Question

Find the value of resistor to be connected between $\text{C}$ and $\text{D}$, so that the resistance of the entire circuit between $\text{A}$ and $\text{B}$ does not change with the number of elementary sets.

• A. $R$
• B. $R(\sqrt{3}-1)$
  
• C. $3R$
• D. $R(\sqrt{3}+1)$
  

Answer 1

The correct option is B $R(\sqrt{3}-1)$


Let the resistor to be connected across $CD$ is $x$.

Then the equivalent resistance across $EF$ should be $x$ and also across $AB$ should be $x$.

The equivalent circuit is shown in the figure below.


So, from the above circuit,

$x=\frac{(2R+x)R}{(2R+x)+R}$

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

Solving quadratic equation, we get

$x=R(-1\pm \sqrt{3})$

We choose the positive value

$x=R(\sqrt{3}-1)$

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

Key concept: In an infinitely long network, addition or removal of one section from the chain still leaves the chain infinite.