Question

An infinite ladder network of resistance is constructed with a $1\Omega$ and $2\Omega$ resistance as shown in figure. What is the value of effective resistance between $\text{A}$ & $\text{B}$?

• A. $1\Omega$
• B. $2\Omega$
• C. $3\Omega$
• D. $4\Omega$

Answer 1

The correct option is B $2\Omega$

Let the effective resistance between points between $\text{C}$ & $\text{D}$ be $R$, then the circuit can be redrawn as shown below.


The effective resistance between $\text{A}$ & $\text{B}$ is

$R_{eq}=1+\frac{2\times R}{R+2}$

This resistance $R_{eq}$ can be taken as $R$ because if we add one identical item in infinite item, then the result will be same. Therefore,

$\therefore R=1+\frac{2\times R}{R+2}$

$\Rightarrow R^2+2R=3R+2$

$\Rightarrow R^2-R-2=0$

$\Rightarrow (R+1)(R-2)=0$

Taking positive value, because resistance can't be negative

$\therefore R=2\Omega$

Hence, option $\left(b\right)$ is correct.