I'm currently working on an extension to modify the built-in symbolic calculator to display its stack entries in rendered LaTeX--it already has various language modes for displaying its s-expression based formulas, including LaTeX, so this honestly isn't that big of a lift.
It's essentially "rich people are richer than me, and that makes me mad." There is no novel idea here to engage with. People have been jealous of wealth since the beginning of time. But now we cloak that jealously with faux moral outrage and elaborate economic "solutions."
Can't it be "poor people are getting poorer because rich people are taking wealth away from them on balance, and that makes us mad". I mean... that's bad right? It should make us mad. We're a wealthier society on average than we were in the 80's, we all should be wealthier, yet we aren't.
I genuinely don't understand the denialism on this. You don't have to be a raving socialist to recognize that an outcome is unfair. Can't you be a free market republican and want to see the poor do better along with the rest of us?
Don't fall for what? The fact that some people move between percentiles during their life doesn't change the fact that the distribution has grown more unequal. What the number means is that, while those who are poor are poorer now in relative terms than 1989, the rich have gotten richer in relative terms.
Taking on debt to finance a public good like high speed rail is perfectly reasonable. Privatization isn't a natural outcome of a public corporation running a deficit--it's the result of a political decision to disinvest in that project.
Agree. Government debt is a good thing - just ask fiscal conservative Ronald Reagan, who was the first president since World War II to increase the national debt relative to the GDP. Government is not a company, and judging it by the standards of a company is a mistake. (If anything, it is the dual of a company.)
Ronald Reagan lowered government revenues with a round of tax cuts and sharply raised government spending. The debt was entirely from increasing the military budget without having taxes to pay for it. I would not describe his presidency as fiscally conservative.
Good question. Under the influence of a magnetic field, charged particles do all sorts of counterintuitive things due to the Lorentz force, which acts at right angles to the particle velocity and the field.
One example is a sideways precession due to the interaction between gravitational acceleration g and the Lorentz force. Fortunately this effect is basically negligible, because gravity is so much weaker than the electromagnetic forces operating on the particles.
The difficulties tend to come from the geometry of the magnetic field itself. For example, in a torus shape, there is an unavoidable drift in the vertical direction due to the combination of centrifugal force in the big circumference, and the force implied by the gradient of the magnetic field.
My understanding is that stellarators are designed in such a way that this sideways drift in one part of the device is canceled by opposite drift in another part due to the twisting of the field.
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.
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.
> 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.
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
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.
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.
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.
Eq. (14) is just the expression for a dipole. Keep reading to see the full solution in the simplest case (circular plates), Eq. (19).
Since you insist: your derivation goes wrong right at the start by, as you say, "ignoring curvature effects", i.e. by considering radial distance only. By doing that, you are effectively imposing spherical symmetry; you are not doing parallel plates, you are doing concentric spherical shells. That makes the whole exercise pointless, since it's obvious from symmetry alone that such a device could never produce a net thrust: there is no preferred direction for the thrust to act along.
To answer your final question, just look at Eq. (19): make the plate radius (R) larger.
That answer should also be perfectly obvious from the limit case of infinite plates. A correct derivation for the general case must reproduce that result in that limit. Yours does the opposite; it gets worse the larger you make the capacitor. In the infinite limit, it is infinitely wrong.
You're right, my derivation is mistaken for failing to take the z component of the force. EQ. 19 is more like it although you'll note, also goes as 1/z^3.
You're also right that all that is moot from sheathing. But an ion still begins it's journey out at the bottom of a large potential well; one which is particularly steep because of the debye length, but still just as deep.
