Expert: Philip A. Stahl - 9/28/2010

Question
QUESTION: In a frame moving with U_E = E cross B/B^2, the "E cross B" velocity, the
motion of ions and electrons are the same (since the flow comes from the E
cross B force which is the same for ions and electrons).  Here B is the
magnetic field and E the electric field.

It would therefore seem that there would be no current perpendicular to the
magnetic field  j = e n (u_i - u_e) = e n (U_E - U_E) = 0, in this frame. Here e
is the fundamental charge, n the density (same for ions and electrons by
quasineutrality), u_i is the ion velocity and u_e the electron velocity.

However, the equations of motion have a Lorentz force term in the
momentum equation, and this comes from a non-vanishing current.... Where
have I gone wrong in this argument?

ANSWER: Hello,

The only place I can see where you might have gone wrong is in not properly accounting for an additional force.


Note also that what you've identified as:

"U_E = E cross B/B^2, the "E cross B" velocity"

is in reality the plasma drift velocity v_d( "E cross B" velocity) also discussed previously here, in a different context:

http://en.allexperts.com/q/Astrophysics-3368/2008/1/Plasma-problem.htm

(Note the vector v_d, or your U_E, is normal to both E and B and has magnitude [E/B].  

In your argument you've obviously considered the Lorentz force (total force on a particle with charge q, moving at velocity v = U_E):

F = q(E + v X B)

and inquired what value v (U_E) must be to make F = 0. And this is found to be the quantity:

v_d =U_E = E cross B/B^2

which also means the current is zero.

(Analogous to a force exerted by moving a wire between magnetic poles say, such that the downward force on the wire F = IB per unit length of wire, and mechanical power per unit length (or work done) is F v, so one needs Fv = EI, where v = EI/F. BUT, if v = 0, then I = 0)

BUT if one has an additional force arising from gravity (F_g = mg) then the equation of motion becomes (again let v = U_E):

m(dv/dt)= mg + q(v X B)

Where we have substituted an "effective E-field" (E')for the original E (e.g. in q(E + v X B))such that:

E' = (mg)/ q

(where we're simply saying to replace E with E' and such that if some gravity-related force enters, it has a magnitude: E'q = mg.) Then the new drift velocity v_d'(U_E) becomes:

v_d' = m(g x B)/ q B^2

and g, B are vectors.

(Note here that if the dot product, g*B is not zero, then we have freely accelerated motion)

This drift velocity v_d' (or U_E' if you will) now depends on both charge and mass, whereas v_d (U_E) didn't.

Here is where things get tricky, but it's also useful here to refer to the diagram given in the link to the earlier problem.

The electrons and positive ions will drift now in opposite directions such that:

^ y
!     ----------> electrons (-q)
!
!    <---------- Ions (+q)          B  (out)
!
!
!--------------------------->x

(E is in +y direction).


And *this* arrangement (unlike for the diagram shown in the link problem) will yield a drift current in the system such that:

j= - ne v_d'(e) + n e v_d'(p)

where e, p denote protons and electrons and q = e, the unit electron charge or 1.6 x 10^-19 C.

Now, on substituting for these velocities using:


v_d' = m(g x B)/ q B^2


viz.

v_d'(e) = m(e)(g x B)/ (-q) B^2

v_d'(p) = m(p)(g x B)/ (q) B^2


We obtain:

j =rho (g x B)/ B^2 = [nm(p) - nm(e)](g x B)/ B^2

~ nm(p) [g x B)/ B^2]

where the net current arises since m(p) >> m(e), i.e. most of the current is carried by the much heavier ions(protons).

A last cautionary point: the net force density on the plasma vanishes, since:

F = (j X B) + rho(g)

but:  j =rho (g x B)/ B^2


So:  F = rho {[(g x B) X B]/B^2 + g}  = 0

which can be verified via appropriate vector formulae.

---------- FOLLOW-UP ----------

QUESTION: Great. Thanks for that.... I think you're probably right that I have neglected
some other forces, however if that's the case then the following consideration
applies:

In a magnetized plasma (larmor radius << mean free path) there are indeed
the type of drifts you have described, and these may indeed be responsible
for the currents. However, in a collisional plasma (one described by MHD)
charged particles do not complete gyro-orbits and hence (I think) there are no
drifts (I think the E cross B drift persists in the limit of vanishing larmor
radius)...

