Single-Particle Electrodynamics - Assassination Science

6.6 Divergence of the point particle fields
In this section, we compute the spatial divergence of the fields generated by
a pointlike particle, in the vicinity of the particle’s worldline, by analysing
the explicit expressions listed in Section G.6.19.
Our reasons for carrying out such a procedure are threefold:
Firstly, it will provide an explicit check of the veracity of the self-field expressions
computed in the previous section, since, from Maxwell’s equations,
we know in advance what the divergence of the fields should be: the source
expressions are simple at the instant that the particle is at rest.
Secondly, such a computation provides the concrete basis on which to add
the extra contact field required for a magnetic dipole, over and above that
of the dual of the electric dipole field, in the case of arbitrary motion of the
particle. (This task was performed, on the basis of the results in this section,
in Section 5.5.15.)
Thirdly, the method used to regularise the point particle expressions provides
us with a suitable framework for carrying out, in the following sections,
subtle but important integrations of divergent expressions in the radiation
reaction calculations.
We begin, in Section 6.6.1, by explaining the particular regularisation procedure
that we will use for the divergent point particle fields expressions; the
explicit substitutions required are detailed in Sections 6.6.2 and 6.6.3. Various
spatial gradients of these basic quantities are computed in Sections 6.6.4
and 6.6.5. These results are then used in Sections 6.6.6 and 6.6.7 to carefully
examine the spatial gradients of the monopolar inverse-square and dipolar
inverse-cube fields, and again in Section 6.6.8 for various inverse-square terms
in the retarded dipole fields of Section G.6.19.
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