Question

Twelve wires, each of equal resistance r, are joined to form a cube, as shown in the figure (32-E20). Find the equivalent resistance between the diagonally-opposite points a and f.

Answer 1

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:
$\frac{i}{3}r+\frac{i}{6}r+\frac{i}{3}r=V$

$$\begin{gathered} \Rightarrow \left(\frac{2ir}{3}+\frac{ir}{6}\right)=V \\ \Rightarrow \left(\frac{5ir}{6}\right)=V \end{gathered}$$
The effective resistance between a and f,
$R_{\mathrm{eff}}\mathit{}=\frac{V}{i}=\frac{5}{6}r$