Fundamentals of Plasma Physics

76 Chapter 3. Motion of a single plasma particle

averaging out the gyromotion. As sketched in Fig.3.4, the particle’s positionand velocity
are each decomposed into two terms

x(t) =xgc(t) +rL(t), v(t) =

dx

dt

=vgc(t) +vL(t) (3.60)

whererL(t),vL(t)give thefastgyration of the particle in a cyclotron orbit andxgc(t),
vgc(t)are theslowlychanging motion of theguiding centerobtained after averaging out
the cyclotron motion. Ignoring any time dependence of the fields for now, the magnetic
field seen by the particle can be written as

B(x(t)) = B(xgc(t) +rL(t))

= B(xgc(t)) + (rL(t)·∇)B. (3.61)

BecauseBwas assumed to be nearly uniform over the cyclotron orbit, it is sufficient to
keep only the first term in the Taylor expansion of the magnetic field. The electric field
may be expanded in a similar fashion.

rL
t

xgc
t

x
t

B

guiding center trajectory

particle

actual

trajectory

Figure 3.4: Drift in an arbitrarily complicated field

After insertion of these Taylor expansions for the non-uniform electric and magnetic