Single-Particle Electrodynamics - Assassination Science

under differentiation, the latter expression generates even more complicated
coëfficients (compare the lengthy Σ -derivative expressions in Section G.4.7
and Section G.4.8 to their much simpler counterparts in Section G.4.9).
Returning, now, to the expressions (5.75), (5.76), (5.77), (5.78), (5.79)
and (5.80), one can immediately recognise familiar faces among the results.
Most obviously, the 1/R 3 electric field (5.77) contains the static dipole field
expression most transparently:
E ′d ∣
3
∣ v=0
= 3(n·σ)n − σ;
the full velocity dependence of the result (5.77) merely incorporates the
Lorentz transformation of these static fields by the velocity v. (The factor
of 1/r 3 ≡ 1/R 3 is of course encapsulated in the notational definitions of
(5.73).) The corresponding 1/R 3 magnetic field of (5.80) is, as one would
expect, simply the cross-product of the three-velocity and the corresponding
electric field (5.77). (That one would expect this result, in advance, is recognised
by noting that the Lorentz-boosted electric charge static fields (5.42)
and (5.44) themselves possess this property; if one considers a static electric
dipole to consist of two spatially separated electric charges in the rest frame
of the particle,—which themselves will be stationary, if the dipole moment
is not precessing,—then the transformation properties of the dipolar static
fields under Lorentz boosts must be identical to those of the electric charge.)
Likewise, the 1/R electric and magnetic fields of (5.75) and (5.78) are,
manifestly, perpendicular to both the normal vector n, and to each other,—
and of the same magnitude,—as must of course be the case for an electromagnetic
radiation field. (Indeed, this follows from the fact that the covariant
field expressions (5.70), (5.71) and (5.70) of Section 5.4.5 are in accord with
the Goldberg–Kerr theorem [93]).
The particular terms contained in the results (5.75), (5.76) and (5.79)
would not, of course, be very familiar to many readers. But the author
suggests that this unfamiliarity is due solely to one’s simply not having been
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