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| Questions for those who know electromagnetism better than I do |
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| EPAIII:
Boy, I really wish I had some references on hand about how this "cloud" of electrons inside a conductor behaves. With no current flow I can easily see how their attraction to the nuclei of the metal atoms will keep them "bound" to those atoms and their charge density will be approximately uniform. But once a current does start to flow, and I am primarily considering a DC current at this point, then those bonds are effectively broken and, as is obvious from the very fact that a current IS flowing, those electrons are much freer to wander about. And they are still feeling both the attraction to those nuclei and the repulsion from each other. They will move from one atom to the next and while in flight between atoms, they will, while feeling that repulsion, be free to act on it and move away from the center. I like the reference to a FET. But perhaps a better one may be to the Hall Effect where any conductor can be used and not just semi-conductors. Relatively moderate magnetic fields can be used to induce a Hall Voltage with only moderate DC currents flowing through the Hall sensor. It is not much of a stretch to see where a moderate electrostatic repulsion would also create a potential difference (Voltage) inside a conductor. Those electrons that make up the current flow in most conductors are not just mobile: they are well lubricated. |
| T3sl4co1l:
--- Quote from: EPAIII on August 29, 2021, 01:39:01 am ---Boy, I really wish I had some references on hand about how this "cloud" of electrons inside a conductor behaves. With no current flow I can easily see how their attraction to the nuclei of the metal atoms will keep them "bound" to those atoms and their charge density will be approximately uniform. --- End quote --- Not quite sure if this is a sarcastic tone or what, but it's again not hard to find from the keywords -- https://en.wikipedia.org/wiki/Fermi_gas In fact, not only does the gas remain a gas in the absence of current flow (which is obvious given its statistics), but it remains so even down to absolute zero, where the Pauli exclusion principle creates a pressure forcing electrons into unbound states. Instead of single atoms with happy filled (well, filled up to neutrality) electron shells, the outer electron shells happen to overlap, indeed so much that they share one or more electrons each, or something like that. Something is also apparent given Ohm's law: there's no stick-slip motion here, it's smooth conduction all the way from zero to very high current densities (though again, not at all exceeding the charge density, because there's a L-O-T of electrons; pretty sure it vaporizes from ohmic loss before you can get close). The gas also proposes a mechanism for ohmic loss: the gas is in thermal equilibrium with the crystal lattice -- that is to say, there is an energy coupling between modes (electron-phonon exchange). Phonons (crystal vibrations) being the predominant mode of heat energy in solids. Effectively, the electrons collide every once in a while, exchanging momentum with ions, including momentum picked up by the feeble electric field inducing drift (i.e., electric current); thus ballistic motion (unimpeded acceleration) is prevented and the excess energy manifests as heat. (I'm not sure what a corresponding mechanism is for superconductors; I suppose the same must still apply, it just happens to be insufficient energy on average to disrupt the Cooper pairs, or whatever form the superconductivity takes.) For point of reference, the thermal velocity of an electron gas is on the order of 10^5 m/s, while the drift velocity of ordinary current density in copper is ~cm/s. It cools down a bit of course, but evidently retains velocity at 0 K (I don't know what average speed, alas). An actual real example of "zero point energy". ;) --- Quote ---But once a current does start to flow, and I am primarily considering a DC current at this point, then those bonds are effectively broken and, as is obvious from the very fact that a current IS flowing, those electrons are much freer to wander about. And they are still feeling both the attraction to those nuclei and the repulsion from each other. They will move from one atom to the next and while in flight between atoms, they will, while feeling that repulsion, be free to act on it and move away from the center. --- End quote --- If it helps, the environment seen by an electron in a crystal, is similar to what a wave sees inside a waveguide: more specifically a periodic potential waveguide, which has a bandstop characteristic (electromagnetic bandgap). Free propagation is allowed above and below the stopband; as it happens, in crystals, electrons are not free to move below the bandgap (valence band), but holes (absence of electrons) are; and electrons above (in the conduction band) are free. This is relevant to semiconductors and most insulators (most insulators are just wide-bandgap semiconductors, though I think there's a technical