-dependent generalizations of Coulomb’s law and the Biot-Savart law.
This is not an easy problem!! The approach we will take is to represent the time-dependent
 
 
E  r , t  and B  r , t  fields in terms of their corresponding time-dependent potentials
 
 


V  r , t  and A  r , t  , which are in turn related to the time-dependent sources tot  r   , J tot  r   .
  
 
 
In electrostatics,   E  r , t   0 enabled us to write: E  r , t   V  r , t  .
 
  
  
B  r , t 
In electrodynamics, we cannot do this because:   E  r , t   0 . {   E  r , t   
}
t
 
  
  
However, in electrodynamics  B  r , t   0 {still}.  B  r , t     A  r , t 
 
where A  r , t  = magnetic vector potential, as we saw in the case of magnetostatics.
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
1
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 11
Prof. Steven Errede

 
  
  A  r, t  
 
  
B  r , t 
   
Insert B  r , t     A  r , t  into Faraday’s Law:   E  r , t   
 t 
t


 
 
  
 
A  r , t  
A  r , t  
Then we see that:    E  r , t  
  0 i.e. the curl of  E  r , t  
 does vanish
t 
t 


 
 
 
A  r , t  

for any/all  r , t  ,  we can define:  E  r , t  
  V  r , t 
t 

  


  



{n.b. If   F  r   0   r  , then: F  r   f  r  since:   f  r   0 always.}
 
 
 
 
 
A  r , t 
Thus: E  r , t   V  r , t  
 n.b. reduces to E  r   V  r  for the electrostatics case.
t
 

 
Since we used  B  0 and   E   B t in deriving the above relation, Maxwell’s equations
 


(2)  B  0 and (3) Faraday’s Law:   E   B t are automatically satisfied. What about
 
 


equation (1) Gauss’ Law:  E   tot  o and (4) Ampere’s Law:   B  o J tot   o o E t ??
(1) Gauss’ Law:
  
1

 E  r , t   tot  r , t 
o
 
 
 
A  r , t 
with: E  r , t   V  r , t  
t
 
   
A  r , t  
   
1


2
 A  r , t   tot  r , t 
=   V  r , t  
   V  r , t  
o
t 
t


Electrodynamics version of the Poisson equation:

   
1


 2V  r , t  
 A  r , t    tot  r , t 
t
o


(4) Ampere’s Law:
 
  
 
E  r , t 
  B  r , t   o J tot  r , t   o o
t
 
 
 
A  r , t 
n.b. These two equations contain all of the information
E  r , t   V  r , t  
with:
that is contained in the four Maxwell equations !!!
t
 
  
B r ,t     Ar ,t 
and:
 

   
 
  V  r , t  
2 A r , t 
    A  r , t   o J tot  r , t   o o  
   o o
t 2
 t 

  
  
but:     A    A   2 A






 

 
 2 
2 A r , t       
V  r , t  







A
r
t
J



,


  A  r , t   o o




o
o
o
tot  r , t 
t 2 
t 


2
n.b. this relation is
actually 3 separate
eqns. for x, y and z!
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 11
Prof. Steven Errede
These two equations may seem “ugly” at this point in time, but watch what we can do with them:


 2V  r , t 
 2V  r , t 
a.) Add: 0   o o
to the first equation, and group terms as follows:
  o o
t 2
t 2


 2 
 2V  r , t       
V  r , t  
1

1 eqn:   V  r , t    o o
    A  r , t    o o
   tot  r , t 
2
o
t
t 

 t 



 
 2 
2 A r , t       
V  r , t  
nd










A
r
t
J
,
2 eqn:   A  r , t    o o





o o
o tot  r , t 
t 2 
t 


st

V  r , t  
  

b.) Define: L  r , t     A  r , t    o o
.
t 

c.) Define the D’Alembertian (aka “Box2”) operator:
2   2   o o
2
1 2
2



t 2
c 2 t 2
Then:

L  r , t 
1


  tot  r , t 
1 eqn:  V  r , t  
t
o


 


2nd eqn: 2 A  r , t   L  r , t    o J tot  r , t 
2
st
d.) Divide 1st eqn. by c, then use  o o  1 c 2 relation:


1 
1 L  r , t 


  o tot  r , t    o c tot  r , t 
1 eqn:  V  r , t  
c
c t
c o o


 


2nd eqn: 2 A  r , t   L  r , t    o J tot  r , t 
2
st
Thus:

1 
1 L  r , t 

  o c tot  r , t  “time-like”
1 eqn:  V  r , t  
c
c t

 
 

2
2nd eqn:  A  r , t   L  r , t    o J tot  r , t  “space-like”
st
Do you see/can you see any interesting
parallels between these two equations???
If you can, it’s in fact not coincidental!!!
2
We shall see later on in the semester (when we get to relativistic electrodynamics), that
relativistic four vectors exist {valid in any inertial reference frame}. The relativistic 4-vector
 


potential: A  r , t   V  r , t  c , A  r , t  {where   0,1, 2,3 ;   0 is the temporal component


of the relativistic 4-vector, and   1, 2,3 are e.g. the x, y, z spatial components of the 4-vector}.
 

 
The relativistic 4-current density: J tot
 r , t   ctot  r , t  , J tot  r , t  .



 1  

,   , whereas

x   c t 

 1  
 
,  .
the contravariant relativistic 4-gradient operator:   
x  c t 
The covariant relativistic 4-gradient operator:   
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
3
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 11
Prof. Steven Errede
The D’Alembertian {aka “Box2”} operator can be written in relativistic 4-vector notation as:
1 2
 
 v
 vv
2
2
x xv
c t


 1 V  r , t      A  r , t 


  A  r , t   
   A  r , t 
We also can write: L  r , t    2

t
x
c

2   2 
We can also write:


L  r , t 
1 L  r , t   


 1   
 L  r , t     
   L r ,t   
   L  r , t      v Av  r , t 
c t
x
 c t

Thus, the two equations:

1 
1 L  r , t 

  o c tot  r , t 
1 eqn:  V  r , t  
c
c t
 
 
 
2
nd
2 eqn:  A  r , t   L  r , t    o J tot  r , t 
2
st
can be written as a single equation, very compactly (& elegantly!) in relativistic electrodynamics as:


 
 v  v A  r , t      v Av  r , t    o J tot
r,t 
Very shortly, we will learn that as a consequence of the gauge-invariant nature of the
electromagnetic interaction, we can choose to work in the so-called Lorenz gauge, namely that:


 1 V  r , t      Av  r , t 


L r,t    2
  A  r , t   
  v Av  r , t   0
v
t
x
c

 




v 
If L  r , t    v A  r , t   0   r , t  , then separately both L  r , t  t  0 and L  r , t   0   r , t  .
Hence:


L  r , t 
1 L  r , t   


 1   
 L  r , t     
   L r ,t   
   L  r , t      v Av  r , t   0   r, t  .
x
c t
 c t


v 
Thus, in the Lorenz gauge   v A  r , t   0 , and hence our single equation (above) becomes:



 
 v  v A  r , t    o J tot
 r , t  or equivalently: 2 A  r , t    o J tot  r , t 
This single relativistic 4-potential equation (actually 4 separate equations, since   0,1, 2,3 )
contain(s) all of the information that is contained in the four Maxwell field equations !!!
 
