Fundamentals of Plasma Physics

3.5 Drift equations 73

3.5 Drift equations

We show in this section that it is possible to deduce intuitive and quite accurate analytic
solutions for the velocity (drift) of charged particles in arbitrarily complicated electric and
magnetic fields provided the fields areslowly changing in both space and time (this re-
quirement is essentially the slowness requirement for adiabatic invariance). Drift solutions
are obtained by solving the Lorentz equation

m

dv

dt

=q(E+v×B) (3.55)

iteratively, taking advantage of the assumed separation of scales between fast and slow
motions.

3.5.1 SimpleE×Band force drifts

Before developing the general method for analyzing drifts, a simple example illustrating
the basic idea will now be discussed. This example consists of an ion starting at rest in a
spatially uniform magnetic fieldB=Bˆzand a spatially uniform electric fieldE=Eyˆ.
The origin is defined to be at the ion’s starting position and both electric and magnetic fields
are constant in time. The assumed spatial uniformity and time-independenceof the fields
represent the extreme limit of assuming that the fields are slowly changing in space and
time.
Because the magnetic forceqv×Bis perpendicular tov, the magnetic force does no
work and so only the electric field can change the ion’s energy (this can be seen by dotting
Eq.(3.55) withv). Also, because all fields are uniform and static the electric field can be
expressed asE=−∇φwhereφ=−Eyis an electrostatic potential. Since the ion lowers
its potential energyqφon moving to largery,motion in the positiveydirection corresponds
to the ion “falling downhill”. Since the ion starts from rest aty= 0whereφ= 0, the total
energyW =mv^2 /2 +qφis initially zero. Furthermore, the time-independence of the
fields implies thatWmust remain zero for all time. Because the kinetic energymv^2 / 2
is positive-definite, the ion can only attain finite kinetic energy if itfalls downhill, i.e.,
moves into regions of positivey.If for any reason the iony-coordinate becomes zero at
some later time, then at such a time the ion would again have to havev= 0because
W=mv^2 −qEy= 0.
When the ion begins moving, it will initially experience mainly the electric forceqEˆy
because the magnetic forceqv×B, being proportional to velocity, is negligible. The
electric force accelerates the ion in theydirection so the ion develops a positivevyand
also moves towards larger positiveyas it “falls downhill” in the potential. As it develops
a positivevy, the ion starts to experience a magnetic forceqvyyˆ×Bzˆ=vyqBˆxwhich
accelerates the ion in the positivex−direction causing the ion to develop a positivevx
in addition. The trajectory now becomes curved as the ion veers in thexdirection while
moving towards largery.The positivevxcontinues to increase and as a consequence a new
magnetic forceqvxxˆ×Bzˆ=−vxqBˆydevelops and, being in the negativeydirection, this
increasing magnetic force counteracts the steady electric force, eventually causing the ion
to decelerate in theydirection. The velocityvynow decreases and ultimately reverses so
that the ion starts to head in the negativeydirection back towardsy= 0. As a consequence
of the reversal ofvy,the magnetic forceqvyˆy×Bˆzwill become negative and so the ion