Single-Particle Electrodynamics - Assassination Science

clearly, for any ε, this parametrisation of the principal value function vanishes
around r = 0, and hence also does so in the point limit. In terms of (5.89),
the point particle dipole results, from (5.83), (5.85), (5.86) and (5.87), can
be written
E d 3(n·d )n − d
point(r) = P − 1 4πr 3 3 d δ(r),
B µ point(r) = P 3(n·µ)n − µ + 2 µ δ(r). (5.90)
4πr 3 3
There is, however, somewhat of a philosophical objection to the “gutting”
procedure used to define the principal value function, expression (5.89), which
can be understood as follows: The electric dipole result for a small sphere,
equation (5.83), was obtained by considering the superposition of a continuous
volume of infinitesimal dipoles. Unlike the magnetic case, no extra fields
were added, by hand, to the result. Thus, one could, conceptually, consider
shrinking the dipole generating the field (5.83) itself to a point—and then
use this as a constituent in an a larger finite sphere dipole; and so on, ad
infinitum. It therefore seems arguably natural to simply define the dipolar
inverse-cube function to be the limit of the result (5.83), in the point limit.
Let us refer to this as the regularised value of the function (5.88), and denote
it by prefixing the function by the symbol ‘R’:
R 3(n·σ)n − σ
4πr 3
⎧⎪ − σ
⎨ 4πε , r < ε,
3
≡ lim
ε→0 3(n·σ)n − σ
⎪ ⎩ , r > ε;
4πr 3
(5.91)
The reason for christening it so is that, regardless of which way one “regularises”
the function (5.88)—in other words, modifying its definition so that it
remains finite, rather than mathematically divergent,—one invariably finds a
result equivalent to (5.83), in the point limit, rather than that of the principal
value (5.89). (See Chapter 6 for an example of this.)
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