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Potential due to a moving charge
Suppose that a particle carrying a charge moves with uniform
velocity through a frame . Let us evaluate the
vector potential, , and the scalar potential, , due to
this charge at a given event in .
Let us choose coordinates in so that
and
. Let be that frame in the standard configuration
with respect to in which the charge is (permanently) at rest
at (say) the point . In , the potential at is the
usual potential due to a stationary charge,
where
. Let us now transform
these equations directly into the frame . Since
is a contravariant 4vector, its components transform according to the
standard rules (1397)(1400). Thus,



(1520) 



(1521) 



(1522) 



(1523) 
since in this case. It remains to express the quantity
in terms of quantities measured in . The most physically meaningful
way of doing this is to express in terms of retarded values
in . Consider the retarded event at the charge for which, by definition,
and . Using the standard Lorentz transformation,
(1346)(1349), we find that

(1524) 
where
denotes the radial
velocity of the change in . We can now rewrite Eqs. (1520)(1523) in the
form
where the square brackets, as usual, indicate that the enclosed quantities
must be retarded. For a uniformly moving charge, the retardation of
is, of course, superfluous.
However, since

(1527) 
it is clear that the potentials depend only on the (retarded) velocity
of the charge, and not on its acceleration. Consequently, the expressions
(1525) and (1526) give the correct potentials for an arbitrarily moving charge.
They are known as the LiénardWiechert potentials.
Next: Fields due to a
Up: Relativity and electromagnetism
Previous: Transformation of fields
Richard Fitzpatrick
20060202