Single-Particle Electrodynamics - Assassination Science

Now, let us consider such a spin-half particle, in its rest frame. The power
into the dipole, (4.10), is then
P = ( ˙ d ·E);
using the relation (4.12), we may now write this as
P = d (N ·E).
s
But if we substitute the rest-frame torque N from (4.11), we find
P = d E·d×E ≡ 0.
s
Thus, for electric dipoles for which d and s are parallel, we find that the
mass of the dipole is a constant of the motion. This is a non-trivial result,
and will not be found to be true for all systems considered in this chapter.
We also need to consider the rate of change of the spin magnitude s.
Differentiating the definition
s 2 ≡ s 2 ,
we have
ṡ ≡ 1 s (s·ṡ) ≡ 1 s (s·N).
In the general case, the electric dipole torque (4.12) yields some finite ṡ; but,
again, if the dipole moment d and spin s are parallel, we find
ṡ| d‖s
= 0.
We now consider the problem of providing a rigorous relativistic derivation
of the power, force and torque expressions found in this section. We use
the framework described in Chapter 3. Clearly, we need to place the dipole as
a whole at rest, to use the framework described there; the simultaneity of the
two ends of the stick, assumed above, is then appropriate and relativistically
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