  
  
B  r , t 
1

(1) Gauss’ Law:  E  r , t   tot  r , t  (3) Faraday’s Law:   E  r , t   
o
t
 
  
 
  
E  r , t 
(2) No Magnetic Charges:  B  r , t   0 (4) Ampere’s Law:   B  r , t   o J tot  r , t   o o
t
4
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 11
Prof. Steven Errede
Gauge Transformations


 
Using  v  v A  r , t      v Av  r , t    o J tot
 r , t  enables us to reduce the problem of finding


the six components associated with the two vectors E  Ex xˆ  E y yˆ  Ez zˆ and B  Bx xˆ  By yˆ  Bz zˆ

down to four components – the scalar potential V and the vector potential A  Ax xˆ  Ay yˆ  Az zˆ .
 
 
  
 
 
As we saw last semester in P435, B  r , t     A  r , t  and E  r , t   V  r , t   A  r , t  t

do not enable us to uniquely define / specify / determine the scalar and vector potentials V  r , t 
 
 
and A  r , t  ; only potential differences V2 – V1 and A2  A1 are physically meaningful…
 
 
 

We are free to impose extra conditions on V  r , t  and A  r , t  as long as E  r , t  and B  r , t 
remain unchanged by the imposition of these extra conditions…
 

This freedom to impose extra/additional conditions on V  r , t  and A  r , t  without changing
 
 
E  r , t  and B  r , t  is known as gauge freedom or (more formally) as gauge invariance.
 
 


Suppose we have e.g. two sets of potentials V  r , t  , A  r , t  and V   r , t  , A  r , t  that
 
 
correspond to the same physical fields E  r , t  and B  r , t  . However, these two sets of

potentials must be related to each other:
 
 
 
A  r , t   A  r , t     r , t 
and:





V r ,t   V r ,t    r ,t 
Because:
 
 
  
  

 
B  r , t     A  r , t     A  r , t     A  r , t     r , t 
But if:
     r, t    0   r, t  , then since:    f  r, t    0






  
    r , t   0
{always}
 
 
 

 We can always write   r , t  as the gradient of a scalar function   r , t  :   r , t     r , t  .
 
 
 
A  r , t 
E  r , t   V  r , t  
t
Then:
 
 
 

 
 
A  r , t 
A  r , t    r , t 

 V   r , t  
 V  r , t     r , t  

t
t
t
 
 
  r , t 
 

0
The scalar function   r , t  must be related to   r , t  by:   r , t  
t
 

   r , t 
 
 
 
  r , t  
 
  r , t     r , t      r , t  
 0     r ,t  
But:
0
t 
t


  r , t  
 

which must hold for arbitrary/any/all space-time points  r , t  
  r ,t  
 0.
t 



© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
5
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 11
Prof. Steven Errede

 
  r , t  
Note that     r , t  
  0 can also be satisfied if:
t 


  r , t  
 
  r ,t  
   t 
t 


  r , t 

  t  .
i.e. the scalar fcn   t  depends only on time, t. Then we see that:   r , t   
t

However, we can always “absorb”   t  into   r , t  by adding

t t
t  0

  t   dt  to   r , t  ,
t t




i.e.    r , t     r , t       t  dt  . We can then redefine the “new”    r , t  .  “old”   r , t  .
t 0
Note also that since the scalar function   t  depends only on time t, this will not affect the
 
 
 
gradient of   r , t  in any way, and hence   r , t     r , t  is completely unaffected by this!
 
 
 
 
 
 
 
 

Thus: A  r , t   A  r , t     r , t   A  r , t     r , t  or:   r , t   A  r , t   A  r , t   A  r , t 

  r , t 


And: V   r , t   V  r , t  
t

  r , t 



 V   r , t   V  r , t   V  r , t 
or: 
t
 
 

Hence, for any scalar function   r , t  , we can always add   r , t  to A  r , t  provided that


we simultaneously subtract   r , t  t from V  r , t  .
 
 
 This “prescription” will leave the E  r , t  and B  r , t  -fields unchanged / invariant under
this so-called gauge transformation!


  
Gauge transformations can be exploited to adjust  A  r , t  .
  
In magnetostatics, we chose  A  r , t   0 ( = the Coulomb gauge).

In electrodynamics, the situation is not always so clear cut!! The most convenient “gauge”
choice depends on the detailed nature of the problem!  Many gauges to work with….
  
Coulomb Gauge:  A  r , t   0 (Most useful in electro/magnetostatics)
The two most
popular gauges

  
V  r , t 
for use in E&M
  A  r , t     o o
Lorenz Gauge:
(Most useful in electrodynamics)
t
Ludwig Valentin Lorenz, Danish physicist – a contemporary of J.C. Maxwell, ca. 1867 – not to
be confused with Hendrick A. Lorentz, Dutch physicist & contemporary of Albert Einstein…
{See/read J.D. Jackson & L.B. Okun’s “Historical Roots of Gauge Invariance” Rev. Mod. Phys.
73, 663 (2001).
On the web at: http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.73.663
And also: http://arxiv.org/vc/hep-ph/papers/0012/0012061v2.pdf}.
6
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
UIUC Physics 436 EM Fields & Sources II

Fall Semester, 2015
Lect. Notes 11
Prof. Steven Errede
Gauge Transformation(s) are intimately connected to the choice of an {inertial} reference frame.
“Hints” of this connection, in the context of special relativity – length contraction and time 
dilation FX between two {inertial} reference frames – are the spatial gradient   r , t  and
t  t



temporal “gradient”    r , t  t terms {where    r , t     r , t      t   dt  } that can be
t  0
 

added to {subtracted from} the vector potential A  r , t  {scalar potential V  r , t  }, respectively.
  
