Fundamentals of Plasma Physics

3.5 Drift equations 75

This may be solved to give the average drift velocity

vE≡〈v〉=

E×B

B^2

. (3.57)

This steady ‘EcrossB’ drift is independent of both the particle’s polarity and initial ve-
locity. One way of interpreting this behavior is to recall that according to the theory
of special relativity the electric fieldE′observed in a frame moving with velocityuis
E′=E+u×Band so Eq. (3.56) is simply a statement that a particle drifts in such a
way to ensure that the electric field seen in its own frame vanishes. The ‘EcrossB’ drift
analysis can be easily generalized to describe the effect on a charged particle ofanyforce
orthogonal toBby simply making the replacementE→F/qin the Lorentz equation.
Thus, any spatially uniform, temporally constant force orthogonal toBwill cause a drift

vF≡〈v〉=

F×B

qB^2

. (3.58)

Equations (3.57) and (3.58) lead to two counter-intuitive and important conclusions:

1. A steady-state electric field perpendicular to a magnetic field does not drive currents
   in a plasma, but instead causes a bulk motion of the entire plasma acrossthe magnetic
   field with the velocityvE.


2. A steady-stateforce(e.g., gravity, centrifugal force, etc.) perpendicular to the mag-
   netic field causes oppositely directed motions for electrons and ions and so drives a
   cross-field current
   JF=

∑

σ

nσ

F×B

B^2

. (3.59)

3.5.2 Drifts in slowly changing arbitrary fields

We now consider charged particle motion in arbitrarily complicated butslowly changing
fields subject to the following restrictions:

1. The time variation is so slow that the fields can be considered as approximately con-
   stant during each cyclotron period of the motion.


2. The fields vary sograduallyin space that they are nearly uniform over the spatial
   extent of any single complete cyclotron orbit.


3. The electric and magnetic fields are related by Faraday’s law∇×E=−∂B/∂t.


4. E/B << cso that relativistic effects are unimportant (otherwise there would be a
   problem withvEbecoming faster thanc).
   In this more general situation a charged particle will gyrate aboutB, stream parallel to
   B,have ‘E×B’ drifts acrossB, and may also have force-based drifts. The analysis is based
   on the assumption that all these various motions are well-separated (easily distinguishable
   from each other);this assumption is closely related to the requirement that the fieldsvary
   slowly and also to the concept of adiabatic invariance.
   The assumed separation of scales is expressed by decomposing the particle motion
   into a fast, oscillatory component – the gyro-motion – and a slow component obtained by