Lately, I've been reading about techniques to reduce networks and find their equivalent resistance/capacitance. While doing this, I came across the cube resistance problem and many other problems (eg. resistors on tetrahedron etc.), where the authors have argued that certain points on the figure have the same potential. But, none of them have explained a procedure which would allow one to use this technique for other problems. So I've two questions:
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Does the figure need to be symmetrical in some manner if one has to use this technique?
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How should one go about finding points with the same potential?
I've tried a couple of things: Suppose that we were required to find the equivalent resistance across the main diagonal of a cube. Then usually I would distribute the currents and look for branches carrying the same current. From this, I would try to deduce the points having the same potential. But of course, this technique hasn't worked so any hints or suggestions will be valuable.
Best Answer
The general idea is to find and exploit symmetries in the network. A symmetry means that if you change something about the problem, it remains the same. Generalized method for dealing with circuit involving symmetry? links to a basic introduction to a formal procedure for identifying the symmetries of the network, and the effect these symmetries have on the relation between the inputs and outputs of the circuit. However, in many cases the symmetries are most easily identified visually from a diagram of the circuit.
For example, the following infinite ladder of resistors looks exactly the same if you add another "unit" at the front. This suggests a method of finding the total resistance $R_\infty$ which is the same as $R$ in series with $R$ || $R_\infty$ :
$$R_\infty=R+\frac{R_\infty R}{R_\infty+R}$$ This is a quadratic equation which can be solved to find $R_\infty$.
In your cube problem, resistors $a$, $b$ and $c$ are in equivalent positions - ie if you rotate the cube about an axis through AB you can replace $a\to b\to c\to a$ without making any difference to the resistance between A and B. This symmetry means that the points marked $\alpha$ are all at the same potential, as are those marked $\beta$.
Without affecting the circuit we can connect wires between the points marked $\alpha$ - and likewise between those marked $\beta$ - because no current will flow through them. The cube is then equivalent to the following series of parallel resistors :
http://www.rfcafe.com/miscellany/factoids/kirts-cogitations-256.htm