Single-Particle Electrodynamics - Assassination Science

thus, using (4.50), we find
∆p +
(t) = qεE y
v orb
( ){ ( vorb t vorb t
sin −i sin
ε
ε
) ( vorb t
+ j cos
ε
)}
. (4.51)
Now, if we average this expression over one orbit of the charge, and multiply
by the number of positive charges in the loop, we find
∆p +
=
n ∫ τorb
dt ∆p
τ +
(t) = −i qεE y
.
orb 0
2v orb
If we now look at performing the above analysis for a negative charge in the
loop, we must simply change +q to −q, and reverse the sign of θ(t) in (4.50):
this gives us
∆p −
(t) = − qεE y
v orb
( ){ ( vorb t vorb t
sin +i sin
ε
ε
) ( vorb t
+ j cos
ε
)}
, (4.52)
where the negative sign out the front is that due to q → −q, and the first
term in braces has changed sign because sin(−θ) ≡ − sin(θ). The integral of
(4.51), over all of the negative charges, thus also gives
∆p −
=
n ∫ τorb
dt ∆p
τ −
(t) = −i qεE y
.
orb 0
2v orb
Thus, adding the contributions of the negative and positive charges together,
we find
∆p ≡ ∆p +
+ ∆p −
= −i qεE y
v orb
.
Using (4.26), and noting that µ×E is, for our configuration, in the negative-x
direction, we therefore find
∆p = µ×E . (4.53)
vorb
2
Now, this result (4.53) seems to imply that, in the nonrelativistic limit of
the motion of the constituent charges, their net mechanical momentum would
diverge. This is, of course, not the case: if the charges are not moving with
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