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I'm very open to the possibility that I'm missing something important in all this. But I still don't feel like you've made any clear argument about why my energ

I'm very open to the possibility that I'm missing something important in all this. But I still don't feel like you've made any clear argument about why my energy reasoning about the fields outside of the parallel plates fails. Your initial response to me here didn't address the exterior field at all: your linked calculations (which you called "the full derivation") were explicitly just about fields between the plates, and in fact assumed the plates were infinite. You also claimed there that I had "failed to apply conservation of momentum correctly in the general case of two different masses", even though my objection had nothing to do with the mass difference. I do agree that the non-zero field outside of the gap was the subject of the previously linked subthread. But the discussion there was about "ballenarosada"'s (flawed) estimate of the exterior field: my overall claim (and theirs!) is that the specific functional form of that field can't matter, since we know from fundamental principles that its integral (the electric potential) must go to zero at infinity. You claimed there that if one neglects plasma screening, "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." This claim is incorrect. Again, the easy way to prove that is just to note that the electric potential goes to zero at infinity. But if you want to actually consider the electric field due to a finite parallel plate capacitor, you can compute that easily enough along the axis of a pair of circular plates. As it must be, the integral from the midpoint to the positive plate is exactly equal and opposite to the integral from the positive plate to infinity (independent of the plate radius or the separation). Your final claim is that plasma screening means that integrating all the way to infinity isn't necessary. I'll admit that the details of that are well outside my area of expertise, and I'd appreciate hearing more about it. But my strong sense is that if the number of charged particles is finite, the fundamental result from basic electromagnetism must remain true regardless of screening effects: the electric potential at infinity must be zero. (Right??) So as I understand it, screening effects can make significant changes to the shape of the potential curve, but regardless of those details its values at the plate and at infinity remain unchanged. And that's all we need to conclude that there is no net thrust for an ion that doesn't change direction (like the ones that the author claims will produce thrust). In particular, if we imagine a positively charged plate in a plasma, I can easily imagine a negative screening charge accumulating right next to the plate, which would in a very short distance bring the exterior field to zero. But at the same time, that nearby accumulation of negative charge would strengthen the field right next to the plate, almost like a second parallel plate capacitor. The effect would be that positive ions outside of the positive plate would be strongly repelled, and would in turn push the device in exactly the opposite of the claimed direction of thrust. I'd love to learn more about the plasma physics of this situation. But I'm less interested in the specific details there than I am in understanding anything that would interfere with the basic energy analysis that I've outlined here and above.

T-A 7y ago
> Your initial response to me here didn't address the exterior field at all My initial response to you was to your statements about a subthread where I addressed the exterior field. > your linked calculations (which you called "the full derivation") were explicitly just about fields between the plates That was in response to your invocation of angular momentum. > and in fact assumed the plates were infinite. No. This is what I said in that response: "Once it's through the screen, the force changes sign, but quickly drops to zero (for infinitely large screens, it is exactly zero everywhere outside the space bounded by the two screens)." The statement about infinite screens is something which any physicist would understand immediately: a reminder that there is a simple case (the limit of infinite size) which tells you how things must go as you scale up the device. If your calculations don't reproduce the known result in that limit (and ballenarosa's don't; they actually blow up in that limit) you know you're doing something wrong. There is no assumption that the plates are infinite. In fact, thanks to plasma screening, the electric field outside the plates falls off much quicker than you'd expect even from the correctly calculated vacuum expression (which I linked to in that thread). The Debye length in low Earth orbit is measured in millimeters [1][2], so for any device significantly larger than that, the far field which you keep invoking is completely negligible. > You also claimed there that I had "failed to apply conservation of momentum correctly in the general case of two different masses", even though my objection had nothing to do with the mass difference. A look at the derivation should tell you that the mass difference is crucial for the device to work. If the ions being ejected in opposite directions were of equal mass, there would indeed be no net momentum transfer to the device. In other words, I was actually trying to make the most charitable interpretation possible of your argument. (The more severe one being that somebody who professes to teach E&M does not even understand basic properties of a multipole expansion.) > the specific functional form of that field can't matter, since we know from fundamental principles that its integral (the electric potential) must go to zero at infinity No. What matters is how fast it goes to infinity. That determines what you get when you integrate over it, from the plate out to infinity. If the potential's gradient falls to zero right at the plate surface, and then stays there, all you get is zero. If it falls off slower, you get more. How much? Enough to give back all the momentum picked up inside the device? To answer that, you certainly need to know the functional form. > You claimed there that if one neglects plasma screening, "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." This claim is incorrect. This "claim" is correct, and easily verified by simply looking at the referenced closed form expression for the field outside a plate capacitor. > Again, the easy way to prove that is just to note that the electric potential goes to zero at infinity. Again, this is nonsense. It is the functional form of the potential out to infinity which determines how much momentum the ions transfer back to the device on their way out. The potential gradient being zero, at infinity or at any other point, simply tells you that no momentum can be transferred at that point. It tells you exactly nothing about how much momentum was transferred on the way to that point, unless you know the full functional form of the potential along that way. > if you want to actually consider the electric field due to a finite parallel plate capacitor, you can compute that easily enough along the axis of a pair of circular plates Yes, that's done in the reference I provided in the ballenarosa subthread. > As it must be, the integral from the midpoint to the positive plate is exactly equal and opposite to the integral from the positive plate to infinity (independent of the plate radius or the separation). Are you integrating the potential?!? That only tells you that energy is conserved (surprise!). It doesn't tell you how it's distributed. Look again at the derivation: kinetic energy of the ion (exchanged with the potential) is proportional to v^2, momentum is proportional to v (no square). That's what makes the device work: energy and momentum of the system (device + ions) are both conserved (of course) but not transferred between its components at the same rate. You want to integrate the equations of motion of the ions as they move in the potential, not the potential. As for this, > In particular, if we imagine a positively charged plate in a plasma, I can easily imagine a negative screening charge accumulating right next to the plate, which would in a very short distance bring the exterior field to zero. But at the same time, that nearby accumulation of negative charge would strengthen the field right next to the plate, almost like a second parallel plate capacitor. The effect would be that positive ions outside of the positive plate would be strongly repelled it also makes no sense at all. First you say (correctly) that plasma screening "would in a very short distance bring the exterior field to zero", but after just one more sentence you go "positive ions outside of the positive plate would be strongly repelled", a complete contradiction. I am sorry, but at this point I must retract my previous claim that I don't find it hard to believe your claimed profession. [1] https://en.wikipedia.org/wiki/Debye_length#Typical_values https://en.wikipedia.org/wiki/Debye_length#Typical_values [2] https://arxiv.org/abs/0906.3884 https://arxiv.org/abs/0906.3884

