Single-Particle Electrodynamics - Assassination Science

Again noting the relations (4.23) and (4.26), and taking the point limit, we
therefore find
N = µ×B. (4.43)
Equation (4.43) is the (completely uncontroversial) expression for the torque
on an electric-current magnetic dipole. To see why it is uncontroversial,
we only need note that—unlike the power and force expressions—this result
(4.43) is the same as the result (4.19) found for the magnetic-charge dipole
in the previous section.
In fact, Hraskó [108] has shown why one must obtain this torque law
for any object generating a magnetic dipole field, regardless of the nature
of the object. He likewise showed why the Lorentz force must be obtained
for any electric monopole. His proofs considered the mechanical momentum
contained in the electromagnetic field surrounding the position of the charge
or magnetic dipole, using essentially the same concept of “conservation of
mechanical four-momentum” as the derivation in Section 2.3.8 of the Lorentz
force. In fact, his proofs are much more satisfactory: he essentially uses the
Dirac [68] method of considering the flow of mechanical four-momentum (via
the mechanical stress-energy tensor) through the surface of a small volume
surrounding the particle; this has the great theoretical advantage that the
proof is now manifestly local, and independent of retardation effects, etc.
Now, Hraskó found that the contributions to the integrals of the interior of
the small volume surrounding the point particle vanish, for the monopole
force and dipole torque, when the volume is shrunk to zero. However, for the
power and force on a dipole, this integral over the small volume surrounding
the particle does not vanish: it is, in fact, finite; and hence the internal field
differences between the magnetic-charge dipole and electric-current magnetic
dipole can (and do) lead to different power and force expressions. (The reason
that these integrals do not vanish as with the monopole case is that the
dipole field contains one extra power of 1/r over that of the monopole field;
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