question_answer

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

Correct Answer: C

Solution :

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\[x\Omega \] \[\therefore \]  \[{{R}_{AB}}=1+\frac{x}{1+x}\] \[\therefore \]  \[x=1+\frac{x}{1+x}\] \[\Rightarrow \]               \[x+{{x}^{2}}=1+x+x\] \[\Rightarrow \]               \[{{x}^{2}}-x-1=0\] \[\Rightarrow \]               \[x=\frac{1+\sqrt{1+4}}{2}=\frac{1+\sqrt{5}}{2}\Omega \]