Single-Particle Electrodynamics - Assassination Science

the former generates solely an electric field, and the latter solely a magnetic
field.
However, the mechanical self-momentum density as a function of r
does not vanish: it is given by
⎧
⎪⎨
p qµ
ρ (r) =
⎪⎩
+ 2qrn×µ
(4π) 2 ε 6 , r < ε,
− qn×µ
(4π) 2 r 5 , r > ε.
Now, these expressions are both odd in r, and hence vanish when integrated
over all space; there is no net mechanical self-momentum. However, when
we cross the vector r into p qµ
ρ (r), to compute s qµ
ρ (r), we find
⎧
⎪⎨
s qµ
ρ (r) =
⎪⎩
+ 2qr2 {(n·µ)n − µ}
(4π) 2 ε 6 , r < ε,
q{(n·µ)n − µ}
− , r > ε,
(4π) 2 r 4
which are of course even in r. Upon integration, we thus find that
∫
s qµ
e.m. = d 3 r 2qr2 ∫
{(n·µ)n − µ}
− d 3 q{(n·µ)n − µ}
r
rε (4π) 2 r 4
= qµ
10πε
= 1 3 qµη 1. (5.100)
Thus, a charged magnetic dipole has an electromagnetic contribution to its
mechanical spin angular momentum, in the direction of the magnetic moment
(which, for spin-half particles, will be in the direction of its bare mechanical
spin), of value qµη 1 /3. This repesents an infinite renormalisation of the spin
of the body.
5.5.13 Charged electric dipole
For a body possessing both electric charge and electric dipole moment, there
is again only an electric field generated, and hence the mechanical self-
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