Single-Particle Electrodynamics - Assassination Science

(we shall reïnsert the new redshift force shortly), and the expression (4.54)
for the mechanical momentum ∆p for the stationary electric-current magnetic
dipole with ultra-relativistic constituents, into (4.55), we thus find the
equation of motion
d t (mγv) = ∇(µ·B) − d t (µ×E)
= (µ·∇)B + µ×(∇×B) − ˙µ×E − µ×d t E,
by using a three-vector identity on the first part. Now, the loop we are
considering in this section is stationary, so the convective derivative
d t E ≡ ∂ t E + (v·∇)E
is in this case simply equal to ∂ t E. Using Maxwell’s equations for ∇×B, we
thus find
d t (mγv) = (µ·∇)B + µ×(∂ t E + J) − ˙µ×E − µ×∂ t E.
Thus, for the ultra-relativistic current-loop model of a fixed magnetic dipole,
we find
d t (mγv) = (µ·∇)B − ˙µ×E + µ×J. (4.56)
Apart from the last term, this is identical to the equation of motion for
the magnetic-charge dipole. The last term in (4.56) is a contact force between
the magnetic dipole µ and any external current J that is generating
a magnetic field. It is beautifully appropriate that the magnetic-charge and
ultra-relativistic current-loop dipoles only behave differently if one probes
the internal structure of the magnetic dipole, by making it collide with an
external current J; if, on the other hand, we keep our external currents J
away from the position of the dipole—and hence our external currents only
“see” the external magnetic dipole field, identical for either model—then we
find that the two types of dipole also move identically. One could not hope
for a more æsthetically pleasing result.
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