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Registered Member #2063
Joined: Sat Apr 04 2009, 03:16PM
Location: Toronto
Posts: 352
so my physics teacher thought it would be funny to give us a hard question just to make us feel dumb. 12 resistors have the same resistance of R ohms each. a 3D cube is constructed with those resistors, find the equation for resistance across any 2 diagonaly oposite corners. after getting mind blown for an hour after school i decided to build my own resistor cube and measure the resistance my self :) I thought it would just be 3 resistors in a series string, then 6 string in parallel. but turns out that wasn't the correct answer.
the resistors I used were 220K each and the resistance across a diagonally opposite corner is 180K :(
Registered Member #2099
Joined: Wed Apr 29 2009, 12:22AM
Location: Los Altos, California
Posts: 1716
I applaud your experimental approach. And generally take a dim view of analytic solutions to interesting problems that aren't verified empirically, in cases where doing the latter is easy and fun.
The trick for solving this class of problems (like infinite grids of resistors) is to look for symmetry.
In the cube case, by symmetry, the three intermediate nodes adjacent to the start node must be at the same potential, call it A. So they could be shorted together, without changing any R currents or node voltages.
And the other three intermediate nodes, all adjacent to the end node, are at another common potential, call it B. So -they- could be shorted.
From start to A we have 3 R's in parallel, equivalent to R/3. From A to B we have 6 R's in parallel, equivalent to R/6. From B to end we have 3 R's in parallel, equivalent to R/3. Total between opposite corners of cube = 5R/6.
Please give credit to the Internet if you present this to your physics teacher without having thought of it independently.
[edit] and for extra fun: connect a voltage source between the opposite corners and measure voltage across various R's. What percentage will be across the R's adjacent to start and end points? What percentage will be across the middle rank of R's? What are the individual resistor currents, with respect to total current?
Registered Member #2099
Joined: Wed Apr 29 2009, 12:22AM
Location: Los Altos, California
Posts: 1716
For better empirical accuracy, you might want to measure some of your left-over resistors from the same batch, to see if they are consistently a little below their nominal value.
Since you already made the cube, you can check a couple of other simple exercises. 1. resistance between adjacent nodes. 2. resistance between nodes separated by 2 resistors
Symmetry makes their solutions particularly easy. Not as trivial as the cube-diagonal case, but no calculator is needed. Do you measure (or figure) around 128 K for case 1, and 165 K for case 2?
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how to solve a 3D resistor cube?
