Infinite Resistor Ladder

Consider the ladder of resistors, each of resistance r, shown in Figure 1.1. What is the resistance seen between terminals A and C?

Figure 1.1

SOLUTION

Define equivalent resistances R and R' as shown in Figure 1.2a. By symmetry, the equivalent resistances attached to points A and C are both R. The total resistance between terminals A and C is the sum of the series resistance

(1)

Now, rewrite the ladder one rung back, as in Figure 1.2b. The new resistance R_A'C' should equal the original R_AC, given that an infinite ladder of resistors has a finite resistance between the terminals in the first place. Without calculating loops, it can be seen that R' does not contribute to the resistance R_A'C' (Remove R'. What would be the potential difference between its terminals?). Ignoring B', we have the equivalent circuit in Figure 1.2c. The loop on the right may be replaced by

 Figure 1.2a                       Figure 1.2b                          Figure 1.2c

so

(2)

But, from (1),

We arrive at the quadratic equation

whose positive solution,

gives a resistance of   for R_AC. Calculation of R_AB can be done in the same manner as for R_AC.