T-A 7y ago

> Your initial response to me here didn't address the exterior field at all My initial response to you was to your statements about a subthread where I addressed the exterior field. > your linked calculations (which you called "the full derivation") were explicitly just about fields between the plates That was in response to your invocation of angular momentum. > and in fact assumed the plates were infinite. No. This is what I said in that response: "Once it's through the screen, the force changes sign, but quickly drops to zero (for infinitely large screens, it is exactly zero everywhere outside the space bounded by the two screens)." The statement about infinite screens is something which any physicist would understand immediately: a reminder that there is a simple case (the limit of infinite size) which tells you how things must go as you scale up the device. If your calculations don't reproduce the known result in that limit (and ballenarosa's don't; they actually blow up in that limit) you know you're doing something wrong. There is no assumption that the plates are infinite. In fact, thanks to plasma screening, the electric field outside the plates falls off much quicker than you'd expect even from the correctly calculated vacuum expression (which I linked to in that thread). The Debye length in low Earth orbit is measured in millimeters [1][2], so for any device significantly larger than that, the far field which you keep invoking is completely negligible. > You also claimed there that I had "failed to apply conservation of momentum correctly in the general case of two different masses", even though my objection had nothing to do with the mass difference. A look at the derivation should tell you that the mass difference is crucial for the device to work. If the ions being ejected in opposite directions were of equal mass, there would indeed be no net momentum transfer to the device. In other words, I was actually trying to make the most charitable interpretation possible of your argument. (The more severe one being that somebody who professes to teach E&M does not even understand basic properties of a multipole expansion.) > the specific functional form of that field can't matter, since we know from fundamental principles that its integral (the electric potential) must go to zero at infinity No. What matters is how fast it goes to infinity. That determines what you get when you integrate over it, from the plate out to infinity. If the potential's gradient falls to zero right at the plate surface, and then stays there, all you get is zero. If it falls off slower, you get more. How much? Enough to give back all the momentum picked up inside the device? To answer that, you certainly need to know the functional form. > You claimed there that if one neglects plasma screening, "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." This claim is incorrect. This "claim" is correct, and easily verified by simply looking at the referenced closed form expression for the field outside a plate capacitor. > Again, the easy way to prove that is just to note that the electric potential goes to zero at infinity. Again, this is nonsense. It is the functional form of the potential out to infinity which determines how much momentum the ions transfer back to the device on their way out. The potential gradient being zero, at infinity or at any other point, simply tells you that no momentum can be transferred at that point. It tells you exactly nothing about how much momentum was transferred on the way to that point, unless you know the full functional form of the potential along that way. > if you want to actually consider the electric field due to a finite parallel plate capacitor, you can compute that easily enough along the axis of a pair of circular plates Yes, that's done in the reference I provided in the ballenarosa subthread. > As it must be, the integral from the midpoint to the positive plate is exactly equal and opposite to the integral from the positive plate to infinity (independent of the plate radius or the separation). Are you integrating the potential?!? That only tells you that energy is conserved (surprise!). It doesn't tell you how it's distributed. Look again at the derivation: kinetic energy of the ion (exchanged with the potential) is proportional to v^2, momentum is proportional to v (no square). That's what makes the device work: energy and momentum of the system (device + ions) are both conserved (of course) but not transferred between its components at the same rate. You want to integrate the equations of motion of the ions as they move in the potential, not the potential. As for this, > In particular, if we imagine a positively charged plate in a plasma, I can easily imagine a negative screening charge accumulating right next to the plate, which would in a very short distance bring the exterior field to zero. But at the same time, that nearby accumulation of negative charge would strengthen the field right next to the plate, almost like a second parallel plate capacitor. The effect would be that positive ions outside of the positive plate would be strongly repelled it also makes no sense at all. First you say (correctly) that plasma screening "would in a very short distance bring the exterior field to zero", but after just one more sentence you go "positive ions outside of the positive plate would be strongly repelled", a complete contradiction. I am sorry, but at this point I must retract my previous claim that I don't find it hard to believe your claimed profession. [1] https://en.wikipedia.org/wiki/Debye_length#Typical_values https://en.wikipedia.org/wiki/Debye_length#Typical_values [2] https://arxiv.org/abs/0906.3884 https://arxiv.org/abs/0906.3884