Infinite Resistor Ladder
المؤلف:
Sidney B. Cahn Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 1 , p 58
9-8-2016
1741
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 RA'C' should equal the original RAC, 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 RA'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 RAC. Calculation of RAB can be done in the same manner as for RAC.
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