Single-Particle Electrodynamics - Assassination Science

densities and the conservation laws contain a redundancy: given any three
definitions, the fourth actually follows as a consequence.
The choice of which definition is “consequential” is of course arbitrary. In
practice, either the mechanical field density or the Lorentz force is generally
derived from the other three. Let us illustrate the latter, with a simplified but
indicative outline of the argumentation required: If one considers a system
of just two electric charges, of finite separation, then, when they are both at
rest, it is simple to show that the excess in the mechanical field energy, compared
to the case when the two charges are infinitely separated, is simply the
Coulomb-gauge scalar potential of either particle, evaluated at the position
of the other particle, and multiplied by this other particle’s charge:
∆W field = q 1 ϕ 2 (z 1 ) = q 2 ϕ 1 (z 2 ).
(See, e.g., Jackson [113, Sec. 1.11]; a copy of this proof is supplied in Section
C.2.1.) In the general case, the second charge may be replaced by an
arbitrary number of generating charges (see Section C.2.1); one then has
W excess = qϕ(z),
where we write q for the charge being brought into the midst of the other
charges generating ϕ(r); z is the position of the charge q. Similarly, when a
charge is brought into the vicinity of a magnetic field distribution B(r), the
excess in the mechanical field momentum can be shown to be the Coulombgauge
vector potential of the sources generating the B(r), evaluated at the
position of the charge brought in:
∆p field = qA(z).
(A proof of this result is supplied in Section C.2.2.) Now, the elementary
“conservation of mechanical energy” derivation of the force on an object (see,
e.g., [96]) shows that the mechanical momentum p of the object must satisfy
d t p + ∇W excess = 0,
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