Single-Particle Electrodynamics - Assassination Science

= 1 8( 4
3 π(2ε)3 ) 2
, (6.23)
which is the expected result—the factor of 1/8 = 1/2 3 taking care of the
“other half” of the factor of 2 6
= 64 apparent increase in volume of the
6-dimensional integration space (the Jacobian providing the other factor of
1/8).
We defer a computation of the integral (6.21) to Section 6.4.5; we shall
first need to derive integrals over V d more complicated than (6.23).
6.4.4 Explicit outer integrals
While we have predicted that the r s -dependencies of the terms in our radiation
reaction expressions will only be of one of the two forms (6.20) or
(6.21), the r d -dependencies of the terms do not have such simplicity. In fact,
they involve an almost arbitrary number of factors of r d , as well as factors
r n d of the magnitude r d of r d . (The latter can be understood in terms of
the discussion of Section 6.4: they arise as the leading-order term of single
powers of R, the magnitude of R, which is also the negative of the value of
the retarded time t ret (r, r ′ ).)
Now, since the region of integration V d is spherically symmetric, the angular
and radial integrals may be decoupled by the use of spherical coördinates.
We shall consider the angular integrations shortly. For the case where the integrand
in question is independent of r s , we simply need to consider integrals
of the form
η m ≡ 1 ( ) 4 −2 ∫ ∫
4π 3 πε3 d 3 r d 3 r ′ 1
,
r≤ε r ′ ≤ε rd
m
where we are here defining the symbols η m , promised in Chapter 5. As with
(6.23), these may be computed by elementary integration; by using the result
(6.22), and simplifying somewhat, one finds that
η m =
72
(2ε) m (3 − m)(4 − m)(6 − m) η 0. (6.24)
246