An infinite ladder is constructed with 1 and 2 resistors. It consists of repeating sections, each containing a 1 resistor in the top branch and a 2 resistor connecting the top and bottom branches. The bottom branch is a zero-resistance wire. A 6 V battery is connected across the input terminals. Calculate the current passing through the 2 resistor located closest to the battery.

Step-by-step solution

Let the equivalent resistance of the infinite ladder across the input terminals be R_ eq . Since the ladder is infinite, removing the first repeating unit leaves the equivalent resistance unchanged. Thus, the circuit can be modeled as a 1 resistor in series with a parallel combination of a 2 resistor and R_ eq . The equivalent resistance is given by: R_ eq = 1 + 2 R_ eq 2 + R_ eq R_ eq (2 + R_ eq ) = 2 + R_ eq + 2 R_ eq R_ eq ^2 + 2 R_ eq = 2 + 3 R_ eq R_ eq ^2 - R_ eq - 2 = 0 (R_ eq - 2)(R_ eq + 1) = 0 Since resis