The Coulomb Gauge:  A  r , t   0

Here, the scalar potential V  r , t  is “easy” to calculate, but in comparison, the vector potential
 

A  r , t  is “difficult” to calculate. The inhomogeneous differential equation for V  r , t  is:


1
1
   




 2V  r , t  
 A  r , t    tot  r , t    2V  r , t    tot  r , t 
o
o
t
The “usual” 3-D
Poisson’s equation
1




Then for:  2V  r , t    tot  r , t  if we can set: V  r  , t   0 {i.e. tot  r , t  is a local chg. dist’n}
o

the solution to 3-D Poisson’s equation is: V  r , t  
1
4 o


tot  r , t 
v
r
d 
  
with: r  r  r  and:

r  r  r 2  r 2
 
 

But V  r , t  alone does not determine E  r , t  in electrodynamics – we must also know A  r , t 
 
as well! In the Coulomb gauge, the differential equation for the vector potential A  r , t  is:
 

 
 
2 A r , t      
V  r , t  
 A  r , t   o o
     A  r , t    o o
   o J tot  r , t 
2
t
t





 
 
  V  r , t  
2 A r , t 
n.b. Also depends
 2 A  r , t    o o
  o J tot  r , t   o o  
 
2
on V(r,t)!
t

t


2
becomes:

In the Coulomb Gauge, the scalar potential at time t, V  r , t  is determined by the distribution of
electric charge at the “right now” time t – which is acausal, because EM signals/information cannot
propagate faster than the speed of light c! However, changes in the electric charge density distribution



tot  r   at the source point r  take a finite time to be observed at the observation point r !!!

V r ,t  
1
4 o

v

tot  r , t 
r
d 

Thus, in the Coulomb Gauge, the scalar potential V  r , t  instantaneously reflects all changes


in tot  r , t  . However, recall that V  r , t  by itself is not a physically measurable quantity – only





  t   V  r , t   V  r , t  or V  r , t   V  r , t   V  r , t  are
potential differences such as V21
2
1
1
1 2
1 1
physically measurable quantities!
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
7
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 11
Prof. Steven Errede
 
An astronaut standing on the surface of the moon can only e.g. directly measure E  r , t  ,
 
 
 
 

which manifestly involves both V  r , t  and A  r , t  : E  r , t   V  r , t   A  r , t  t , thus in

the Coulomb gauge, while the scalar potential V  r , t  instantaneously reflects all changes in
 
 

tot  r , t  , the combination V  r , t   A  r , t  t does not have this instantaneous, “right 
 
now” behavior in the Coulomb gauge, i.e. the combination V  r , t   A  r , t  t is causal in
 
the Coulomb Gauge, and thus E  r , t  changes in a causally-connected manner only after EM
“news” / information arrives at the appropriate / causally-related time interval t , as a direct
consequence of the propagation speed (= c in free space/vacuum) of this EM “news” /
information. Thus, in the Coulomb gauge, the causal behavior is carried by / encrypted into the
 
 
vector potential A  r , t  , in satisfying the above differential equation for A  r , t  !

  
V  r , t 
The Lorenz Gauge:  A  r , t    o o
t
Here we obtain:


  
V  r , t 

L  r , t     A  r , t   o o
0
t

L  r , t 
1


1


2
  tot  r , t    V  r , t    tot  r , t  and:
(a)  V  r , t  
o
t
o
2
 
 
1 2
2
2
  A  r , t    o J tot  r , t  where:     2 2 .
c t
 
 


Thus, we see from 2 V  r , t    tot  r , t   o and 2 A  r , t    o J tot  r , t  that the

Lorenz gauge puts the {time-like} scalar potential V  r , t  and the {space-like} vector potential
 
 

A  r , t  on an equal footing! (i.e. a “democratic” treatment of V  r , t  and A  r , t  ).
 
 
 
(b)  A  r , t   L  r , t    o J tot  r , t 
2
2
 

In special relativity, V  r , t  c and A  r , t  respectively are the temporal and spatial
 



components of the relativistic 4-vector potential A  r , t   V  r , t  c , A  r , t  , and c tot  r , t 
 
and J tot  r , t  respectively are the temporal and spatial components the relativistic 4-vector
 

 
current density J tot
 r , t   c tot  r , t  , J tot  r , t  , where the index   0 : 3  {0,1, 2,3} .







 
In tensor notation: 2 A  r , t    o J tot  r , t  or equivalently:  v  v A  r , t    o J tot
r ,t  .


1 2
 
2



.
xv x v
c 2 t 2
This is the 4-D {space-time} Poisson eqn (aka inhomogeneous wave eqn) in the Lorenz gauge!

 c tot , J totx , J tot y , J totz
where: A  V c , Ax , Ay , Az  and J tot
and 2 
n.b. Convention: repeated indices (here = v) are summed over (i.e. summed over v = 0:3).
n.b. Griffiths claims that he will use the Lorenz gauge for the remainder of his book {we will also!}.
The whole of electrodynamics thus reduces to solving the {inhomogeneous} 4-D Poisson’s



 
equation: 2 A  r , t    o J tot  r , t  or:  v  v A  r , t    o J tot
 r , t  for specified sources!
8
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 11
Prof. Steven Errede
Griffiths Example 10.1
 

Find the electric charge and current density distributions tot  r , t  and J tot  r , t  that produce:

V r ,t   0
2
Az  x, t 
1
4  o kct
2
 o k
@t  t
 
ct  x  zˆ for x  ct
where k = constant


A  r , t    4c
x
and c  1  o o
 0 for x  ct

ct

ct

@t  0
2
2
2
Note that the parabola:  ct  x    ct   2ct x  x  0 at ct  x .
At t = 0: Az  x, t  0  
o k
x but this is also subject to x  ct , hence: Az  x, t  0   0  x .
2
4c
2
 o k
ct  x  , for x  ct

k
1
2

For t > 0: Az  x, t  0    4c
and: Az  x  0, t  0   o  ct   o kct 2
4c
4
 0, for x  ct





Since V  r , t   0   r , t  , we know that tot  r , t   0   r , t  .
 
However, A  r , t   zˆ tells us that there must be some kind of current present, and in the ẑ -direction.
In the process of solving this problem, we will determine what kind of current is present…


Note that A  Az zˆ i.e. note that A is an explicit fcn(x) only, and points (only) in the ẑ -direction.
 
 
 
First, we determine E  r , t  and B  r , t  from A  r , t  :
 
 
 
A  r , t 
k
  o  ct  x  zˆ for x  ct , and: E  r , t   0 for x  ct .
E r ,t   
t
2
0 
0 
 0
 0
 0

   Az Ay 
A

A

A 

A

A

y
xˆ   x  z  yˆ  
 x  zˆ   z yˆ since Ax  Ay  0

And:   A  

 z
x
y 
z 
x 
 y
 x






Hence:
For x > 0
 
  

2
k 
k
B r ,t     Ar ,t    o
ct  x  yˆ   o  ct  x  yˆ for x  ct , and B  0 for x  ct .

