Single-Particle Electrodynamics - Assassination Science

in fact, proportional to a three-dimensional Dirac delta-function in r-space.
To determine the precise coëfficient of the delta-function, we integrate the
expression (6.86), for arbitrary w, over all three-space:
∫
d 3 r
using the change of variable
whence
we find that
hence,
3w 2 ∫ ∞
4π(r 2 + w 2 ) = 5/2 0
∫
d 3 r
u ≡ r˜r ≡
d r u =
4πr 2 dr
3w 2
4π(r 2 + w 2 ) 5/2 ;
r
, (6.87)
(r 2 + w 2 )
1/2
w 2
(r 2 + w 2 ) 3/2 ,
3w 2
∫ 1
4π(r 2 + w 2 ) = 3 5/2 0
lim
w→0
du u 2 = 1;
{ }
3
1 − r2
≡ δ(r). (6.88)
4π˜r 3 ˜r 2
Returning to (6.85), the identification (6.88) means that
∇·
n
= δ(r), (6.89)
4πr2 the elementary result (where we have discarded the tilde notation in the final
expression).
We now return to the general expression (6.84), and consider contracting
it with an arbitrary three-vector a. We find
hence,
(a·∇) n˜r
4π˜r 2
≡ a δ ij − 3n˜ri n˜rj
i
4π˜r 3
= a j − 3(n˜r·a)n˜rj
4π˜r 3 ;
−(a·∇) n˜r
4π˜r 2 = 3(n˜r·a)n˜r − a
4π˜r 3 . (6.90)
269