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IPhO 1967 Problem 2

Problem 2
Consider an infinite network consisting of resistors (resistance of each of them is r) shown in Fig. 3. Find the resultant resistance RAB between points A and B.

Fig.3
(Taken from: I IPhO (Warsaw 1967-Theoretical problems-Problem 2)


Solution
It is easy to remark that after removing the left part of the network, shown in Fig. 4 with the dotted square, then we receive a network that is identical with the initial network (it is result of the fact that the network is infinite).

Fig.4
Thus, we may use the equivalence shown graphically in Fig. 5.

Fig.5
Algebraically this equivalence can be written as


Thus


This equation has two solutions:


The solution corresponding to “-“ in the above formula is negative, while resistance must be positive. So, we reject it. Finally we receive


Sources :
-Waldemar Gorzkowski, Institute of Physics, Polish Academy of Sciences, Warsaw, Poland
-The Olympic home page www.jyu.fi/ipho

 

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