Download Retarded Times and Potentials

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Retarded Times and Potentials
An infinitely long wire has a current in the +y direction that is turned on
abruptly at t  0 . The function I t  describing this current is:
0
I t   
I0
for t  0
for t  0
I 0  const.
The x and y dimensions on the graph below are scaled so that each tick
mark represents the distance light would travel in one time unit. The
wire is parallel to the y-axis, and located at x  3 .
A. Indicate on the graph, where
in space, observers are aware of
a non-zero current in the wire at
time t = 2.

0
B. Remember A  r , t  
4
J  r , t R 
 r  r  d 
Write the integral expression for the vector potential, for the case of this
wire, using y as your variable of integration.
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C. Remember tR  t 
r r '
c
Suppose the time is t = 5, and you are
at the origin, r = 0. Indicate, roughly,
the retarded time for various points
along the wire.
For t = 5 and observer at origin (r =0)
indicate what parts of the wire have
non-zero current. That is, where
2
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D. For an observer at the origin ( r  0 ), indicate on the graph the points
r
in space r  where the retarded time t R is greater than zero at t  4 .
r
r
tR  t 
 0
c
Indicate on the graph the length of wire that contributes to the magnetic
vector potential at the origin at t  4 .

r
A r  0, t   0
4
r
J r , t R 
 rr  d 
4
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E. If the current-carrying wire is electrically neutral, then the electric
field at the origin is given by just the time-derivative of the vector
potential:
r
A
E r0  
r0
t
By taking the time-derivative of the expression from the previous page
for the vector potential at the origin, show that
E
r0

0
4

1  r r
J r , t R d 
r 
r  t R
[Hint: Recall the results from the first page.]
We may write the current density evaluated at the retarded time in
terms of a step function:
r r
r
J r , t R   J 0 r   t R 
where
0
 t R   
1
for t R  0
for t R  0

 t R    t R  , where  t R  is the Dirac
t R
delta function evaluated at the retarded time.
It can also be shown that
Using the information on this page, which points on the currentcarrying wire are contributing to the electric field at the origin ( r  0 )
at any given time after t  3 ?
For an observer at the origin ( r  0 ), for which points in space r  is the
r
current density J r , t R  non-zero at t  2 ? Note that this is the same
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function for the current density as before, but evaluated at the retarded
time t R instead of at t .
A. Because electromagnetic effects propagate at the speed of light c , an
event taking place at the point r  and at the time t  can’t cause an effect
at the point r until enough time t has passed for light to travel the
distance between the two points:
t  t '
r  r '
c
This can be re-written as:
t't
r r '
c