Single-Particle Electrodynamics - Assassination Science

tum of ±µ×E. Firstly, one can show that the mechanical field momentum
excess for an electric dipole in a magnetic field is in fact given by
p excess = −d×B; (4.59)
we will shortly find that this same result can be obtained from the Lagrangian
description of the electric dipole, as the difference between the canonical and
mechanical momentum of the particle, in the same way that we found the
quantity qA for an electric charge. One might therefore expect the dual of
the result (4.59), namely,
p excess = µ×E, (4.60)
to be applicable to the magnetic dipole. But this ignores the fact that there
is an extra delta-function field B M ≡ µ δ(r) at the position of the magnetic
dipole, over and above the dual of the electric dipole field (see Chapter 5);
thus, we get an extra field contribution for the current loop, of value
∫
p excess = d 3 r E×µ δ(r) ≡ −µ×E.
But this mechanical field momentum contribution cancels that of (4.60). In
other words, the current loop has no net mechanical field momentum excess
at all; this is again verified by the Lagrangian analysis, which finds no difference
between the canonical and mechanical momentum for the current loop.
(One can intuitively understand this result by recalling that in an electric
dipole, Faraday lines of electric field have beginnings and ends; but in a magnetic
dipole they do not: they are closed paths, since there are no magnetic
charges.)
Finally, we of course have the third contribution to the consideration of
the current loop, of value +µ×E: the Penfield–Haus mechanical momentum
of the constituents themselves. Unlike the mechanical field momentum
excesses, this contribution, being due not to the fields but to the motion of
the particles themselves, does need to be taken into account over and above
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