Twelve wire, each having resistance r, are joined to form a cube as shown in figure.Find the equivalent resistance between the end of a face diagonal such as `a and c`.

Explanation

Suppose a potential difference V is applied between the points a and c so that a current I enters at a and the same current leaves at c. The current distribution is shown in figure.ltbygtBy symmetry, the paths ad and ab are equivalent and hence will carry the same current `i_(1).The path ah will carry the same current `i-2i_(1)(using Kirchhoff's junction law).Similarly at junction c, currents coming form dc and bc will be `i_(1)`each and fron fc will be `i-2i_(1)`.Kirchhoff's junction law at b and d shows that currents through be and dg will be zero and hence may be ignored for further analysis. Omitting these two wires, the circuit is redrawn in figure.
The wire hef and hgf are joined in parallel and have equivalent resistance`((2r)(2r))/((2r)+(2r))=r`between h and f.This is jioned in series with ah and fc giving equivalent resistance`r+r+r=3r`.This 3r is jioned in parallel with adc (2r)and abc(2r) between a and c,
the equivalent resostance R between a and c is, therefore, given by
`1/R=1/(3r)+1/(2r)+1/(2r),`
`giving R=3/4 r.`