# Electric field due to charged plates moving through a plasma

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1. Feb 23, 2017

### timelessmidgen

Hi folks, this is not an actual HW question, but rather one of personal interest. I'm basically trying to understand your classic idealized parallel plate capacitor, but moving face-on through a plasma.

1. The problem statement, all variables and given/known data

We have a diffuse neutral plasma made of free protons and electrons (each particle type with number density n and charges +e and -e respectively). Through the plasma we have a set of charged parallel plates traveling at high velocity, v0. The plates lie in the X-Y plane and the velocity vector is in the the direction of +z (IE it's moving through the plasma face on.) The parallel plates are not solid sheets, but made of diffuse mesh wire grids, so plasma particles can move through the charged plates without hitting solid material. The square root of the area of the charged plates is much larger than their plate separation, which itself is much larger than the separation of the wires which make up the mesh grid. In other words, we can approximate it as an ideal, infinitely large parallel plate capacitor. The charged plates are held at a constant potential difference, V, with the leading plate charged positively, and the trailing plate negatively. Note that the capacitor is moving fast. It's traveling much faster than the mean thermal velocity of the protons, and faster or of order the mean thermal velocity of the electrons. BUT, 0.5 mev02<V e (IE, the electrons that flow into the parallel plates don't make it across the potential gap, they get shot out the top first.)

After an initial equilibration, what is the electric field above, between, and below the charged plates? What is the surface charge density, σp, of the plates?

2. Relevant equations
v0>>sqrt(2kbT/mp)
v0≥sqrt(2kbT/me)
0.5 mev02<V e
standard equations regarding Gauss's law and parallel plates.

3. The attempt at a solution
The geometry I apply: stick the bottom plate at z=0, the top plate at z=h, and let the plasma flow downward through it.

Initially, the electric field will just be that of a classic parallel plate capacitor: Ez=-σ/ε between the plates, 0 above and below the plates. After a moment, however, charged particles start streaming through, adding a volume charge density. This volume density will essentially be the total amount of charge swept up times the fraction of time each particle spends at each layer in between the plates. In other words, for protons and electrons the charge volume density is:

ρ=±n V e / v(as a function of z)
ρ=±n V e / sqrt( V^2 + 2 a (z-h))

where a is the particle acceleration. For protons ap=-V e/(h mp), for electrons ae=V e/(h me).
The total charge density is just the addition of these two (though we have to be a little bit careful working with them, since ρ for the electrons is only defined down to the electron turnover point, h-v/(2 ae), and below that it's zero.)

It gets a little bit hairy, but I can actually integrate up little slices of ρ(z) dz and find the electric field everywhere due to these new charge densities. So now we have a new electric field. Of course the problem doesn't stop there, because particles keep flowing in through our capacitor, and now they'll be feeling this NEW electric field. I could, perhaps, throw in some numbers and iterate: figure out the velocity of the particles under this new field, construct a new charge density, integrate slices to find electric field, repeat, and pray for convergence. But I'm not sure that's a great idea...

Is there a simpler way to approach this? Does this sound like any familiar problems? I'm not so hot with plasma physics - should I be learning about Debye shielding? Does Debye shielding even work (or work the same way) when you're dealing with macroscopic velocities that are larger than the thermal velocities?

2. Feb 28, 2017

### Staff: Mentor

What is the mean free path of your particles compared to the capacitor plate spacing? If it is small, you can model everything like a gas flow, just with positive and negative charges separately. If it is large, you also have to keep track of the velocity distribution at every point.

I'm not sure if your problem is perfectly well-defined: every charge accumulation will lead to a global electric field that does not go to zero for increasing distances.