4c x
2c
For x < 0
B y  x, t 
E z  x, t 
@t  0
ct
@t  0
ct
x
 12 o kct
ct
@t  t
 
 
n.b. For x  ct , E  r , t  and B  r , t   0
@t  t
 o kt
 12 o kt
@t  t
ct
@t  0
x
1
2

n.b. By  x, t  has a discontinuity for t > 0 at x = 0 !!


A discontinuity in B   a free surface current K free !!
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
9
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 11
Prof. Steven Errede
  
  
  
It can be shown that indeed  E  r , t   0 ,  B  r , t   0 {and  A  r , t   0 }
{n.b. please explicitly check/work these out yourselves!}, and that indeed:
  
k
  E  r , t    o yˆ
2
 
B  r , t 
k
  o yˆ
2
t


  
k
  B  r , t    o zˆ
2c
 
E  r , t 
 kc
  o zˆ
t
2
Maxwell’s eqn’s
all satisfied, with

tot  r , t   0 and
 
J tot  r , t   0 .
 
However, we have seen before {déjà vu!} that the discontinuity in B  r , t  @ x = 0 for t  0


heralds/signals the presence of a free surface current K free  r , t  .
 


In our case {here}, the free surface current K free  r , t  lies in the y-z plane {n.b. A  K free  zˆ !}
The boundary condition associated with a free surface current (see P435 Lect. Notes 24, p. 13-17
and/or Griffiths equation 7.63 (iv), p. 333) is:
H1  H 2 
1
1
B1 
1
2

B2  K free  nˆ
1  2  o {here}
with:
Region 1: x < 0
Region 2: x > 0

 B1  B2  o K free  nˆ at x  0
i.e.: By  x  0, t 
x 0
 By  x  0, t 
x 0
 xˆ  nˆ
ẑ
x 0
n.b. n̂ points from medium 2 → 1 ( nˆ   xˆ )
See Griffiths p. 331-2, figs. 7.46 and 7.47
medium 1 (x < 0)

K free

K free
xˆ   nˆ

 o K free  nˆ


ŷ  n.b. K free  r , t  lies in the y-z plane
medium 2 (x > 0)  
o kt
2
yˆ 
o kt
2

yˆ   o K free  xˆ


But: xˆ  yˆ  zˆ , yˆ  zˆ  xˆ , ẑ  xˆ  yˆ  ktyˆ  K free  xˆ  K free  t   ktzˆ at x = 0 in y-z plane.

Physically, this corresponds to a uniform, but time-dependent free surface current K free  t   ktzˆ
which flows in the  ẑ direction at x = 0 in the y-z plane. It starts up from zero at time t = 0, and

its strength {magnitude} K free  t   kt increases linearly with time t. The EM “news” travels


outward at speed c, note that E and B are still zero for points x  ct !!!
 
 
 
Note also that Ar  r , t  , Er  r , t  , Br  r , t    for x  ct as t   because we have an {  }

extended (not finite) free surface current source K free  t   ktzˆ in this problem…
10
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 11
Prof. Steven Errede
Continuous Electric Charge and Current Density Distributions
We derived {see above} the following relations for the potentials in the Lorenz gauge:

  
V  r , t 



L  r , t     A  r , t   o o
 0 2 A  r , t    o J tot  r , t   4-D Poisson equation
t
 





 
with: A  r , t   V  r , t  c , A  r , t   V  r , t  c , Ax  r , t  , Ay  r , t  , Az  r , t  
 





 
and: J tot
 r , t   c tot  r , t  , J tot  r , t   c tot  r , t  , J totx  r , t  , J tot y  r , t  , J totz  r , t 


1


2 V  r , t    tot  r , t 
Or:
o

 
where: 2 
 
 
2 A  r , t    o J tot  r , t 

 
1 2
2
2
2








o o
x x
t 2
c 2 t 2

 2V  r , t 
1


 V  r , t    o o
  tot  r , t 
o
t

2
 
 
 V r ,t 
 2 A  r , t    o o
  o J tot  r , t 
t
2
Or:
For static situations (no time dependence) these reduce to the familiar “3-D” Poisson equation:

tot  r  
1
1



2
V r  
d 
 V  r    tot  r 
 solution is:
o
4 o v r
 



J

 r  d 



 2 A  r    o J tot  r 
A  r   o  tot
 solution is:
4 v r
where:
 
r  r  r
  
r  r  r

     
rˆ  r r  r r   r  r   r  r 
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
11
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 11
Prof. Steven Errede
 

Now consider what happens if the sources tot  r , t  and J tot  r , t  in the volume v’ are
time-dependent:

An observer at the field point P  r , t  detects changes in the potentials and/or the EM fields

at the time t. However, those changes observed at the field point P  r , t  result from changes in
 

tot  r , tr  and / or J tot  r , tr  that occurred at earlier time(s) tr  t  r c , because it takes a finite

amount of time for EM “news” (i.e. changes) occurring at a source point S  r , tr  to propagate

to the observation/field point P  r , t  .
In terms of a space-time light-cone diagram (for propagation of EM news in free space = vacuum):
ct  c  t  tr   r
or:
t   t  tr   r c
or:
t  tr  r c
or:
tr  t  r c = retarded time
with:
r  r  t   r   tr 
 

r  r  t   r   tr 
and:
and:
12

The Light-Cone in Space-Time:
c  time
ct


 
rˆ  r r  r r




  r  t   r   tr   r  t   r   tr 
r
Field/Observation

Point P  r , t 
r 2   ct 
ct
 c  t  tr 
2
 r 2  c 2  t  tr 
2
  2
2
  r  r    c 2  t  tr 
ctr
Source

Point S  r , tr 
space
 0, 0 
r
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 11
Prof. Steven Errede


Due to causality, it takes a finite time t  t  tr  r c  r  t   r   tr  c for a change e.g. in


the electric charge density tot  r , tr  at the source point S  r , tr  at the earlier, retarded time tr

to propagate to the observation/field point P  r , t  at the later time, t > tr: t  tr  r c . In free
space (the vacuum) this EM “news” / information propagates with speed c = 3  108 m/sec.

