Single-Particle Electrodynamics - Assassination Science

(One should carefully note that the spin derivatives ˙σ ′ and ¨σ ′ appearing
here are the lab-time derivatives of the FitzGerald spin definition (5.74), in
accordance to the precedence-of-operations rules of Section A.3.8; they are
not the “FitzGerald–Lorentz contractions” of the standard spin derivatives
˙σ and ¨σ.)
To compute the magnetic field expressions B ′d n , we can again use the
properties (5.34) and (5.35). Inspection of (5.70) and (5.72) shows that B ′d
1
satisfies (5.35) and B ′d
3 satisfies (5.34). However, (5.71) is more problematical,
mixing terms of both types, and indeed having one term of neither
type. If one inspects the detailed expressions carefully, and performs some
algebraic shuffling, one finds that B ′d
2 can ultimately be written as the sum
of a term like (5.35), and two additional terms. The complete results are
then
B ′d
1 = n×E ′d
1 , (5.78)
B ′d
2 = n×E ′d
2 + κ 2 σ ′ × ˙v + κ 3 n ′ × [ n×(σ ′ × ˙v) ] , (5.79)
B ′d
3 = v×E ′d
3 . (5.80)
The corresponding results for a point particle carrying a magnetic dipole
moment are, of course, simply the electromagnetic duals of these expressions—plus
an extra Maxwell term at the position of the particle, which will
be discussed in Section 5.5.
5.4.7 Simplicity of the author’s expressions
The equations (5.75), (5.76), (5.77), (5.78), (5.79) and (5.80) are the author’s
final results for the retarded fields from an electric dipole. They are, arguably,
as simple as one can get.
An illustration of this assertion is the fact that, even though the expressions
above are in fact true for an arbitrary velocity v of the particle, the
quantity v itself only appears once, in one of the terms of E ′d
2 ; and the cor-
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