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Question
Twelve wires, each of equal resistance r, are joined to form a cube, as shown in the figure. Find the equivalent resistance between the diagonally-opposite points a and f.
Solution
Let V be the potential difference between the points a and f. Let current i enter a and leave from f. The distribution of current in various branches is shown in figure.
To calculate the potential difference between a and f, consider the path abcf and apply Kirchofff's Law:-
\[\frac{i}{3}r + \frac{i}{6}r + \frac{i}{3}r = V\]
\[\Rightarrow \left( \frac{2ir}{3} + \frac{ir}{6} \right) = V\]
\[ \Rightarrow \left( \frac{5ir}{6} \right) = V\]
The effective resistance between a and f,
\[R_{eff} = \frac{V}{i} = \frac{5}{6}r\]
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