Single-Particle Electrodynamics - Assassination Science

a sufficient amount of inertia, the cyclical changes to their velocity during
each orbit, due to the force of the electric field, may be as large as the
designed orbital velocity: in fact, in the nonrelativistic limit it completely
dominates their motion; hence, the assumption of roughly constant orbital
speed, underlying the expression (4.50), is invalidated in the nonrelativistic
limit, and the expression (4.53) cannot be applied to that regime.
On the other hand, we see that, in the ultra-relativistic limit—where the
speed of each charge rigorously remains at practically the speed of light,—the
result (4.53) gives
∆p = µ×E. (4.54)
In this limit, the magnitude of the magnetic dipole moment is constant, the
induced electric dipole moment vanishes, and there is a model-independent
contribution, (4.54), to the mechanical momentum of the electric-current
magnetic dipole.
Now, if we are considering the ultra-relativistic current loop as a model for
a fixed magnetic dipole moment, then we clearly need to take account of the
extra mechanical momentum (4.54) when computing the equations of motion
for the loop as a whole. If the quantity µ×E is changing—for whatever
reason—then the ultra-relativistic constituent charges will effectively absorb
or relinquish some of the extra mechanical momentum (4.54) that they are
in possession of. If m is the mechanical rest-energy of the loop as a whole,
and v is its velocity, we thus see that
F ≡ d t p = d t (mγv + ∆p).
In other words, we find
d t (mγv) = F − d t ∆p. (4.55)
Inserting the expression for the force on an arbitrary stationary electriccurrent
magnetic dipole, found in the previous section:
F = ∇(µ·B)
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