Single-Particle Electrodynamics - Assassination Science

For two electric charges q 1 and q 2 , at positions z 1 and z 2 , their Coulomb
fields yield
so the resulting field energy (C.1) is
E(r) = q 1(r − z 1 )
4π |r − z 1 | 3 + q 2(r − z 2 )
4π |r − z 2 | 3 ,
W field = 1 q 2 ∫
1
d 3 1
r
2 (4π) 2 |r − z 1 | 4 + 1 q 2 ∫
2
d 3 1
r
2 (4π) 2 |r − z 2 | 4
+ q ∫
1q 2
d 3 r (r − z 1)·(r − z 2 )
(4π) 2 |r − z 1 | 3 |r − z 2 | 3 .
The first two terms are the field mechanical energy expressions for the charges
alone, and are independent of the relative separation of the charges. The
excess field mechanical energy is therefore given by the last term. A change
to the integration variable
yields
where
Using the identity
one finds
and hence
W excess =
ρ ≡ r − z 1
|z 1 − z 2 |
∫
q 1 q 2
d 3 ρ ·(ρ + n)
ρ
(4π) 2 |z 1 − z 2 | ρ 3 |ρ + n| 3 ,
n ≡ z 1 − z 2
|z 1 − z 2 | .
ρ + n
|ρ + n| 3 ≡ −∇ 1
ρ
|ρ + n| , (C.2)
∫
d 3 ρ·(ρ + n)
ρ
ρ 3 |ρ + n| 3 = 4π,
W excess =
q 1 q 2
4π |z 1 − z 2 | .
(C.3)
(C.4)
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