Physics-
General
Easy

Question

The equivalent resistance between points A and B of an infinite network of resistances, each of 1blank capital omega, connected as shown is

  1. Infinite    
  2. 2blank capital omega    
  3. fraction numerator 1 plus square root of 5 over denominator 2 end fraction capital omega    
  4. zero    

The correct answer is: fraction numerator 1 plus square root of 5 over denominator 2 end fraction capital omega


    Let x be the equivalent resistance of entire network between A and B. Hence, we have

    R subscript A B end subscript equals 1 plus resistance of parallel combination of 1capital omega and x capital omega
    therefore blank R subscript A B end subscript equals 1 plus fraction numerator x over denominator 1 plus x end fraction
    therefore blank x equals 1 plus fraction numerator x over denominator 1 plus x end fraction
    ⟹ x plus x to the power of 2 end exponent equals 1 plus x plus x
    ⟹ x to the power of 2 end exponent minus x minus 1 equals 0
    ⟹ x equals fraction numerator 1 plus square root of 1 plus 4 end root over denominator 2 end fraction
    equals fraction numerator 1 plus square root of 5 over denominator 2 end fraction capital omega

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