Single-Particle Electrodynamics - Assassination Science

fastest at the position θ = −π/2. This implies that the net charge densities
will no longer balance at each point of the loop; in particular, we will find
that it now possesses an induced electric dipole moment; the integrity of its
magnetic dipole moment may also be lost. Essentially, we are feeling the
ramifications of letting our constituent charges retain a degree of freedom:
they are now “sympathetic” to the external fields.
Now, if the physical situation that one would like to consider is, indeed,
made up of a number of real electric charges in circulation, with some given
velocities, then this induced electric dipole moment is a reality, and needs
to be considered in the equations of motion of the system. On the other
hand, if one is actually trying to construct an appropriate model of a fixed
magnetic dipole moment, without an induced electric dipole moment (as we
are for the purposes of this thesis, namely, a model that is applicable to the
fixed intrinsic moments of spin-half particles), then this is not acceptable.
We must therefore take steps to ensure that an appropriate limit is taken
that eliminates this unwanted induced moment.
Now, if one considers the situation from first principles, one can see that
if the initial speeds of the circulating charges are small, then the charges will
essentially act as if they are free, and the positive charges will simply tend
to crowd together at one end of the loop, and the negative charges at the
other end, ultimately only constrained by their mutual repulsion. This is of
course what we do not want to happen. So let us, following Penfield and
Haus [171], look at the opposite limit: that of ultra-relativistic circulating
charges. We then of course know that the extra mechanical momentum
absorbed and relinquished during each orbit will lead to only small changes
in speed, due to the relativistic relationship between mechanical momentum
and three-velocity. Moreover, the time that each charge spends in any single
orbit will also be reduced, down to the limiting value (for given ε)
τ min = 2πε
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