Single-Particle Electrodynamics - Assassination Science

6.5.3 Final expression for the retarded radius vector
We now rewrite (6.47) in terms of the variables r d and r s of Section 6.4:
R = r d − 1 2 R2 ˙v + 1 6 R3¨v − 1 4 R2 (r d· ˙v) ˙v + 1 4 R2 (r s· ˙v) ˙v + · · · + O(ε 6 ). (6.48)
To eliminate R 2 , R 3 , R 4 and R 5 , we firstly square equation (6.48) itself to
get
R 2 = r 2 d − R 2 (r d· ˙v) + 1 4 R4 ˙v 2 + 1 3 R3 (r d·¨v) − 1 2 R2 (r d· ˙v) 2
+ 1 2 R2 (r d· ˙v)(r s· ˙v) + · · · + O(ε 7 ).
Substituting this equation back into itself until R is eliminated from the
right-hand side, we thus find that
{
R 2 = rd
2 1 − (r d· ˙v) + 1 2 (r d· ˙v) 2 + 1 4 r2 d ˙v2 + 1 3 r d(r d·¨v) + 1 2 (r d· ˙v)(r s· ˙v)
}
+ · · · + O(ε 5 ) . (6.49)
Employing the unit vectors n d and n s ,
n d ≡ r d
r d
,
n s ≡ r s
r s
, (6.50)
to more explicitly exhibit the dimensionalities of the various terms in each
expression, and using the binomial theorem on (6.49), we thus find
R = r d
{n d − 1 2 r d ˙v + 1 6 r2 d ¨v + 1 4 r2 d (n d· ˙v) ˙v + 1 }
4 r dr s (n s· ˙v) ˙v + · · · + O(ε 5 ) .
6.5.4 The retarded normal vector
(6.51)
The retarded field expressions of Chapter 5 do not explicitly use the threevector
R; rather, they are somewhat simplified by using n. We can compute
256