Question

Each resistor shown in the figure is an infinite network of resistance $1\Omega$. The effective resistance between points $\text{A}$ and $\text{B}$ is
($\sqrt{3}=1.73$)

• A. $1\Omega$
• B. $3\Omega$
• C. Less than $1\Omega$
• D. More than $1\Omega$ but less than $3\Omega$

Answer 1

The correct option is D More than $1\Omega$ but less than $3\Omega$


We can see that, section $\text{FACE}$ is similar to section $\text{HBDG}$.

For section $\text{FACE}$, let us assume equivalent resistance between $\text{A}$ and $\text{C}$ is $x$.

Since, section $\text{FACE}$ is an infinite ladder network, so we may consider the given circuit of $\text{FACE}$ as

From diagram, effective resistance between $\text{A}$ and $\text{C}$ is given by

$R_{AC}=\frac{(x+2)\times 1}{x+2+1}=\frac{x+2}{x+3}$

$\Rightarrow \frac{x+2}{x+3}=x[\because R_{AC}=x]$

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

$\Rightarrow x^2+2x-2=0$

$\Rightarrow x=\sqrt{3}-1$ or $x=-\sqrt{3}-1$

Since, value of resistance can't be negative.

$\therefore R_{AC}=\sqrt{3}-1=0.73\Omega$

Further, from symmetry, circuit become


$\therefore R_{AB}=0.73+1+0.73=2.46\Omega$

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

$$\begin{array}{c}\text{Why this question ?}\\ \text{Key Idea-This question help students}\\ \text{to understand that how we can combine}\\ \text{two or more infinite resistor series and}\\ \text{find the equivalent resistance.}\end{array}$$