Fundamentals of Plasma Physics

3.8 Motion in small-amplitude oscillatory fields 109

where〈x(t)〉is the particle’s time-averaged position andδx(t)is the instantaneous devia-
tion from this average position. If the amplitudes ofE 1 (x,t)andB 1 (x,t)are sufficiently
small, then the fields at the particle position can be approximated as

E(〈x(t)〉+δx(t),t) ≃ E 1 (〈x(t)〉,t)

B(〈x(t)〉+δx(t),t) ≃ B 0 +B 1 (〈x(t)〉,t) (3.192)

This is the opposite limit from what was considered in Section 3.7.3. TheLorentz equation
reduces in this small-amplitude limit to

m

dv

dt

=q[E 1 (〈x〉,t) +v×(B 0 +B 1 (〈x(t)〉,t))]. (3.193)

Since the oscillatory fields are small, the resulting particle velocity will also be small
(unless there is a resonant response as would happen at the cyclotron frequency). If the
particle velocity is small, then the termv×B 1 (x,t)is of second order smallness, whereas
E 1 andv×B 0 are of first-order smallness. Thev×B 1 (x,t)is thus insignificant com-
pared to the other two terms on the right hand side and therefore can be discarded so that
the Lorentz equation reduces to

m

dv

dt

=q[E 1 (〈x〉,t) +v×B 0 ], (3.194)

a linear differential equation forv. Sinceδxis assumed to be so small that it can be
ignored, the average brackets will be omitted from now on and the first order electric field
will simply be written asE 1 (x,t)wherexcan be interpreted as being either the actual or
the average position of the particle.
The oscillatory electric field can be decomposed into Fourier modes, each having time
dependence∼exp(−iωt)and since Eq.(3.194) is linear, the particle response to a field
E 1 (x,t)is just the linear superposition of its response to each Fourier mode. Thus it is
appropriate to consider motion in a single Fourier mode of the electric field, say

E 1 (x,t) =E ̃(x,ω)exp(−iωt). (3.195)

If initial conditions are ignored for now, the particle motion can be found by simply assum-
ing that the particle velocity also has the time dependenceexp(−iωt)in which case the
Lorentz equation becomes

−iωmv=q

[

̃E(x) +v×B

0

]

(3.196)

where a factorexp(−iωt)is implicitly assumed for all terms and also anωargument is
implicitly assumed forE ̃.Equation (3.196) is a vector equation of the form

v+v×A=C (3.197)

where
A=

ωc

iω

ˆz

ωc=

qB 0

m

C=

iq

ωm

̃E(x)

(3.198)