We point out {here, for completeness’ sake} that both the source point S  r , tr  and

observation/field point P  r , t  are at rest (e.g. in the lab frame – an inertial {i.e. a non-accelerating}
reference frame). The electrodynamics of this situation will be different e.g. if the observer is moving
relative to the source, or if both the source and the observer are moving with respect to a chosen
reference frame (e.g. the lab frame). Special relativity deals with these situations…
Thus, for non-static source volume charge density and/or current density distributions
 
 


tot  r , tr  and J tot  r , tr  , the scalar and vector potentials V  r , t  and A  r , t  at the

observation/field point P  r , t  at the later, causal time t  tr  r c (t > tr) are causally related
 


to the sources tot  r , tr  and J tot  r , tr  at the source point(s) S  r , tr  at the earlier, so-called
retarded time, tr  t  r c by the following relations:

Vr  r , t  
Retarded
Scalar and
Vector
Potentials:
1

tot  r , tr 
4 o

 

Ar  r , t   o
4

v
r
 
J tot  r , tr 
v
r
d  with tr  t  r c
d 
and


r  r  t   r   tr 
These expressions for the potentials are known as retarded potentials because changes in the


source volume charge density and/or current density distributions tot  r , tr  and J tot  r , tr  at

source point(s) S  r   occurring at the (earlier) “retarded time”, tr < t, take a time interval
 

t  t  tr  r c  r  r  c to propagate from the source point S  r , tr  to the observation/field



point P  r , t  arriving there at the later, causal time t  tr  r c , where r  r  t   r   tr  .

This is exactly the situation where an observer is looking out into the night sky. Light


  tr  away, arriving on Earth
(= EM radiation) from a star a distance r star  robs  t   rstar
tr  t  r star c .
at time t had to have left the surface of that star at an earlier time:
The transit/propagation time of the light from the star to the Earth is: t  t  tr  r star c .

From our own star (the sun), this time interval is:
t 
r
c

r Earth Sun 1.496 1011 m
c

3 108 m /s
 500 seconds = 8.3 minutes
Thus, we see that causality over astronomical distances is significant, but it is also
important even for laboratory/everyday distance scales.
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
13
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 11
Prof. Steven Errede
In the previous semester’s E&M course (P435) we saw that, for static sources:

 
 
 1

tot  r  

 
E  r   V  r   
d  where: r  r  r  and   fcn  r  only


v
r
4 o

 
  tot  r   
1



E r  
d 
where: tot  r    fcn  r   only




4 o v
 r 

 
tot  r  
1
E r   
rˆ d 
4 o v r 2

 
  
 o J tot  r 
B r     Ar    
d 
4 v r
 
 
o   J tot  r   

B r  
d 
 r 
4 v


 
 
J  r    rˆ

B  r   o  tot 2
d 
4 v
r
 

where: J tot  r    fcn  r   only
However, we cannot simply “generalize” these to the time-dependent case merely by adding t



and tr arguments to the E and B and tot and J tot (field and source) variables respectively!!

i.e.
 
Er  r , t  
1
4 o
 

Br  r , t   o
4

v

v


tot  r , tr 
r2

rˆ d 
 
J tot  r , tr   rˆ
r
2
d 

Nyet !!!
Nyet2 !!!
The reason why these expressions are not correct is simple: The causal connection between
t and tr has not been properly taken into account in the above two formulae: tr  t  r c with


r  r  t   r   tr  .
Properly taking into account this causal connection we need to realize that:


tot  r , tr   tot  r , t  r c 
 
 
J tot  r , tr   J tot  r , t  r c 
i.e.
tot

and J tot are now also implicit functions of
 
r  r  r
tr  t  r c and hence are implicit


because r  r  t   r   tr   ct  c  t  tr  !!
via the causal relation
functions of

r
 
 
Thus, in order to correctly / properly calculate E  r , t  and B  r , t  we need to back up and
calculate these relations much more carefully!!!
14
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 11
Prof. Steven Errede

In calculating e.g. the Laplacian of the retarded scalar potential V  r , t  , it is critical

tot  r , tr 
1


d  depends on r in two places:
to realize that the integrand in Vr  r , t  


v
4 o
r


Explicitly, in the denominator of the integrand, because: r  r  t   r   tr  , and also
 
Implicitly, in the numerator of the integrand, because: tr  t  r c  t  r  r  c .
Since:
1

Vr  r , t  
4 o


Then since   fcn  r  only:

 1

Vr  r , t    
 4 o


v

tot  r , tr 
v
r

tot  r , tr 
r
d  

1
4 o

 
tot  r , t  r  r  c 
v
 
r  r
d 
 
tr  t  r c  t  r  r  c

d  

  tot  r, tr  
1


 d  


v
r
4 o
4 o


1

 
  tot  r , t  r  r  c  
 d 
 
v  


r
r



of the scalar
potential at the
field/observation
point P( r ).
 



But tot  r , tr  is an explicit fcn  r   and also an implicit fcn  r  because tr  t  r c  t  r  r  c .

  tot  r , tr  
1

By the chain rule:
 d  


v
4 o
4 o
r


1
And:
  1 
1 







r
,
t
r
,
t






tot
r
tot
r
   d 
v  r
 r 





tot  r , tr  
tot  r , tr  

tot  r , tr  
t r 
t r
tr
t
 
n.b. In the last step we used the fact that tr  t  r c with r  r  r   fcn  t , tr  (because {here}
the source and the observer are not moving relative to each other – i.e. they are at fixed/stationary
points, e.g. in the lab reference frame), therefore: tr  t and thus:  tr   t {here}.

 r 



1
1
1  
1

What is tr ?? Since   fcn  r  only: tr    t    t  r   r    r  r    rˆ
c
c
c
c
 c
0

  
 1  1
rˆ
where: r   r  r   rˆ and:          2
r  r
r
r
See Appendices
A&B




 rˆ 
tot  r , tr  
tot  r , tr   r 
1



t r 
Thus: tot  r , tr  

   tot  r , tr        tot  r , tr  rˆ
tr
t
c
 c 
 c

  r , tr 

where:  tot  r , tr   tot
t
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
15
UIUC Physics 436 EM Fields & Sources II


Vr  r , t  
Thus:
Fall Semester, 2015
Lect. Notes 11
Prof. Steven Errede
  1 
1 




,
,

r
t

r
t







tot
r
tot
r
   d 
4 o v  r
 r 
1

 1
1
rˆ


but: tot  r , tr     tot  r , tr  rˆ and:      2 {from above}
r
c
r

   tot  r , tr 
 ˆ 
ˆ  tot  r, tr   r2   d 
r
Therefore:


rc
4 o v 
 r  



If we now take the divergence of Vr  r , t  , i.e. the Laplacian of Vr  r , t  :


