Single-Particle Electrodynamics - Assassination Science

The position of the mechanical centreof energy, z CoE , is, of course, simply
the energy-weighted expectation value of the position operator r:
∫ d 3 r rW ρ (r)
z CoE ≡ ∫ d 3
r W ρ (r) . (5.104)
If one goes back and examines the expressions carefully, one finds that the mechanical
self-energy densities considered in previous sections were all even in
r (automatically ensured, of course, if the density is simply a sum of squared
fields, as it was in those sections); application of (5.104) then trivially yields
z CoE = 0, the centre of our sphere. But since the expression (5.103) for the
interference contribution ∆Wρ
qd (r) is odd in r, the subsequent contribution
to (5.104) then yields a non-zero result; specifically,
∫
(m q e.m. + m d e.m.)z qd
CoE = − d 3 r qr2 ∫
(n·d )n
+ d 3 2q(n·d )n
r
rε (4π) 2 r 4
= 3qd
20πε
= 1 2 qdη 1. (5.105)
We can see, from the result (5.105), that the mechanical centre of energy
of the uniform-density spherical charged electric dipole is offset from the
centre of the sphere, in the direction of the electric dipole moment d. On
the other hand, this offset vanishes in the point limit, since m d e.m. is of order
ε −3 , whereas the right-hand side of (5.105) is only of order ε −1 .
This effect has serious ramifications for the radiation reaction calculations
of Chapter 6 (which uses just such a model to regularise the pointlike charged
electric dipole); we clearly need to add a “mass dipole” to the system to
return the mechanical cenre of energy to the centre of the spherical body.
Note that, since the radiation reaction calculation proceeds on the basis of
finite ε, with the point limit only being taken at the end of the calculations,
we cannot simply rely on the vanishing of the centre-of-energy offset in the
point limit for the radiation reaction calculations themselves.
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