Consider an infinte ladder network shown in figure. A voltage $V$ is applied between the points $A$ and $B$ . This applied value of voltage is halved after each section. Then value of $\frac{R_{2}}{R_{1}}$ is

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Solution

Let the potential at $A$ be $V$ , potential at next two junctions can be considered as $\frac{V}{2}, \frac{V}{4}$

Using Kirchhoff's Junction Law
$i = i_{1} + i_{2}$
$\frac{\frac{V}{2} - V}{R_{1}} = \frac{\frac{V}{4} - \frac{V}{2}}{R_{1}} + \frac{0 - \frac{V}{2}}{R_{2}}$
$- \frac{V}{2 R_{1}} = - \frac{V}{2 R_{2}} - \frac{V}{4 R_{1}}$
$\frac{V}{2 R_{1}} = \frac{V}{2 R_{2}} + \frac{V}{4 R_{1}}$
$\frac{1}{R_{1}} = \frac{1}{R_{2}} + \frac{1}{2 R_{1}}$
$\Rightarrow \frac{R_{2}}{R_{1}} = 2$

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