Single-Particle Electrodynamics - Assassination Science

(i.e., 2πε/v orb with v = 1, the speed of light), and so the effects of the electric
field on its orbital motion will be minimised further, since the impulse given
to the charge on its “downhill leg” is of order qEτ. Thus, we find that we in
fact need to take the constituent charges to be in the ultra-relativistic limit
for our current purposes.
Let us, however, retain the quantity v orb for the remainder of this derivation,
and only set it to unity at the end; this will show us quantitatively
why the ultra-relativistic limit is necessary. Now, the angular position of our
chosen charge, as a function of time, may be obtained quite simply from its
orbital motion; if we assume its speed to be approximately constant throughout
its orbit, then
θ(t) = v orbt
ε , (4.50)
since it makes one complete orbit in a time period
and hence
τ orb = 2πε
v orb
,
ω orb ≡ 2π
τ orb
≡ v orb
ε .
Using (4.50) in (4.48), we therefore find a differential equation for the zerothorder
contribution to ∆p θ (t):
( ) vorb t
d t ∆p θ = qE y cos ,
ε
with the initial condition (4.49). This is trivially integrated:
∆p θ = qεE y
v orb
( vorb t
sin
ε
We now reïnstate the full vectorial nature of ∆p θ , by using the identity
ˆθ ≡ −i sin θ + j cos θ;
)
.
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