Sum

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.

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#### 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,

\[R_{eff} = \frac{V}{i} = \frac{5}{6}r\]

Concept: Kirchhoff’s Rules

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