Single-Particle Electrodynamics - Assassination Science

ground our understanding of the quantities involved: the case of a stationary
particle. Now, from the discussion of Section 6.1, we know that this example
is trivial: there is no self-interaction at all. But the fundamental quantities
involved in obtaining this null result are, naturally, the zeroth-order terms
of the corresponding quantities when arbitrary motion is considered, and
hence will provide us with a leading-order intuitive understanding of the
complicated algebraic expressions to be considered shortly.
Let us first consider the retarded field experienced by the receiving constituent
r due to the retarded sending constituent r ′ . The relative separation
three-vector from r ′ to r is, of course, simply
r − r ′ .
This will, in the static case, be the quantity R n of the retarded field expressions,
according to the notation of Chapter 5. Let us make the simple
notational definition
R ≡ Rn. (6.5)
Then, from the above, we have
R(r, r ′ )| static
= r − r ′ . (6.6)
Now, the time that it takes for the electromagnetic field to propagate from
the sending constituent r ′ to the receiving constituent r is simply given (in
naturalised units) by R, the distance that the light has to travel. We can
extract R from (6.6) by taking the magnitude of the vector:
R(r, r ′ ) ≡ |R(r, r ′ )| ;
in the static case, (6.6) then yields
R(r, r ′ )| static
= |r − r ′ | ≡ { (r − r ′ ) 2} 1/2
. (6.7)
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