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Interesting Physics Problem (Resistors)?
Follow this link:
Please, no comments on the nature of xkcd.com. My friend told me about this site, and I was bored with an hour to kill, so I gave in. A fair portion of the comics are actually quite clean, though-provoking and entertaining.
Anyways - the Physics Problem! I took AP Physics and know much of the basic laws of resistors and the Diff EQ's that govern inductors and capacitors, but Kirchoff's rule is failing me. The current is supposed to follow the path of least resistance - if we partition the grid into two infinite sets of parallel resistors towards the top and bottom, does the effective resistance become just 1? Or would it be 0?
@LeAnne:
Are you sure it's that simple? After all, there are only 4 resistors extending from each point. I have a hard time beleiving that these finite extensions can be quaffed by an "added" infinite parrallel resistor. Can you prove this, perchance?
3 réponses
- Scythian1950Lv 7il y a 1 décennieRéponse favorite
I recall this sort of problem, which in fact will give you a resistance value between 0 and 1, depending on which 2 nodes are picked. Let me check the web to see if I can find something for you. It's far from a "classroom" problem. Whole papers have been devoted to this subject, as the first link indicates.
I know, for example, for a infinite square network of 1 Ω resistors, the effective resistance between two diagonally opposite nodes in a square is 2/π, which is an interesting example of how π has a way of popping up in unexpected places. What's stranger is that this is the same value in Buffon's Needle Problem where the spacing of parallel lines equals the length of the needle, 2/π.
This is not an "idle" poser, since it's related to finding the effective resistance between two points where electrodes touch an infinite sheet of a semi-conductor.
Edit: Okay, I have found your answer. The effective resistance between the 2 nodes as in the cartoon is:
-(1/2) + 4/π = 0.77324...
You can check this and other values in the PDF paper in the 2nd link. See 2nd page.
Source(s) : http://www.springerlink.com/content/l6x16827531887... http://atkinson.fmns.rug.nl/public_html/resist.pdf - LeAnneLv 7il y a 1 décennie
The key word here is "infinite" - an infinite number of resistances in parallel will absolutely result in zero resistance - or a number infinitely close to zero.
- Anonymeil y a 5 ans
What is the EMF? The voltage given by the battery is it shown on the diagram??