Vr  r , t  
1
 


 Vr  r , t    Vr  r , t  
2

Using the
chain rule:



   tot  r , tr 
 rˆ  

ˆ


r
t
r



,




 2   d 
r
tot
rc
4 o v 
 r  
1
1
4 o
  rˆ  

 1  rˆ    
  tot  r , tr   tot  r , tr    

v
 c  r 
 r  

  rˆ   
 rˆ  


  2 tot  r , tr   tot  r , tr   2   d 
 r   
 r 

1


Since: tot  r , tr     tot  r , tr  rˆ {from above}
c
 





r  r 
r
1
1









Then: tot  r , tr   tot  r , t    tot  r , t 
   tot  r , tr  r   tot  r , tr  rˆ
c
c 
c
c




 tot  r , tr   2 tot  r , tr 


where: tot  r , tr  
t
t 2

  rˆ    r  1
    2   2
r  r
r
and:
See
Appendix D

and: r  rˆ
See
Appendix A
  rˆ 

3
and:  2   4  r 
r 

See
Appendix C
  
 3  r  = 3-D delta fcn for r  r  r 


 1  tot  r , tr  

3



r
4

r
,
t




 d 



tot
r
v  c 2  r 




tot  r , tr   1
1 2  1

3


 2 2

d
r
,
t
r






 d 

r
v tot 
c t  4 o v
 o 
r




tot  r ,t 

1

 Vr  r , t  
4 o
2
Thus:
Vr  r ,t 
Or:
16

2
1  Vr  r , t 
1



2
 Vr  r , t    Vr  r , t   2
  tot  r , t 
2
o
c
t
2
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 11
Prof. Steven Errede

Thus, we see that the retarded scalar potential Vr  r , t  does indeed satisfy the inhomogeneous
wave equation / the 4-D Poisson’s equation!
 
The same methodology can be carried out for the retarded vector potential Ar  r , t  with the
same results {please work through this yourselves !!!}, such that:

2
1  Vr  r , t 
1



2
 Vr  r , t    Vr  r , t   2
  tot  r , t 
2
o
c
t


2
 
 
 
1  Ar  r , t 
2
2
 Ar  r , t    Ar  r , t   2



J
o tot  r , t 
c
t 2
2
where:

Vr  r , t  
1

tot  r , tr 
4 o

 

Ar  r , t   o
4

v
v
r
 
J tot  r , tr 
r
d 
Retarded potentials associated with
the retarded time: tr  t  r c .
d 
1 2
explicitly involves the second
c 2 t 2
derivative with respect to time,  2 t 2 (i.e. note that it is quadratic {not linearly } dependent in
the time variable t ), therefore the D’Alembertian operator a.) manifestly obeys time-reversal
invariance (t → t) and b.) nor does it distinguish past from future!
Note that because the D’Alembertian operator 2   2 
Thus, there exist equally mathematically-acceptable, but physically unacceptable , acausal
solutions (i.e. ones which violate causality), known as the so-called advanced potentials
(n.b. the above proof(s) are also valid for the advanced potential solutions) where:
Advanced Time: ta  t  r c with ta > t and thus: t  ta  r c .
and:
with:
1

Va  r , t  
4 o

 

Aa  r , t   o
4

v
v

tot  r , ta 
r
 
J tot  r , ta 
r
d 
d 
with the advanced time: ta  t  r c .

2
1  Va  r , t 
1



2
 Va  r , t    Va  r , t   2
  tot  r , t 
2
o
c
t


2
 
 
 
1  Aa  r , t 
2
2
 Aa  r , t    Aa  r , t   2



J
o
tot  r , t 
c
t 2
2
The advanced potentials are entirely consistent with Maxwell’s equations, but violate
causality – because they predict potentials now (at time t) that depend on the charge and current
distributions at a future time ta  t  r c  We do not observe such things in our universe!
{n.b. this has not stopped physicists from seriously looking for such things as tachyons, etc.}
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
17
UIUC Physics 436 EM Fields & Sources II


Fall Semester, 2015
Lect. Notes 11
Prof. Steven Errede
In our universe, direct/empirical observation tells us that electromagnetic influences /
changes / disturbances propagate with time going forward, not going backward in time
– i.e. the universe that we live in behaves causally.
The macroscopic theory of electrodynamics must be manifestly time-reversal invariant
(i.e. under the operation of time reversal, t → t) because at the microscopic/elementary
particle physics level, the electromagnetic interaction manifestly obeys time-reversal
invariance. This is not a trivial point, because e.g. the microscopic weak interaction violates
time-reversal invariance in certain situations, e.g. the weak decays of neutral strange,
charmed and b-mesons  K 0 K 0 , D 0 D 0 , B 0 B 0 , Bs0 Bs0  !!
Griffiths Example 10.2:
An infinitely long straight wire carries a time-dependent current I(t) = 0 for t < 0, I(t) = Io for t  0.
I(t)
Io
n.b. t = 0 is the time at the wire
 thus: t = 0 is tr = 0 !!!
0
t



Find/determine the resulting E and B fields at an observation point P  r , t  at a radial distance ρ
from the axis of the wire. We choose the current-carrying long wire to lie along the ẑ -axis as
shown in the figure below:
ẑ
Source
Point

S  r , tr 

r   tr 

r  z
x̂
I t 
 

r  r  t   r   tr 
  
r  r  r  r   2  z 2
ŷ

r t 

r 
Field/Observation

Point P  r , t 
n.b. We assume that the ∞-long line current is {always} electrically neutral, hence the retarded
scalar potential Vr   , t   0 everywhere    , t  .

 z  I  tr 
The retarded vector potential is: Ar   , t   o 
dz zˆ where: r   2  z 2
z

4
r
and t  tr  r c , where tr = 0 is the retarded time that the current is switched on.
18
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 11
Prof. Steven Errede
If t  tr  r c and r   2  z 2 , then for times t  r c , from the observer’s perspective, at the

position  r  ˆ  the current I(t) has not yet been switched on. For t  r c , only the segment
z 
 ct 
2
  2 contributes (because outside of this range the retarded time tr < 0, so I(tr < 0) = 0).
Thus, for observer times t  r c (i.e. tr  0 ) the retarded vector potential is:


Ar   , t   o
4
z 
 ct 2   2
I o dz
z 
 ct 2   2
 z

2

I
 o o ln
2
2
 z z
2


o I o  ct 
ln
Ar   , t  

2

 ct 
2


2
z
zˆ 
z 
o I o
 2
0
4
 ct 2   2
dz
 2  z2
zˆ
 ct 2   2
zˆ
z 0
 2 
 zˆ  n.b. A   , t  has no explicit z or φ-dependence.
r


1 u  x 

ln u  x  
, after carrying out the needed differentiation(s) and ensuing
x
u  x  x

algebra, the retarded electric and magnetic fields at the observer’s position P  r  ˆ  for times
Noting that:
t  r c (i.e. tr  0 ) are:


