Single-Particle Electrodynamics - Assassination Science

The significance of this result is that it is precisely what we would obtain
by replacing n and r in the static field result of Chapter 5 by the appended
quantities n˜r and ˜r; the result is thus the same no matter which way we
choose to proceed, as we should demand on the basis of consistency.
6.6.7 The dipolar inverse-cube field
We now provide another brief illustration of utility and simplicity of ˜r-space
by employing it in the computation of the divergence of the inverse-cube
static dipole field, namely,
We find
hence,
∇· 3(n˜r·a)n˜r − a
4π˜r 3 .
∇· 3(n˜r·a)n˜r − a
= 3
4π˜r 3 4π˜r (n˜r·∇)(n˜r·a) + 3 (n˜r·a)(n˜r·∇) 3 4π
1˜r
3
+ 3(n˜r·a)
4π˜r 3 (∇·n˜r ) − 1 (a·∇)
4π
1˜r
3
= 3(n˜r·a) { }
1 − r2
− 9(n˜r·a) r 2
4π˜r 4 ˜r 2 4π˜r 4 ˜r 2
∇· 3(n˜r·a)n˜r − a
4π˜r 3
+ 3(n˜r·a) { }
3 − r2
+ 3(n˜r·a) ;
4π˜r 4 ˜r 2 4π˜r 4
= 15(n˜r·a) { }
1 − r2
. (6.91)
4π˜r 4 ˜r 2
To make sense of this result, we take the gradient in the direction of a of the
regularised representation (6.88) of the delta-function δ(r):
{
3
(a·∇)δ(r) ≡ (a·∇)
4π˜r 3
1 − r2
˜r 2 }
= 3 { }
1 − r2
(a·∇) 1˜r 4π ˜r − 3
2 3 4π˜r 3 (a·∇)r2 ˜r 2
270