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A Cube of Resistors |
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Consider a cubic arrangement of eight vertices, with electrical resistance R along each edge connection as shown below. |
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What is the resistance between the vertices marked 000 and 111? If we suppose a voltage V is applied to vertex 111 and a voltage 0 (ground) is applied to vertex 000, then by symmetry the other vertices have one of just two values, which we will call v1 and v2, as shown below. |
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The current flowing out of the 111 vertex is I = 3(V−v2)/R, and the current flowing into the 000 vertex is I = 3v1/R. Also, the current flowing from the vertices with voltage v2 to the vertices with voltage v1 is I = 6(v2−v1)/R. These relations imply v1 + v2 = V and 3v1 = v2, from which we get v1 = 2V/5 and hence I = (6/5)V/R. Therefore, the effective resistance between the 000 and 111 vertices is 5R/6. |
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We could take a similar approach, exploiting symmetries, to solve for the resistance between two adjacent vertices, or between two vertices diagonal on a face, but its interesting to consider a more general approach. For convenience we designate the verticies with the letters a,b, ,h as shown below. |
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If there were no external connections, the net current into (or out of) each node would be zero, so letting each letter denote the voltage at the respective vertex, and assuming unit resistance on each edge, we have the eight equations |
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In matrix form this can be written as |
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The coefficient matrix is singular, but if we set a = 0 and b = 1, we can delete the first row and column, and replace the b row with the equation b=1, and then, noting that the current into vertex a (which is the reciprocal of the effective resistance) is given by b+c+h, we have |
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Hence the resistance between adjacent vertices is 7R/12. Likewise to find the resistance between vertices at opposite diagonals of a face, we can replace the d row with d=1, which gives |
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This shows that the resistance between d and a is 3R/4. Lastly, to find the resistance between opposite vertices of the cube, we can replace the f row with f=1, which gives |
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Hence, as we showed previously, the resistance between f and a is 5R/6. By symmetry, these three cases cover all the vertex-to-vertex resistances in the cube. |
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