Single-Particle Electrodynamics - Assassination Science

to particles carrying dipole moments, and obtain new, simplified expressions
for the results. Finally, in Section 5.5, we review various aspects of the static
fields from such particles, insofar as are required for the radiation reaction
calculations to be undertaken in Chapter 6; as a by-product, we obtain the
expression for the extra delta-function field required on the worldline of the
particle, in the case of a magnetic moment, in order that the Maxwell equations
be correctly satisfied.
5.2 History of the retarded dipole fields
That the fields generated by a particle carrying a dipole moment are richer
(and, correspondingly, more mathematically complicated) than those generated
by an electric charge can be appreciated before even writing down an
equation. Firstly, one knows that a dipole must have internal degrees of freedom
describing the orientation of the dipole in the rest frame of the particle;
these degrees of freedom (and, in particular, their rates of change) will enter
into the equations for the generated fields, in addition to those quantities
already present in the electric charge case. Secondly, the static fields of a
dipole fall off like R −3 (rather than R −2 as for an electric charge), which
means that two time derivatives of the velocity and/or spin must be present
to generate the “radiation” fields. (This can be seen on dimensional grounds:
the four-velocity and unit four-spin are themselves dimensionless; there are
no other kinematical quantities available apart from the proper time; and
the radiation fields must, by definition, fall off like R −1 so that the energy it
carries may propagate out indefinitely.)
Perhaps due to this premonitory warning of extra complexity, the question
of obtaining the general retarded fields for a particle with a dipole moment
has not attracted much attention over the decades. Bhabha and Corben [40]
appear to have been the first to make a substantial attack on this problem,
in 1941, using methods developed two years earlier by Bhabha [39]. (See
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