difference in some materials and I forget why), however conductors happen to have electrons naturally sitting in the conduction band, or the bandgap happens to be zero or negative energy width, in either case populating the conduction band. I also forget the technicals of how this comes to be, but it involves the earlier mentioned effects. --- Quote ---I like the reference to a FET. But perhaps a better one may be to the Hall Effect where any conductor can be used and not just semi-conductors. Relatively moderate magnetic fields can be used to induce a Hall Voltage with only moderate DC currents flowing through the Hall sensor. It is not much of a stretch to see where a moderate electrostatic repulsion would also create a potential difference (Voltage) inside a conductor. Those electrons that make up the current flow in most conductors are not just mobile: they are well lubricated. --- End quote --- Indeed, the Hall effect also gives a nonuniformity in a conductor. In this case, it's the effect of crossed magnetic and current fields. It also depends substantially on carrier density: metals are terrible because they have so many carriers they just swamp out the effect, and sensitive measurements are needed to demonstrate it. Semiconductors have few carriers that are strongly affected, and therefore semiconductor sensors are predominantly used. There's even an effect due simply to mechanical acceleration: there is a voltage across a disc (axis to rim) under rotation, even without a magnetic field. Tim |
| bsfeechannel:
--- Quote from: CatalinaWOW on August 24, 2021, 04:19:00 am ---You realize that you can use Maxwell's equations to do circuit analysis. But most folks find the simplified equations of Kirchoff are adequate and far easier to solve. --- End quote --- This is a misconception. KVL and KCL are not simplified Maxwell equations for circuit analysis. They are solutions to Maxwell’s equation for when there are no varying magnetic fields enclosed by—and no varying electric fields along—the path you’ve chosen to analyze your circuit. When that doesn’t happen, you have the following options. 1. Find analytical solutions to Maxwell’s equations. 2. Find numerical solutions to Maxwell’s equations. 3. Try to do away with the “offending” varying fields by employing techniques like shielding, grounding, impedance matching, decoupling, avoiding loops and excessive long lines, etc. 4. Pray. |
| CatalinaWOW:
--- Quote from: bsfeechannel on August 30, 2021, 03:00:49 am --- --- Quote from: CatalinaWOW on August 24, 2021, 04:19:00 am ---You realize that you can use Maxwell's equations to do circuit analysis. But most folks find the simplified equations of Kirchoff are adequate and far easier to solve. --- End quote --- This is a misconception. KVL and KCL are not simplified Maxwell equations for circuit analysis. They are solutions to Maxwell’s equation for when there are no varying magnetic fields enclosed by—and no varying electric fields along—the path you’ve chosen to analyze your circuit. When that doesn’t happen, you have the following options. 1. Find analytical solutions to Maxwell’s equations. 2. Find numerical solutions to Maxwell’s equations. 3. Try to do away with the “offending” varying fields by employing techniques like shielding, grounding, impedance matching, decoupling, avoiding loops and excessive long lines, etc. 4. Pray. --- End quote --- There are two misconceptions (at least) in your response. I did not say that kcl and kvl are simplified forms of Maxwell's equations. I was snidely suggesting that while circuits could be analyzed using Maxwell's, they represent special cases where simpler methods are adequate. And unless you have a different interpretation of enclosed and along, capacitors and inductors violate two of your conditions for use of k l and kcl. The whole point of the initial comments was that Maxwell's in all their glory are not required for the initial posters problem. In a nod to another post I will say that only one of the four is required, and not all of the terms in that one are non zero simplifying it further. This, plus the symmetries of the problem reduce it to algebra. No differential or integral calculus required. Engineers and physicists beyond there first couple of years of college should be able to handle the full form for a variety of simple geometries, but as you say, closed form solutions for more general problems range from difficult to currently unsolvable. Numerical solutions on today's world are tedious but difficult only in geometries that are so complex they tax computational resources. Much like computational aerodynamics but easier since many em problems are at least linear while the aero guys always have to deal with non linearity. |
| bsfeechannel:
--- Quote from: CatalinaWOW on August 30, 2021, 06:05:24 am ---capacitors and inductors violate two of your conditions for use of k l and kcl. --- End quote --- Not if you treat them as lumped components, in which case the circuit path will be between their terminals, away from their internal varying fields. |
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