Timeline for Resistance between two points on a conducting surface
Current License: CC BY-SA 2.5
29 events
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| Oct 17, 2015 at 9:57 | answer | added | Roger44 | timeline score: 0 | |
| Feb 19, 2011 at 3:34 | vote | accept | TROLLHUNTER | ||
| Feb 19, 2011 at 3:34 | vote | accept | TROLLHUNTER | ||
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| Feb 19, 2011 at 3:34 | history | bounty ended | TROLLHUNTER | ||
| Feb 19, 2011 at 3:33 | vote | accept | TROLLHUNTER | ||
| Feb 19, 2011 at 3:34 | |||||
| Feb 17, 2011 at 9:30 | history | bounty started | TROLLHUNTER | ||
| Feb 16, 2011 at 10:23 | comment | added | Luboš Motl | Moreover, the logarithm could also be a constant, and one or both answers could also be strictly infinite because of the singular point endpoints (requiring an infinite current density around it). | |
| Feb 16, 2011 at 10:21 | comment | added | Luboš Motl | Dear @genneth, just to be sure, I am not too sure about my scalings, either. The dimensional analysis depends on figuring out which place is the most characteristic one to get the right conductance or resistance. The vicinity of the two endpoints themselves seem to be giving the biggest resistance - a lot of current per a very small area - while the space in between where there's enough room seem to dominate the conductance. Now, who wins? ;-) I would prefer to follow or do the full calculation. Is someone doing it in the answers? | |
| Feb 16, 2011 at 9:53 | history | edited | TROLLHUNTER | CC BY-SA 2.5 |
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| Feb 16, 2011 at 9:39 | history | edited | TROLLHUNTER | CC BY-SA 2.5 |
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| Feb 16, 2011 at 9:28 | history | edited | TROLLHUNTER | CC BY-SA 2.5 |
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| Feb 16, 2011 at 8:55 | answer | added | Martin Gales | timeline score: -2 | |
| Feb 15, 2011 at 21:33 | answer | added | Kostya | timeline score: 5 | |
| Feb 15, 2011 at 16:48 | answer | added | Omega Centauri | timeline score: 1 | |
| Feb 15, 2011 at 16:02 | answer | added | Bossavit | timeline score: -3 | |
| Feb 15, 2011 at 13:09 | comment | added | genneth | Although Luboš and I differ in our calculations, I think we're thinking the same thing. In n-D, a point contact causes the current goes out in a ball, and distributes according to $1/r^{n-1}$; at the same time it sets up a potential which goes as $1/r^{n-2}$ (or $log r$ in 2D). Potentials and currents are additive, so you can calculate the potential difference and the total current which flows, thus getting the conductance/resistance. Incidentally, I believe Luboš is correct, and I was wrong. | |
| Feb 15, 2011 at 11:54 | comment | added | Georg | Even for a infinite surface or a infinite half-space the resistance is finite. That is why "earth" was used in early telegraph lines as a conductor. Resistance is located in the vicinity of the contact "points". If distance between two metal rods in earth exceeds some dozen of the poles diameter, the resistance becomes constant. Still today an issue for lightning protection | |
| Feb 15, 2011 at 11:43 | comment | added | TROLLHUNTER | @Luboš Im not sure I follow, the relevant area could be infinite. | |
| Feb 15, 2011 at 11:26 | comment | added | Luboš Motl | Hi @kakemonsteret, good. For $d$ much smaller than $l$, the resistance between the points is of course much smaller than the whole $R$ but it is not zero. To derive genneth's scalings, imagine that you cut a plane in between the two points, at distance $d/2$ from each point. The relevant radius of the area where the current is flowing goes like $d$. So in the 2D or 3D material, you have about $O(d)$ or $O(d^2)$ parallel branches, and each of them has resistance going like $d$, I think, i.e. conductance $1/d$. This fast guess: the total conductance goes like $\ln(d/l)$ or $d$ for 2D, 3D case. | |
| Feb 15, 2011 at 10:43 | comment | added | TROLLHUNTER | @Luboš Yes, exactly, the two points are far from the boundary | |
| Feb 15, 2011 at 10:40 | comment | added | Georg | Defining a 2D conductor with some surface resistance would have been much easier to understand :=( I see one problem in the ansatz: You define two "points" where current is to go in or out. Close to that point current density is infinite. I think a more realistic ansatz is a 2D surface and two circle contacts. | |
| Feb 15, 2011 at 10:39 | comment | added | Luboš Motl | If the two points are close enough to the boundary of the disk (at radius $r$), you surely need to know how close both of them are. If both of them are much further from the boundary of the disk than $d$ as well as $l$, then the boundary is irrelevant and you approximate the situation by an infinite material. For $d\gg l$, it is a 2D material; for $d\ll l$, it is a 3D material. | |
| Feb 15, 2011 at 10:34 | history | edited | Luboš Motl | CC BY-SA 2.5 |
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| Feb 15, 2011 at 10:09 | comment | added | genneth | If I'm right about the infinite (3D) block above, then in the limit $r \gg d \gg l$ it behaves like an infinite 2D block, and the conductance goes as $1/d$. | |
| Feb 15, 2011 at 10:05 | comment | added | genneth | In the other limit of small $d$, it's never going to be zero; in that limit you could approximate things by assuming an infinitely large conductor, in which case the conductance drops as something like $1/d^2$ (I think... no calculations!) | |
| Feb 15, 2011 at 10:04 | history | edited | TROLLHUNTER | CC BY-SA 2.5 |
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| Feb 15, 2011 at 10:03 | comment | added | TROLLHUNTER | Yes, like a surface | |
| Feb 15, 2011 at 10:01 | comment | added | genneth | $r \gg d \gg l$ means that it's a really short cylinder? | |
| Feb 15, 2011 at 8:14 | history | asked | TROLLHUNTER | CC BY-SA 2.5 |