Eight identical resistors, each of resistance r, are connected along edges of a pyramid having square base ABCD as shown in the figure. Calculate equivalent resistance.
(a) between A and D.
(b) between A and O

Explanation

(a) When a battery is connected across terminals A andD. circuit becomes symmetric about a plane passing through mid points of edges BC and AD and vertex O. Therefore, current through different resistors will be as shown in fig(i)
Since currents in AO and OD is equal, therefore they10 may be assumed in series with each other. Similarly, BO and OC may be assumed to be in series
Hence, the circuit may be assumed as shown in Fig. (ll). Therefore, equivalent resistance of each series combination (BO and OC, AO and OD) is equal to 2r. Series combination of BO and OC is parallel with BC Equivalent resistance of this combination
`=(rxx2r)/(r+2r)=(2r)/(3)`
Now, this combination is in series with resistors AB and CD. Therefore, equivalent resistance of this series combinationFor mesh ODAO`i_(1)r+(i_(1)+(i_(2))/(2))r-(i-2i_(1)-i_(2))r=0`or`4i_(1)+(3)/(2)i_(2)=i`
From Eqs. (1) and (2)`i_(1)=(1)/(5)i`and, `i_(2) =(2)/(15)i`
is For mesh OBAVO
`i_(1)r+(i_(1)+(i_(2))/(2))r-V=0`
Substituting values of `l_(1)`and `i_(2)`
`(7)/(15)ir=V`But equivalent resistance of the circuit =`(V)/(i)`Equivalent resistance=`(7r)/(15)`