Single-Particle Electrodynamics - Assassination Science

G.6.6
Infinitesimally small spherical bodies
We now consider rigid bodies that are infinitesimally small three-spheres, of
radius ε, and compute the quantities required to calculate the self-fields. By
taking r as well as t to be of order ε in (G.29), and relabelling r by r ′ , we
have
z r ′(t) = r ′ + 1 2 t2 ˙v + 1 6 t3¨v − 1 2 t2 (r ′· ˙v) ˙v + 1
24 t4 ...
v − 1 3 t3 (r ′· ˙v)¨v
− 1 6 t3 (r ′·¨v) ˙v + 1 2 t2 (r ′· ˙v) 2 ˙v + 1 ....
120 t5 v − 1 8 t4 (r ′· ˙v) v
...
− 1 8 t4 (r ′·¨v)¨v − 1 24 t4 (r ′·... v) ˙v − 1 8 t4 ˙v 2 (r ′· ˙v) ˙v + 1 2 t3 (r ′· ˙v) 2¨v
+ 1 2 t3 (r ′· ˙v)(r ′·¨v) ˙v − 1 2 t2 (r ′· ˙v) 3 ˙v + O(ε 6 ). (G.30)
G.6.7
The retarded radius vector
The three-vector from the retarded constituent r ′ to the constituent r at
t = 0 is denoted R; hence,
where by definition
R ≡ r − z r ′(t ret ),
t ret ≡ −R.
(G.31)
(G.32)
From (G.30), (G.31) and (G.32), we find
R = r − r ′ − 1 2 R2 ˙v + 1 6 R3¨v + 1 2 R2 (r ′· ˙v) ˙v − 1 24 R4 ...
v − 1 3 R3 (r ′· ˙v)¨v
− 1 6 R3 (r ′·¨v) ˙v − 1 2 R2 (r ′· ˙v) 2 ˙v + 1 ....
120 R5 v + 1 8 R4 (r ′· ˙v) v
...
+ 1 8 R4 (r ′·¨v)¨v + 1 24 R4 (r ′·... v) ˙v + 1 8 R4 ˙v 2 (r ′· ˙v) ˙v + 1 2 R3 (r ′· ˙v) 2¨v
+ 1 2 R3 (r ′· ˙v)(r ′·¨v) ˙v + 1 2 R2 (r ′· ˙v) 3 ˙v + O(ε 6 ). (G.33)
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