Single-Particle Electrodynamics - Assassination Science

the electric field. There is, however, a force on the electric dipole moment:
we have
F (M)
µd = 3d d
4πε
∫r≤ε
3 r 3µ
3 4πε ˙σ×σ 3
= − 3dµ σ× ˙σ
4πε3 ≡ −3dµη ′ 3. (6.123)
For the corresponding “moment-arm” torque contribution, we note that the
above expressions are exact (there is no factor of λ since the electric dipole
force depends on ˙σ), and they are even in r; hence, when we cross r into
this expression, the resulting odd integrand vanishes upon integration over
the three-sphere of r. There is therefore no torque on the electric dipole due
to the Maxwell field.
For force on the magnetic dipole moment itself , due to its extra Maxwell
field, we must be a little more careful. Since the Maxwell field is of a contant
value inside the sphere, and zero outside the sphere, we find the only nonzero
gradient of the field is of a Dirac delta-function form, around the surface
of the sphere. Now, the subtlety arises because the magnetic dipole moment
density itself has a sharp transition around this surface. However, this step
function in moment density is of course the same step function that leads to
the Dirac delta function in the gradient of the Maxwell field itself (as can be
seen by noting its source, equation (6.121)). We are therefore led to consider
integrals over (generalised) functions of the form
∫
dt δ(t)ϑ(t)f(t) ≡ 1 f(0), (6.124)
2
where the right-hand side of this identity may be trivially verified by integrating
by parts.
Now, the gradient of the Maxwell field (6.122) is given, from first principles,
by
(σ·∇)B M = − 3µ (σ·n)σ δ(r − ε). (6.125)
4πε3 285