A   , t 
o I o c
Er   , t   

zˆ
2
2
t
2  ct   
and:

(n.b. Er   , t  is anti-║ I(t))
0 
0

 0


 

A
A Az 

A
1
1
 


z
ˆ
ˆ
Br   , t     Ar   , t   




 z
 

z 
 




Only surviving term:

Az   , t 
I
ˆ  o o
Thus: Br   , t   
2

ct
 ct    2
2
0 

0 

A 



A
ẑ





 
 




ˆ

Note that Ar   , t    (logarithmically) as t   because we have an {  } extended (not finite)
source in this problem… However, note also that as t → ∞:

Er   , t     0


These are E and B fields associated with a steady

I
current Io flowing down wire – i.e. the static fields!!
Br   , t     o o ˆ
2
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
19
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 11
Prof. Steven Errede
Jefimenko’s Equations:
Time-Dependent Generalization of Coulomb’s Law and the Biot-Savart Law
Given the retarded potentials:

Vr  r , t  
1
4 o
 

Ar  r , t   o
4


tot  r , tr 
r
v

 
J tot  r , tr 
r
v
d 
where: t  tr 
d 
and:
r
c


r  r  t   r   tr 
We can determine the corresponding retarded electric and magnetic fields from:
 
 

Ar  r , t 

Er  r , t   Vr  r , t  
t
 
  
Br  r , t     Ar  r , t 



However, the (gory) micro-details of obtaining Er and Br from Vr and Ar are not completely
trivial, and must be done/carried out with great care/attention to detail…


We have previously/already calculated Vr  r , t  {on pages 14-16 of these lecture notes}:

1

Vr  r , t  
4 o

  tot  r , tr 
 rˆ  
r

v  r c rˆ  tot  r , tr   r 2   d  (p. 16 at top) with tr  t  c


 
Ar  r , t 
Calculating
is easy {assuming  no relative motion of source vs. observer}:
t
 
Ar  r , t    o
 
t
t  4
 
J tot  r , tr 

v
r
 
d    o
 4

 

1 J tot  r , tr 
v r t d   4o

v
 
J tot  r , tr 
r
d 
 
  
r
J  r , tr  with tr  t 
where: J  r , tr  
t
c


Thus, the Time-Dependent Generalization of Coulomb’s Law is:
 
Er  r , t  
 

   r, t 
 tot  r , tr 
J tot  r , tr  
tot
r

 d 
rˆ 
rˆ 
2

r
r
r
4 o v 
c
c2


1
with: c 2 
1
 o o
in free
space

Note that in the static limit {   J  0 }, this expression reduces to the familiar form:
 
E r  
20
1
4 o

v

tot  r  
r
2
 
d  with r  r  r  .
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
 
  
 

Let’s work on obtaining Br  r , t  : Br  r , t     Ar  r , t   o
4

 

 
From Griffiths Product Rule # 7:   fA  f   A  A f

 

Br  r , t   o
4
Lect. Notes 11
Prof. Steven Errede

  J tot  r, tr  
v    r  d 


  
 
  1 
1   
  J tot  r , tr   J tot  r , tr       d 

v r
 r 

 
Let’s look at just a single component of the curl of J tot  r , tr  :
Then:

  
  J tot  r , tr 

x


J totz  r , tr 
And:
y

J tot y  r , tr 
And:
z
  
   J tot  r , tr 


x

r  rˆ
But:

J totz  r , tr 
y





J tot y  r , tr 
z


J z  r , tr  J z  r , tr 
tr
1 
r








 J totz  r , tr 
  J totz  r , tr 

where: J z  r , tr  
c
y
y
tr
t
r
t
1
r


 Jtot y  r , tr  r   Jtot y  r , tr 
since: tr  t 
z
z
c
c

1
r
r  1  


   Jtotz  r , tr   Jtot y  r , tr     J  r , tr   r 
x
c
y
z  c 
 
Only if there is
no relative
motion
between source
& observer!
See Appendix A
  
1  
   J tot  r , tr   J tot  r , tr   rˆ
c
Now:

 1  1
rˆ
rˆ
          2   2
r  r
r
r
r  r
 

Br  r , t   o
4
See Appendix B


 J  r, t   rˆ J  r, t   rˆ 
tot
r
r
 d 
 tot
2
v 

c
r
r


Thus, the Time-Dependent Generalization of the Biot-Savart Law is:


 J  r, t  J  r, t  
r


tot
r
tot
r
v  r 2  rc   rˆ d  with r  r  t   r   tr  and tr  t  c



Note again that in the static limit { J  0 }, this expression also reduces to the familiar form:
 
 

 
o J tot  r    rˆ
 
B r  
d  with r  r  r  and: rˆ  r r  r r .
2


v
4
r
 

Br  r , t   o
4
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
21
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 11
Prof. Steven Errede
The retarded electric and magnetic field relations:
 

   r, t 
 tot  r , tr 
J tot  r , tr  


tot
r
v  r 2 rˆ  rc rˆ  rc 2  d  with: r  r  t   r   tr 


 
 
 J  r , t  J  r , t  
 

 

r
  rˆ d  and: tr  t  r and: rˆ  r r  r r .
Br  r , t   o   tot 2 r  tot
r
rc 
4 v 
c

 
1
Er  r , t  
4 o
are known as Jefimenko’s equations (in honor of Oleg Jefimenko, who first worked these out in
1966 – n.b. he also has recently written some new E&M books – Google these, if interested!)
 
 
We can use Jefimenko’s equations for retarded Er  r , t  and Br  r , t  to obtain specializations
 
of these formulae for a point electric charge q moving with retarded velocity v  r   tr   .



Let:   r   tr    q 3  r   tr   where r   tr  = instantaneous position of the electric charge q
 

 

J  r   t r      r   t r   v  r   tr  
at the source point r   tr  at the retarded time tr .

 
 q 3  r   tr   v  r   tr  
It can be shown {n.b. after much work!} for a moving point electric charge q:

 
1   rˆ  1   v  tr   
q  rˆ
Er  r , t  


 


4 o   r 2 c t   r  c 2 t   r  


 
o q  v  tr   rˆ 1   v  tr   rˆ  
Br  r , t  


  with:

4   r 2
c t   r  
tr  t 
r
c
t


r  t   r   tr 
c

 



where:   1  v  tr  rˆ c = retardation factor, with: r  r  t   r   tr  and: rˆ  r r  r r .





