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Let us, first of all, consider the motion of charged particles
in spatially and temporally uniform electromagnetic fields.
The equation of motion of an individual particle takes the form

(36) 
The component of this equation parallel to the magnetic field,

(37) 
predicts uniform acceleration along magnetic fieldlines. Consequently,
plasmas near equilibrium generally have either small or vanishing .
As can easily be verified by substitution, the perpendicular component of
Eq. (36) yields

(38) 
where is the gyrofrequency, is the gyroradius,
and
are unit vectors such that (, , ) form a
righthanded, mutually orthogonal set, and is the initial
gyrophase of the particle. The motion consists of gyration around the
magnetic field at frequency , superimposed on a
steady drift at velocity

(39) 
This drift, which is termed the EcrossB drift by plasma physicists,
is identical for all plasma species, and can be eliminated entirely by
transforming to a new inertial frame in which
.
This frame, which moves with
velocity with respect to the old frame, can properly be regarded as the rest frame of the plasma.
We complete the solution by integrating the velocity to find the particle
position:

(40) 
where

(41) 
and

(42) 
Here,
. Of course, the trajectory of the particle
describes a spiral. The gyrocentre of this spiral, termed
the guiding centre by plasma physicists, drifts across the magnetic
field with velocity , and also accelerates along the
field at a rate determined by the parallel electric field.
The concept of a guiding centre gives us a clue as to how to proceed. Perhaps,
when analyzing charged particle motion in
nonuniform electromagnetic
fields, we can somehow neglect the rapid, and relatively uninteresting, gyromotion,
and focus, instead, on the far slower
motion of the guiding centre? Clearly, what we
need to do in order to achieve this goal is to somehow average the equation of motion over gyrophase, so
as to obtain a reduced equation of motion for the guiding centre.
Next: Method of Averaging
Up: Charged Particle Motion
Previous: Introduction
Richard Fitzpatrick
20110331