So, how does one account for these current inducing drifts in MHD? There's
obviously there because the MHD equations contain the Lorentz force....

P.S. I just found something of an explanation in Hazeltine and Waelbroeck,
The Framework of Plasma Physics: section 5.2: MHD fluid, page 115. They say
that to lowest order (in the expansion parameter Larmor radius/Gradient Scale
Length) the current is indeed zero (i.e. the E cross B drifts cancel). However,
at next order there is a contribution which is the current.

I am a bit puzzled by because the first order correction to the E cross B
velocity should vanish in a collisional plasma...  

Answer
Hello,

Okay, first thanks for letting me know you are examining a situation in the *MHD* regime. This changes the complexion considerably!

Now, first, I wouldn't get too obsessive over the term "collisional" since there's some subtlety afoot here. (Also bear in mind - though your text may not have stated it, one can have two forms of MHD: the more traditional, and "kinetic" (which combines one dimensional particle motion along field lines with 2-dimensional fluid theory)

Plasma motions for both types are confined to parallel displacements  (e.g. to B-field) for what we call magnetosonic waves. While motion perpendicular to B incites Alfven waves.  In this sense, only magnetosonic waves are of interest if one is considering the collisional regime. But it one is restricting interest to Alfven dynamics solely (i.e. perpendicular motion) then technically the MHD eqns. are a reasonable approximation to a *collisionless* plasma. (In any case, to test this is so, one will always compute the plasma beta = 2u(o)p/B^2 where u(o) is the magnetic permeability of free space, B is the field strength in Tesla(T), and p is the pressure, in Pa. If then, the beta << 1 the MHD collisionless regime is applicable to a good approximation)

Another reason I don't particularly like making much ado about "collisional" v. "collisionless" is based on the properties that emerge in the limit of: r(L) << l(mfp)

Namely that the electron inertia can now be subsumed into the ion momentum eqn. to yield a single fluid picture. (Which also necesssitates: (T(i) = T(e)). Now, the paradox that arises is that - on account of obtaining a "fluid picture" (even single) the plasma must in some sense be "collisional" or collision-dominated, However, in the classical IDEAL MHD sense since the resistivity n-> 0, the Ohm's law becomes: E + v X B = 0, so the plasma is sufficiently "collisionless". (By contrast, if one has E + v X B = nJ, one has resistive MHD).

I hope this isn't too confusing! What I'm trying to get at is that one needs to take care when these terms are tossed around, to look at the exact contexts. Also, understand what exactly the term "collisional" means when it is being applied to plasma in specific conditions. If in MHD, are Alfven waves present? Is the plasma beta << 1? Then why exactly is the term "collisional" being used at all?

Now, in terms of the drift aspect, it isn't true that all drift ceases to exist. What is true is that different orders of drift and conditions apply. Specifically, in the (classical-conventional) MHD approximation, the Larmor gyrations (of the electrons) are neglected and there is just the E X B drift alone. The primary difference between the conditions of drift (the one we looked at earlier and now in the MHD limit) is that the E field magnitude is different. In the MHD limit, one has {E/B] ~ v_perp, while in the standard drift limit (we saw already): [E/B] ~v(d).

The first is strongly E-field dependent while the latter is weakly E-field dependent.


What about "current-inducing drifts" in MHD? I strongly recommend getting hold of the text 'Introduction to Plasma Physics' by Francis F. Chen, who has an excellent presentation (including diagrams) - in his section (4.18 in my edition) on hydromagnetic (MHD) waves. (The terms 'hydromagnetic' and 'magnetohydrodynamic" are used interchangeably)  Using a specific diagram (Fig. 4-48) he shows exactly how the Lorentz force which he denotes j1 X B_o arises. (Copyright restrictions prevent the duplication of any of his diagrams or material).

The book, however, ought to be available at any university library. What I would do is read carefully through his presentation and make sure you understand how that Lorentz force arises and why. The key details are embedded in his carefully working out the assorted *components* for all the vectors, quantities and then proceeding from there using the relevant eqns.

Good luck!