Due to the explicit r   tr  time-dependence associated with the moving charge q, {e.g. r   tr   v  tr  tr }
we must be very careful in evaluating the time derivatives! The results (after much additional careful
 
work) are Richard Feynman’s expression for the retarded electric field Er  r , t  and Oliver Heaviside’s
 
expression for the retarded magnetic field Br  r , t  associated with a moving point charge q:


 
r  t   r   tr 
q  rˆ r   rˆ  1  2 rˆ 
r
Er  r , t  
with: tr  t   t 
 

 
4 o  r 2 c t  r 2  c 2 t 2 
c
c
 
 
 

 
o q  v  r , tr   rˆ 1   v  r , tr   rˆ  


Br  r , t  


  with: r  r  t   r   tr  and: rˆ  r r  r r

2 2

rc t 
4   r
 



where:   1  v  tr  rˆ c = retardation factor.
22
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 11
Prof. Steven Errede
In the static limit, we (again) see that Feynman’s expression for the retarded electric field
associated with a moving point charge q:
 
Er  r , t  

 


q  rˆ r   rˆ  1  2 rˆ 
ˆ

r

r
t

r
t
r

r
r

r
r

with
and




 2

 
r
c t  r 2  c 2 t 2 
4 o  r


reduces to the familiar form of Coulomb’s Law:
 
q rˆ
E r  
4 o r 2
In the quasi-static/non-relativistic limit (i.e. v  c ), we also see that Heaviside’s expression
for the retarded magnetic field associated with a moving point charge q:
 
 
 
o q  v  r , tr   rˆ 1   v  r , tr   rˆ  



Br  r , t  

  with r  r  t   r   tr 

2 2

4   r
rc t 
 


 
 
and rˆ  r r  r r and retardation factor   1  v  r , tr  rˆ c


also reduces to the familiar Lorentz formula:
  
B r    o
 4
 
 qv  r   rˆ
n.b. for v  c , the retardation factor   1

r2

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
23
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 11
Prof. Steven Errede
Appendices:
Appendix A:

Show: r  rˆ

 
 



where: r  r  t   r   tr  , r  r  t   r   tr  and: rˆ  r r  r r


In Cartesian coordinates: r  r  t   r   tr  
 x  x    y  y     z  z  
2
2
2
Thus:

 

 
2
2
2
r   xˆ 
yˆ  zˆ   x  x    y  y    z  z  
y
z 
 x
1
 2  x  x  xˆ   y  y  yˆ   z  z  zˆ   xxˆ  yyˆ  zzˆ 

 2
2
2
2
x 2  y 2  y 2



x  x   y  y  z  z 
But:
 

r  r  t   r   tr    x  x  xˆ   y  y  yˆ   z  z   zˆ  xxˆ  yyˆ  zzˆ
And:
r  r  t   r   tr    x  x    y  y    z  z    x 2  y 2  y 2


2
2
2


r rrˆ ˆ
r
Thus: r  
r
r
Appendix B:
 1
rˆ
Show:      2
r
r
In Cartesian coordinates:
 1  

 
     xˆ 
yˆ  zˆ 
y
z 
 r   x

1
 x  x    y  y     z  z  
 12  2  x  x  xˆ   y  y  yˆ   z  z  zˆ 
 xxˆ  yyˆ  zzˆ 
2
 x  x  2   y  y   2   z  z   2 



r
rrˆ
rˆ
 3  3  2
r
r
r
2
32
2

2
2
2
 x  y  y 
32
 1
rˆ
Thus:      2
r
r
24
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 11
Prof. Steven Errede
Appendix C:
Following on from the result obtained in Appendix B above, note that in fact:
  rˆ 

1   1   r 
 2          2    2   4  3  r 
r
r
 r 
r 
From the divergence theorem, we know that:

v
2
  1
 1 
1
 rˆ  2
  d  v    d   s   da   s   2 r d rˆ  4
r
r
r
 r 
Which implies that:

Thus, we also have:

2
v
v
2
1
3 
  d  v 4  r  d  4
r

1
3
  d  v 4  r  d  4
r


 
Indeed, if r  r  , then:



 r 
 1     1    r    rrˆ    r 
          2     2     3    3 
r
r
 r 
 r 
r 
 r 
2
 

   x  x  xˆ   y  y  yˆ   z  z   zˆ 
   xˆ 
yˆ  zˆ  
y
z   x  x 2   y  y 2   z  z  2  3 2
 x


Work on just the x-component:



 r 
x  x 

 

 3    
32
2
2
2
x   x  x    y  y    z  z    
 r x
 



3
 2  x  x 
1


2
 


3
2
5
2
2
2
2
 x  x  2   y  y   2   z  z   2  
  x  x    y  y     z  z   

 


 2



 x  y 2  z 2  3x 2 
 y 2  z 2  2x 2 
 





2
2
2 52
2
2
2 52
  x  y  z 

  x  y  z  
Thus, we see that:

 y 2  z 2  2x 2
 r 
z 2  x 2  2y 2
x 2  y 2  2z 2 

 3    


 0 !!!
52
52
2
2
2 52
 x 2  y 2  z 2 
 x 2  y 2  z 2  
r 
  x  y  z 

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
25
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 11
Prof. Steven Errede
 
However, if r  r  then we see that the denominator in the above expression is also
52
 
simultaneously equal to zero: r 5 2   x 2  y 2  z 2   0 , and thus when r  r 
we actually have:

 1     1    rˆ  0
 2          2    4  3  r 
r
r
 r  0
Appendix D:
  rˆ  1
Show that:    2
r r
In Cartesian coordinates:

  rˆ    r   

   x  x  xˆ   y  y  yˆ   z  z   zˆ 
yˆ  zˆ  
    2    xˆ 
y
z   x  x 2   y  y 2   z  z  2 
 r   x
r


Work on just the x-component:

2


 r 
x  x 

 

 2   
2
2
2 
 r  x x   x  x    y  y    z  z    



1
 x  x  2   y  y   2   z  z   2 



2  x  x 
2
 x  x  2   y  y   2   z  z   2 


2
2
2
2
2
1
2x
x  y  z  2x



2
2
2
2
2
 x  y  z   x 2  y 2  z 2 
 x 2  y 2  z 2 


y 2  z 2  x 2

2
 x 2  y 2  z 2 
2
Thus, we see that:
  rˆ 
y 2  z 2  x 2
z 2  x 2  y 2
x 2  y 2  z 2
  


2
2
2
 x 2  y 2  z 2 
 x 2  y 2  z 2 
 r   x 2  y 2  z 2 
1
1
x 2  y 2  z 2


 2
2
2
2
2
2
2 2
 x  y  z  r
 x  y  z 
Thus:
26
  rˆ  1
   2
r r
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois