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An infinite ladder network of resistances is constructed with 1\Omega and 2\Omega resistances as shown in fig. Battery between A and B has negligible internal resistance. The effective resistance between A and B is

Option: 1

2\Omega

 


Option: 2

3\Omega


Option: 3

4\Omega


Option: 4

5\Omega


Answers (1)

best_answer

Let R be the equivalent resistance between points A and B. Here we assume that one more set of resistances is connected between A and B as shown in fig. The connection of one additional set will not affect the resistance R because there are infinite number of such sets connected between A and B.

Resistance between A{}' and B{}'

\begin{aligned} & R^{\prime}=\frac{2 R}{R+2}+1 \\ & =\frac{3 R+2}{R+2} \end{aligned}

But \: \mathrm{R}=\mathrm{R}^{\prime}

\begin{aligned} & \text { Hence } R=\frac{3 R+2}{R+2} \\ & R^2+2 R=3 R+2 \\ & R^2-R-2=0 \\ & R=\frac{+1 \pm \sqrt{1+8}}{2}=-1 \text { or } 2 \end{aligned}

Hence R=2\Omega.

Posted by

Divya Prakash Singh

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