[ballenarosada]

This won't work. The basic idea about the protons gaining more momentum than the electrons is valid. But the dipole creates an opposite field outside of the charged plates. Protons will be decelerated until they pass the first, positively charged plate, then accelerated through the plates, then decelerated back towards the negative plate.

This is all clear if you consider the ions falling through a potential field. The potential is 0 at infinity, positive at the first plate and negative at the second. An incoming ion starts at 0 potential, climbs a big hill to get through the first plate, then falls down below 0. Then on the way out it has to climb back to 0 potential at infinity. So the ions gain energy inside the plates but lose it all back on either side.

[andbberger]

Thanks for this, excellent simple argument cutting right to the core of it.

I'm a fool. Don't speculate about E&M before you've had your coffee kids.

[T-A]

https://physics.stackexchange.com/questions/12309/electric-f...

[ballenarosada]

The electric field outside an infinite capacitor is zero. For a finite capacitor, there is a nonzero field. The importance of the infiniteness assumption can't be understated--such a capacitor cuts the universe in half, and every point of one half has the same electric potential.

On the other hand, if the capacitor is finite, then the surface integrals over the plates are not equal.

[T-A]

> The importance of the infiniteness assumption can't be understated

No, but as you have shown it can be grossly overstated. The field outside a finite plate capacitor falls off as a power of distance >= 2 (details depend on the geometry), while the field inside it is constant. It can therefore safely be ignored for a first order estimate of the effect.

If you want to get fancy and claim that higher order corrections invalidate Zubrin's argument, you need to actually prove it. Also, don't forget to include other effects like plasma shielding.

[ballenarosada]

Well, before breaking out a higher order analysis, I'd like to at least see a real first order analysis. The argument from potential at infinity is dispositive, but let's do some practice anyway:

Let A be the area of the capacitor, and dr the distance between the plates. Let c be the appropriate electrostatic constant for the coulomb force between a proton and the charge density on the plate. At a point a distance r from the capacitor, the field effect from the negative side is, ignoring curvature effects, about

cA/r^2

The repelling charge from the other plate will be about

cA/(r+dr)^2 = cA/(r^2 + 2rdr + dr^2) ~ cA(1/r)(1/(r+2dr))

So the difference between the coulomb forces, i.e. the net force, will be approximately

(cA/r)(1/r - 1/(r+2dr)) = (cA/r)((r+2dr - r)/(r^2 + 2rdr)) ~ cA(2dr/r^3)

So the net force drops off approximately as the third power of the distance, to a first order approximation. Integrating over the radius, we have that the potential goes as -1/r^2, with the approximation breaking down near r=0.

Actually inserting appropriate constants of integration would make this argument robust, but would also just reduce to the argument from potential at infinity. Either way it's clear that the effect can't just be ignored out of hand.

[T-A]

According to your derivation, the net force grows linearly with capacitor area. Alas, the external field of a plate capacitor with infinite area is exactly 0. You can look up the correct way to do a multipole expansion in any introductory EM textbook, or google up nice a exposition like [1].

What really matters here is that with the force on the charge falling off as a power of distance, even if you integrate force * displacement from the screen out to infinity (which you shouldn't do in a plasma, because [2]), you get a finite contribution which can be made arbitrarily small relative to the work done inside the capacitor, where the force is constant, simply by increasing the size of the capacitor.

[1] http://student.ndhu.edu.tw/~d9914102/Teaching/EM/Paper/data/...

[2] https://en.wikipedia.org/wiki/Electric-field_screening

[ballenarosada]

If you read the exposition you linked, equation 14 gives an expression for the field which is linear in the area. Again, it's pretty important that the capacitor be infinite in extent, otherwise it behaves differently.

What's the dimension that you're proposing to increase of the capacitor? The total work done across the capacitor will be fixed regardless of distance across.