118 karma · joined September 20, 2017
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.
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.
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.
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.
On the other hand, if the capacitor is finite, then the surface integrals over the plates are not equal.
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.
Capitalism really doesn't have anything to do with tribalism though. Workers don't have any allegiance whatsoever to their employer beyond that dictated by the imperative of not starving to death. The structural constraint under critique is the one where capital seeks to maximize profits regardless of the human cost. So, when a grower exports food for money instead of giving it to starving people for free, that's what I'm talking about.
- "the only alternative is this particular strawman policy, which would make things worse"
- "there's nothing absurd about people starving due to preventable causes"
It's true that individuals are all acting rationally, and that imposing limits on exports would not be a panacea. However, at the most basic level, it is monstrous that food exports are happening in the context of local starvation. The rationality of the outcome within capitalism shouldn't blind us to the absurdity of the outcome itself.
Based on the article, I don't think this is right. http://astronomy.swin.edu.au/cosmos/S/Spin-flip+Transition, linked, describes the 21cm emission as occurring once every 10 million years for a single atom.