Single-Particle Electrodynamics - Assassination Science

By using the elementary result v = r ˙θ for circular motion, and noting that
the direction of ˙σ is the direction of this extra velocity for q 1 , and antiparallel
to it for q 2 , we thus find
v 1 = v + 1 2 ε ˙σ,
v 2 = v − 1 ε ˙σ. (4.4)
2
We now consider the external electric and magnetic fields that are acting
on our electric dipole. Clearly, since the dipole will eventually be shrunk to
infinitesimal size, it is appropriate for us to expand the fields E(r) and B(r)
as three-dimensional Taylor series around the position of the centre of the
dipole:
E(r) = E(0) + (r·∇)E(0) + O(r 2 ),
B(r) = B(0) + (r·∇)B(0) + O(r 2 ). (4.5)
Let as refer to E(0) and B(0) as simply E and B in the following, and
likewise for the gradients of these fields, on the understanding that these
quantities and derivatives are to be evaluated at the position of the centre of
the dipole. Using (4.1) and (4.2) in (4.5), we thus find
E 1 = E + 1 2 ε(σ·∇)E + O(ε2 ),
E 2 = E − 1 2 ε(σ·∇)E + O(ε2 ),
B 1 = B + 1 2 ε(σ·∇)B + O(ε2 ),
B 2 = B − 1 2 ε(σ·∇)B + O(ε2 ). (4.6)
We now have all the ingredients required to find the pre-relativistic equations
of motion for our electric dipole. The force on each electric charge q 1 or
q 2 is of course simply the Lorentz force (2.17); the power into each charge is
likewise the Lorentz result (2.18). Let us consider first the Lorentz force. We
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