Single-Particle Electrodynamics - Assassination Science

Now, since the Coulomb-gauge potential ϕ generated by charge 1 or 2 at the
position of charge 2 or 1 is given by (see, e.g., [113, Ch. 1])
q 1
ϕ 1 (z 2 ) =
4π |z 1 − z 2 | ,
q 2
ϕ 2 (z 1 ) =
4π |z 1 − z 2 | ,
we find that the excess field mechanical energy (C.4) can be written
W excess = q 2 ϕ 1 (z 2 ) = q 1 ϕ 2 (z 1 ).
(C.5)
The linearity of Maxwell’s equations, and the quadraticity of the mechanical
field energy expression (C.1), then shows that, for any given electric charge q,
the excess mechanical field energy caused by the bringing of this charge into
any arbitrary distribution of electromagnetic scalar potential ϕ(r) is given
by
W excess = qϕ(z),
where z is the position of the charge q.
C.2.2
Electric charge mechanical momentum excess
The electromagnetic field mechanical momentum is given by the following
integral over all space:
∫
p field = d 3 r E(r)×B(r).
(C.6)
Let us consider placing an electric charge, at rest, amidst some arbitrary
magnetic field distribution B(r). The Coulomb electric field of the charge,
together with the magnetic field distribution B(r), will then yield a contribution
to (C.6). The Coulomb field of the charge is given by
E(r) =
q(r − z)
4π |r − z| 3 ,
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