Determine the resistance R AB between points A and B of the frame made of thin homogeneous wire (as shown in figure), assuming that the number of successively embedded equilateral triangles (with sides decreasing by half) tends to infinity. Side AB is equal to a, and the resistance of unit length of the wire is ρ .
Text Solution
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R AB = 
Sol. It follows from symmetry considerations that the initial circuit can be replaced by an equivalent one (as shown).
We replace the inner triangle consisting of an infinite number of elements by a resistor of resistance R A B / 2, where the resistance R AB is such that R AB = R x and R AB = a ρ . After simplification, the circuit becomes a system of series and parallel connected conductors. In order to find R x , we write the equation
R x = R 
Solving the equation, we obtain
R AB = R x = 
= 
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