# Confinement of Thermalized plasmas: Why not E instead B?

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I'm pondering the density limits realizable in thermalized plasmas (as in Tokamaks) and it seems that confinement by a rotating electric field (something like a Paul trap) would theoretically allow much higher densities than the current magnetic field techniques. No doubt I'm missing something but I hope Ill be excused for thinking out loud. Here's the reasoning:

Magnetic Limit ----------------------------------------------------------
The confinement limit for a mag. field is the Brillouin density:
$$\eta=\frac{\epsilon_{0}B^2}{2Mion}$$ in ions per M^3
For deuterons and assuming the highest achievable(?) steady state field:
B=20 Tesla (ITER is 13T last I looked)
Mion=2*AMU in Kg

Gives:
$$\eta=10^{12}$$
ions per cm^3

E Field Limit(?) -------------------------------------------------------
Here I'm considering the force to be overcome is the de-focusing effect of the space charge and neglecting and self confining mag. field. To keep it simple I'm using a cylindrical geometry of a non-neutral plasma. In that case the radial E field created by the space charge is found from Poisson:
$$\nabla E=\rho / \epsilon_{0}$$
solving in gauss's law form:
$$Er = r\eta q_{e} / (2 \epsilon_{0})$$

Then using the density found for the mag limit above at 1M:
Er @ d=1e12cm-3 = 8millivolt / cm
a trivial field. So a confining field producing 8mV/cm at the outside edge is needed to hold a cold(?) plasma. A rotating E-field would be required that produced a time averaged potential well with that Er, something like a Paul trap.
http://en.wikipedia.org/wiki/Paul_trap
I'm guessing that Er could be increased by ~10^8 before running into equipment limits.

Observations:
-The mass variable in the Brillouin limit makes mag. fields a lousy way to contain ions in particular; electrons
would get 2*3600x more density than deuterons, hence Bussard and the Polywell.
-E fields are orders of magnitude better at confinement. So why isn't electric
field confinement of plasma under more investigation?

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Gold Member
Have I posted in an inappropriate section? Apologies if so, please advise and off I'll go.

It's rather a specialised topic. Maybe High energy section ?

vanesch
Staff Emeritus
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Have I posted in an inappropriate section? Apologies if so, please advise and off I'll go.

The most appropriate section would be classical physics, but here is not a bad place either. (I will remove the post in the high energy section, that has nothing to do with it).

I would PM ZapperZ if I were you. He is an expert in things of this nature and I'd be very surprised if he did not know the answer to this.

If that does not work, go to the ITER website, find a experimenter's e-mail and write a succinct question. Most professionals will help you if you keep your e-mail short.

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vanesch
Staff Emeritus
Gold Member
I would PM ZapperZ if I were you. He is an expert in things of this nature and I'd be very surprised if he did not know the answer to this.

If that does not work, go to the ITER website, find a experimenter's e-mail and write a succinct question. Most professionals will help you if you keep your e-mail short.

Yes, I have to say that I'm not knowledgeable enough to give a good answer. I know of electrostatic confinement, (which is used: look up "fusor" systems), but the problem with that - as I understand it - is that the plasma will always be in contact with an electrode which defines the field. I vaguely know of a theorem which forbids "electrostatic confinements", but I'm not very into all this...

siddharth
Homework Helper
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I vaguely know of a theorem which forbids "electrostatic confinements", but I'm not very into all this...

Is that Earnshaw's theorem?

"A collection of point charges cannot be maintained in an equilibrium configuration solely by the electrostatic interaction of the charges."

scienceworld.wolfram.com/physics/EarnshawsTheorem.html

vanesch
Staff Emeritus
Gold Member
Is that Earnshaw's theorem?

"A collection of point charges cannot be maintained in an equilibrium configuration solely by the electrostatic interaction of the charges."

Yes, that's what I had in mind. But I don't know up to what point this is a problem for plasma confinement.

Astronuc
Staff Emeritus
Electrostatic repulsion would require a plasma to be surrounded by a surface of the same charge - and the plasma itself would require some net - (surplus of electrons) or + (deficiency of electrons). This would be difficult if not impossible (see siddharth's post).

I think the charge density would be prohibitive, and would introduce the problem of repulsion in the plasma (and potentially large local instabilities), which would be in addition to its thermal pressure (nkT). Then there is the matter of discharge.

Gold Member
-Electrostatics: Yes 3D electrostatic confinement from an external field is impossible (Earnshaw, or otherwise trying to get local minimums from Laplace's equation). As per my OP Im suggesting a rotating field, ala a Paul trap, for which confinement is clearly possible. The question then: whats the density limit?

-Charge Density prohibitive. Yes I'm proposing a non-neutral, nuclei only plasma. But per the numbers I showed in the OP, its not prohibitive. The field from a plasma space charge at ITER densities is small, millivolts/cm. (At least from my derivation, am hoping you guys will any find holes in that). And so the confinement field is small.

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vanesch
Staff Emeritus
Gold Member
-Charge Density prohibitive. Yes I'm proposing a non-neutral, nuclei only plasma. But per the numbers I showed in the OP, its non prohibitive. The field from a plasma space charge at ITER densities is small, millivolts/cm. (At least from my derivation, am hoping you guys will any find holes in that). And so the confinement field is small.

Looking at your numbers, there's something strange. A charge density of 10^12 elementary charges per cm^3 gives us 10^18 elementary charges per m^3 which means about 0.1 Coulomb per m^3.
Now, a charge density of 0.1 Coulomb per m^3 gives me:
div E = 10^10 V/m^2 if I fill in Gauss' law.

I don't quickly see emerging your mV/cm field...

Am I wrong somewhere ?

Gold Member
Arg! No you are not wrong. I bungled the cm^3 to M^3 conversion (wrong way) Apologies all 'round for the trouble. Ill double check next time before posting.

Then, with the density per _meter_ ^3 of 10^18, using even only a plasma volume of 1x10-6 _meter_ ^3 the field at the edge w/ r~0.01 _meter_ is still Er=90 MegaVolts/Meter. That takes us beyond in your garage fusion plasma confinement.