Question

The equivalent resistance of the figure i.e., infinite network of resistors between the terminals $A$ and $B$ is

• A. zero
• B. infinite
• C. $\frac{R_1+R_2+R_3}{3}$
• D. $\frac{1}{2}\left[\left(R_1+R_2\right)+\sqrt{\left(R_1+R_2\right)(R_1+R_2+4R_3}\right]$

Answer 1

Answer: (D)

Let $R$ be the equivalent resistance. Then addition/subtraction of one more set of resistors $R_1,R_2$ and $R_3$ will not affect the total resistance. Thus,

$R=R_1+($ parallel combination of $R$ and $R_2)+R_3$
$R=R_1+\left(\frac{R{R}_3}{R+R_3}\right)+R_2$
$\Rightarrow R^2+R{R}_3=R{R}_1+R_1{R}_3+R{R}_3+R{R}_2+R_2{R}_3$
$\Rightarrow R^2-R\left(R_1+R_2\right)=\left(R_2+R_1\right)R_3=0$
$\Rightarrow R=\frac{\left(R_1+R_2\right)\pm \sqrt{{\left(R_1+R_2\right)}^2+4\left(R_1+R_2\right)R_3}}{2}$
As $R$ cannot be negative, hence
$R=\frac{1}{2}\left[\left(R_1+R_2\right)+\sqrt{{\left(R_1+R_2\right)}^2+4\left(R_1+R_2\right)R_3}\right]$
$=\frac{1}{2}\left[\left(R_1+R_2\right)+\sqrt{\left(R_1+R_2\right)\left(R_1+R_2+4R_3\right)}\right]$