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Let us calculate the electric and magnetic fields observed at position
and time due to a charge whose retarded position and
time are and , respectively. From now on is termed
the field point and is termed the source point.
It is assumed that we are given the retarded position of the charge as
a function of its retarded time: i.e., . The retarded
velocity and acceleration of the charge are

(1594) 
and

(1595) 
respectively. The radius vector is defined to extend
from the retarded position of the charge to the field point,
so that
.
(Note that this is the opposite convention to that adopted in Sects. 10.18
and 10.19). It follows that

(1596) 
The field and the source point variables are connected by the retardation
condition

(1597) 
The potentials generated by the charge are given by the LiénardWiechert
formulae
where
is a function both of
the field point and the source point variables. Recall that the LiénardWiechert potentials are valid for
accelerating, as well as uniformly moving, charges.
The fields and are derived from the potentials
in the usual manner:
However, the components of the gradient operator are partial
derivatives at constant time, , and not at constant time, .
Partial differentiation with respect to the
compares the potentials at neighbouring points at the same time, but these potential signals originate from the charge at different retarded times.
Similarly, the partial derivative with respect to implies
constant , and, hence, refers to the comparison of the potentials at
a given field point over an interval of time during which the retarded
coordinates of the source have changed. Since we only know the time variation
of the particle's retarded position with respect to we must
transform
and
to
expressions involving
and
.
Now, since is assumed to be given as a function of ,
we have

(1602) 
which is a functional relationship between , , and .
Note that

(1603) 
It follows that

(1604) 
where all differentiation is at constant . Thus,

(1605) 
giving

(1606) 
Similarly,

(1607) 
where denotes differentiation with respect to at constant
. It follows that

(1608) 
so that

(1609) 
Equation (1600) yields

(1610) 
or

(1611) 
However,

(1612) 
and

(1613) 
Thus,

(1614) 
which reduces to

(1615) 
Similarly,

(1616) 
or

(1617) 
which reduces to

(1618) 
A comparison of Eqs. (1615) and (1618) yields

(1619) 
Thus, the magnetic field is always perpendicular to and the
retarded radius vector . Note that all terms appearing in
the above formulae are retarded.
The electric field is composed of two separate parts. The first term
in Eq. (1615) varies as for large distances from the charge.
We can think of
as the virtual present radius vector:
i.e., the radius vector directed from the position
the charge would occupy at time if it had continued with
uniform velocity from its retarded position to the field point.
In terms of , the field is simply

(1620) 
We can rewrite the expression (1538) for the electric field generated
by a uniformly moving charge in the form

(1621) 
where is the radius vector directed from the present
position of the charge at time to the field point, and
. For the case of uniform motion,
the relationship between the retarded radius vector and
the actual radius vector is simply

(1622) 
It is straightforward to demonstrate that

(1623) 
in this case. Thus, the electric field generated by a uniformly
moving charge can be written

(1624) 
Since
for the case of a uniformly moving charge,
it is clear that Eq. (1620) is equivalent to the electric field generated
by a uniformly moving charge located at the position the charge would occupy
if it had continued with uniform velocity from its retarded position.
The second term in Eq. (1615),

(1625) 
is of order , and, therefore, represents a radiation field. Similar
considerations hold for the two terms of Eq. (1618).
Next: The Larmor formula
Up: Relativity and electromagnetism
Previous: The electromagnetic energy tensor
Richard Fitzpatrick
20060202