74 Chapter 3. Motion of a single plasma particle
will also decelerate in thex-direction. Moving with negativevymeans the ion is going
uphill in the electrostatic potential and when it reachesy= 0, its potential energy must
go back to zero. As noted above, the ion must come to rest at this point, because its total
energy is always zero. Because thexvelocity was never negative, the result of all this is
that the ion makes a net positive displacement in thexdirection. The whole process then
repeats with the result that the ion keeps advancing inxwhile making a sequence of semi-
circles in whichvyoscillates in polarity whilevxis never negative. The ion consequently
moves like a leap-frog which bounces up and down in theydirection while continuously
advancing in thexdirection. If an electron had been used instead of an ion, the sign of
both the electric and magnetic forces would have reversed and the electronwould have
been confined to regions wherey≤ 0. However, the net displacement would also be in the
positivexdirection (this is easily seen by repeating the above argument using an electron).
ion
electron
BBẑ
EEy
^Figure 3.3: E x B drifts for particles having finite initial energyIf an ion starts with a finite rather than a zero velocity, it will execute cyclotron (also
called Larmor) orbits which take the ion into regions of both positive andnegativey.How-
ever, the ion will have a larger gyro-radius in itsy > 0 orbit segment than in itsy < 0 orbit
segment resulting again in an average drift to the right as shown in Fig.3.3. Electrons have
larger gyro-radii in they < 0 portions of their orbit, but have a counterclockwise rotation
so electronsalsodrift to the right. The magnitude of this steady drift is easily calculated by
assuming the existence of a constant perpendicular drift velocity in theLorentz equation,
and then averaging out the cyclotron motion:
0 =E+〈v〉×B. (3.56)