Question

An infinite ladder is constructed with 1 Ω and 2 Ω resistors, as shown in the figure. (a) Find the effective resistance between the points A and B. (b) Find the current that passes through the 2 Ω resistor nearest to the battery.

Answer 1

(a) Let the effective resistance of the combination be R. The circuit can be redrawn as shown below.

From the figure,

\[\frac{2R}{R + 2} + 1 = R\]

\[ \Rightarrow 3R + 2 = R^2 + 2R\]

\[ \Rightarrow R^2 - R - 2 = 0\]

\[ \Rightarrow R = \frac{+ 1 + \sqrt{1 + 4 \times 1 \times 2}}{2 \times 1}\]

\[ \Rightarrow R = \frac{+ 1 + \sqrt{9}}{2 . 1} = 2 \Omega\]

(b) Total current sent by the battery

\[= \frac{6}{R} = \frac{6}{2} = 3 A\]

Applying Kirchoff's Law in loop 1, we get:-

\[3 \times 1 + 2i = 6\]

\[ \Rightarrow 2i = 3\]

\[ \Rightarrow i = \frac{3}{2} = 1 . 5 A\]