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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.
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Solution
(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:-
\[3 \times 1 + 2i = 6\]
\[ \Rightarrow 2i = 3\]
\[ \Rightarrow i = \frac{3}{2} = 1 . 5 A\]
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