| Electronics > Beginners |
| Is it possible to know all voltages/amps in a grid or matrix of resistors? |
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| rstofer:
--- Quote from: The Electrician on June 08, 2018, 03:14:43 pm --- --- Quote from: T3sl4co1l on June 07, 2018, 05:16:32 am ---(e.g., the sum of 1/n^2 for n = 1, 2, ..., is exactly 2). Proof of this fact, is, uh, left as an exercise for the student, yes. ;D Tim --- End quote --- Mathematica thinks the sum is not 2: :) --- End quote --- That is correct but the theory is well beyond my skills: https://en.wikipedia.org/wiki/Basel_problem |
| T3sl4co1l:
:palm: I meant to say (1/2)^n. But that too. :P Tim |
| IanB:
--- Quote from: rstofer on June 08, 2018, 03:54:02 pm ---That is correct but the theory is well beyond my skills: https://en.wikipedia.org/wiki/Basel_problem --- End quote --- This may help, if you have not seen it before: https://youtu.be/d-o3eB9sfls |
| Beamin:
--- Quote from: T3sl4co1l on June 07, 2018, 07:03:44 pm ---Graphene doesn't work, because any given electron zipping around in the field of atoms experiences dozens of atoms simultaneously. The electron is a wave confined by a boundary condition; that it ever has resistance is an accident brought on by defects and impurities, encountered with low probability over long distances -- in other words, in the thermodynamic limit. In other other words, it doesn't help to reason about quantum systems from the smallest possible unit, stacked up. There are long range order, and effects, that pull in later. You have to run an integral over the ensemble -- quantum statistical mechanics. A notoriously hard problem, because to calculate that integral, you must account for, not just the effect of each differential unit upon itself, but upon its neighbors as well! Think about taking the frequency response of an amplifier, by looking at just one tiny little blip. You get good information about the high frequency response, but the low frequency response is a slow and minuscule 'tail' that is, at best, inconvenient to read in this way. How much long-term information can you infer from a very short blip? Not much. Much better, use a holistic method, like frequency domain (Fourier) analysis, to find the properties overall. (In QM, this is called k-space.) The equivalent to this principle in linear electronics is: if you cascade a bunch of AC filters, you must account for the interactions between them, which in general won't be well-behaved if you pick everything randomly. In this case, we can use the transmission ('T' or 'ABCD') matrix to solve the problem: a two-port filter is only two zero-dimensional connections, so we don't have to worry about waves or distance or phase or any of that, we simply get four complex numbers describing the input, transmission and output properties of the block. There are six standard, equivalent matrix types used to describe these things, but the handy part about the T-matrix is, you simply multiply them together, and that's your total answer. Suppose you could derive the electron wave function's T-matrix for graphene, for one very small differential unit; now suppose you can cascade infinity of them in all directions. Well, it's not going to be a two-port, because if nothing else, there are two directions to integrate over. Maybe a four-port would work? Unsure. It's... complicated. ;D Tim --- End quote --- It always amazes me how you can put concepts that I would never hope to understand like k-space and how graphene is NOT a resistor matrix into things I can understand. From what I know about the quantum world that answer makes perfect sense. Each bond is a probability not an actual path for an electron "particle" to take, like in the sea of electrons in metal where it conducts because you have so many. A single atom isn't a metal or a gas or a solid etc. Like NaCl is part of an ionic solid because you can't have just a NaCl molecule without the others to keep it inline in the cube. The stuff they teach you in chemistry class rarely goes into that: protons are also charged hydrogen/an acid and act nothing like a proton in a nuclear reactor. I remember learning nuclear in chemistry and the students were totally baffled including the professor with a phd) on why beta decay doesn't just result in a full orbital or a -1 atom; like you would expect in an aqueous solution. The beta particle has way too much energy: same with how alpha isn't just hot ionized helium bouncing around like in a neon light. |
| T3sl4co1l:
Yup, those cases are about equilibrium vs. nonequilibrium conditions, and also energy levels relative to orbits and such. Chemistry isn't very meaningful under nonequilibrium conditions; it can be done, but it takes heroic effort (e.g., studying single-molecule reactions with lasers and a frozen argon matrix). Some beta decays do result in ions directly -- read up on the "electron capture" process. :) I like to say nothing is a stronger Bronsted-Lowry acid than a proton beam, but given that it's more of a mechanical effect (sheer kinetic energy sputters the target away), it's kind of a silly claim. :P Conversely, I suppose vacuum tubes have to deal with the strongest possible Lewis acid/base, but electrons are rather easier to deal with than protons, even at modest energies (100s keV). Insulators may not find it very pleasant (see Lichtenberg figures!), but it seems metals have little problem with that! Tim |
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