The Infinite Resistor Problem
r
r
r
r
r
r
The goal of the problem is to reduce the above circuit into one which replaces the infinite string of resistors
with one equivalent resistance R in terms of r. This circuit then would look like,
R
The key to the problem is recognizing that the fundamental repetitive unit which consists of two resitors,
and the fact that the number of these units is infinite, allows us to equivalently identify the following circuit
as the same as above,
r
r
R
The second and third diagrams is tantamount to the following equation,
R=r+
Rr
,
r+R
(1)
where the left hand side of 1 is the equivalent resistance of diagram 2, and the right hand side is the equivalent
resistance of diagram 3. Solving this equation for R and selecting the positive root gives,
Ã
√ !
1+ 5
R=
r.
(2)
2