Single-Particle Electrodynamics - Assassination Science

momentum and mechanical angular self-momentum vanish trivially.
mechanical self-energy density, on the other hand, is given by
The
W qd
ρ
(r) = 1 2 E2 (r)
= 1 2
{
E q (r) + E d (r) } 2
= 1 2 Eq 2 (r) + 1 2 Ed 2 (r) + E q (r)·E d (r). (5.101)
Now, the first two terms on the last line of (5.101) are just the mechanical
self-energy densities for the electric charge and the electric dipole moment
when they are considered singly, as was done in Sections 5.5.8 and 5.5.9; we
need not analyse these expressions anew. However, the final, “cross” term in
(5.101), arising through the nonlinearity of (5.94), represents an interference
between the charge and dipole fields. Let us concentrate on this interference
term, and denote it by ∆Wρ
qd (r):
∆W qd
ρ (r) ≡ E q (r)·E d (r). (5.102)
Using (5.81) and (5.83), we find
⎧
⎪⎨
∆Wρ qd (r) =
⎪⎩
− qr(n·d )
(4π) 2 ε 6 , r < ε,
+ 2q(n·d )
(4π) 2 r 5 , r > ε. (5.103)
Now, both of the expressions in (5.103) are odd in n; thus, when we integrate
∆Wρ
qd (r) over all space, we obtain no net contribution to the mechanical selfenergy
of the charged electric dipole.
It would be convenient if that were the end of the story. However, the
alert reader will realise that the spatial asymmetry of the expression (5.103)
has a subtle, almost sinister, by-product: the mechanical centre of energy
of the self-field does not coïncide with the centre of charge and polarisation.
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