Single-Particle Electrodynamics - Assassination Science

this is Jackson’s equation (14.13) [113].
With an eye to the next section, we now define the convenient quantities
E ′q
1 , E ′q
2 , B ′q
1 and B ′q
2 :
E q ≡ q { E
′q }
1
4π R + E′q 2
,
R 2
B q ≡ q { B
′q }
1
4π R + B′q 2
, (5.37)
R 2
as well as the (unprimed) quantities E q 1, E q 2, B q 1 and B q 2:
E q n ≡ E′q n
R n ,
B q n ≡ B′q n
R n .
It will also be fruitful to define the two new three-vectors
n ′ ≡ n − v (5.38)
and
n ′′ ≡ n ′ − v×(n×v), (5.39)
as well as the “Doppler factor”
κ ≡
1
1 − (v·n) . (5.40)
In terms of these quantities, the retarded electric charge fields (5.33) and
(5.36) can be written
and
E ′q
1 = κ 3 n×(n ′ × ˙v), (5.41)
E ′q
2 = γ −2 κ 3 n ′ , (5.42)
B ′q
1 = n×E ′q
1 , (5.43)
B ′q
2 = n×E ′q
2 = v×E ′q
2 , (5.44)
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