Question

The circuit diagram shown in figure consists of a large number of elements (each element has two resistors $R_1$ and $R_2$). The resistance of the resistors in each subsequent element differs by a factor of $K=1/2$ from the resistances of the resistors in the previous elements. The equivalent resistance between $A$ and $B$ shown in figure is:

• A. $\frac{R_1-R_2}{2}$
• B. $\frac{(R_1-R_2)+\sqrt{6R_1{R}_2}}{2}$
• C. $\frac{(R_1-R_2)+\sqrt{R_1^2+R_2^2+6R_1{R}_2}}{2}$
• D. $Noneofthese$

Answer 1

The correct option is C $\frac{(R_1-R_2)+\sqrt{R_1^2+R_2^2+6R_1{R}_2}}{2}$

When each element of the circuit is multiplied by a factor k, then the equivalent resistance also becomes k times.

Let the equivalent resistance between A and B be $X$ which is shown in figure.
Now, $X=R_1+\left(R_2||KX\right)=R_1+\frac{R_{2}KX}{R_2+KX}$

$X-R_1=\frac{R_{2}KX}{R_2+KX}$

$K{X}^2-R_{1}KX+R_{2}X-R_1{R}_2=KX{R}_2$

$K{X}^2+(R_2-R_{1}K-R_{2}K)X-R_1{R}_2=0$

$X=\frac{-(R_2-R_{1}K-R_{2}K{)}_{-}^{+}\sqrt{(R_2-R_{1}K-R_{2}K{)}^2-4K(-R_1{R}_2)}}{2K}$

Negative root is invalid for resistance so we can take only positive root.

For $K=\frac{1}{2}$, $X=-(R_2/2-R_1/2)+\sqrt{(R_2/2-R_1/2{)}^2+2R_2{R}_2}$

$X=\frac{1}{2}\left[\right(R_1-R_2)+\sqrt{(R_1-R_2{)}^2+8R_1{R}_2}]$

$X=\frac{(R_1-R_2)+\sqrt{R_1^2+R_2^2+6R_1{R}_2}}{2}$