A cube is constructed by connecting 12 wires of equal resistance as shown in figure. The equivalent resistance between the points A and B shown in the figure is..... . The resistance of each wire is of r `Omega`. A and B are the midpoints of the sides PQ and VU respectively.

Explanation

Note that with reference to the line joining A and B, the pairs AP and UB, AQ and VB, PW and RU, QT and SV, WV and QR are symmetric branches. Hence, current flowing through each of this symmetric pair must be same. e.g. if the current flowing through PW is `(1)/(4)` , the same current (i.e. `(I)/(4) ` ) will flow through RU. With this consideration the proportional currents through the various circuit branches are as assigned their values in figure

Points W and T being symmetric about A are at the same potential, so no current will flow through WT and similarly also through SR.
Applying Kirchoff.s second rule to the closed loop APWVBNMA, taking r as the resistance of each wire.
` - (1)/(2) ((r)/(2)) - (I)/(4) r - (I)/(4) r - (I)/(2) ((r)/(2)) = - epsilon`
`therefore " IR " = epsilon " " ` .... (2)
Comparing equations (1) and (2),
r. = r