Question

The equivalent resistance between points A and B of an infinite network of resistance s each of $1\Omega ,$ connected as shown, is

• A. infinite
• B. 2 $\Omega$
• C. $\frac{1+\sqrt{5}}{2}\Omega$
• D. zero

Answer 1

Answer: (C)

Let $x$ be the equivalent resistance of entire network between $A$ and $B$ . Hence, we have
$R_{AB}=1+$ resistance of parallel combination of $1\Omega$ and $x\Omega$
$\therefore$ $R_{AB}=1+\frac{x}{1+x}$
$\therefore$ $x=1+\frac{x}{1+x}$
$\Rightarrow$ $x+x^2=1+x+x$
$\Rightarrow$ $x^2-x-1=0$
$\Rightarrow$ $x=\frac{1+\sqrt{1+4}}{2}=\frac{1+\sqrt{5}}{2}\Omega$