Single-Particle Electrodynamics - Assassination Science

to us will be on the order of the time required for light to cross the stationary
body—which, in naturalised units, is just 2ε. (From the arguments of
Section 6.1, it is clear that times greater than this static light-distance are
relevant, but they are all larger by a dimensionless factor: they are still of
order ε.) The spatial distances of relevance to us are also, clearly, of order ε.
Thus, in distinction to the analysis of Chapter 3, where we merely expressed
the system’s trajectory in terms of a Taylor series in time for convenience,
for the current application we may expand our expressions out both
temporally and spatially in the small parameter ε—and, more importantly,
we can extract exact self-interaction results, in the point limit ε → 0, since
any terms of order ε or higher in these final expressions will vanish rigorously
in this limit.
6.3.2 Electromagnetic moment densities
There arises the question, in any use of a spherical body in the regularisation
of a point particle, of how one is to distribute the electromagnetic moments
throughout the interior of the volume. The external fields are, for any sensible
choice of distribution, unconcerned about how the sources are arranged (for
the case of the electric charge, this is most simply recognised by considering
Gauss’s law); but the computed mechanical self-field quantities are of course
modified, since they have contributions from the fields both internal and
external to the sphere, which are the same order of magnitude (see, e.g.,
Section 5.5 of Chapter 5).
For the case of simply an electric charge, a spherical charge shell is often
employed (i.e., a sheet of charge around the surface of the sphere). With such
a choice, the charge density is still infinite on the surface, but this infinity is
now one-dimensional, rather than three-dimensional as with a point charge.
This reduction in pathology is, in fact, sufficient for one to analyse the system
without infinities entering the expressions of relevance. It also has the added
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