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Physics-

General

Easy

Question

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

Infinite
2 $Ω$
$\frac{1 + \sqrt{5}}{2} Ω$
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 1 $capital omega$ 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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