Single-Particle Electrodynamics - Assassination Science

Using the fact that t ret ≡ −R, and employing the expression (6.49), we thus
have
v ret = −r d ˙v + 1 2 r2 d ¨v + 1 2 r dr s (n s· ˙v) ˙v − 1 ...
6 r3 d v − 1 8 r3 d ˙v2 ˙v − 1 8 r3 d (n d· ˙v) 2 ˙v
+ 1 12 r3 d (n d·¨v) ˙v − 1 2 r2 d r s (n s· ˙v)¨v − 1 4 r2 d r s (n s·¨v) ˙v − 1 4 r dr 2 s (n s· ˙v) 2 ˙v
+ · · · + O(ε 5 ), (6.56)
˙v ret = ˙v − r d¨v + 1 2 r d(n d· ˙v) ˙v − 1 2 r s(n s· ˙v) ˙v + 1 ...
2 r2 d v − 1 2 r2 d (n d· ˙v)¨v
+ 1 4 r2 d (n d· ˙v) 2 ˙v − 1 2 r2 d (n d·¨v) ˙v + r d r s (n s· ˙v)¨v + 1 2 r dr s (n s·¨v) ˙v
− 1 2 r dr s (n d· ˙v)(n s· ˙v) ˙v + 1 4 r2 s (n s· ˙v) 2 ˙v + · · · + O(ε 4 ), (6.57)
¨v ret = ¨v − r d
...
v + r d (n d· ˙v)¨v + 1 2 r d(n d·¨v) ˙v − r s (n s· ˙v)¨v − 1 2 r s(n s·¨v) ˙v
+ · · · + O(ε 3 ). (6.58)
6.5.6 Constituent retarded FitzGerald spins
For purposes of calculating the retarded fields, we need to use the FitzGerald
spin vector, σ r ′ ′(t), and its derivatives. Using (6.56) and (6.49) in (3.24) and
its derivatives, we find
σ ′ ret = σ − r d ˙σ + 1 2 r2 d ¨σ − 1 2 r2 d ( ˙v·σ) ˙v + 1 2 r dr s (n s· ˙v) ˙σ
+ · · · + O(ε 5 ) (6.59)
˙σ ′ ret = ˙σ − r d ¨σ + r d ( ˙v·σ) ˙v + 1 2 r d(n d· ˙v) ˙σ − 1 2 r s(n s· ˙v) ˙σ
+ · · · + O(ε 4 ), (6.60)
¨σ ′ ret = ¨σ − ( ˙v·σ) ˙v + · · · + O(ε 3 ). (6.61)
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