Two-point resistances in symmetric bracelet resistor networks: accurate estimates and exact expressions

Abstract

Resistor networks, used to model new types of natural or artificial matter, also provide generic examples for practising the methods of physics for obtaining estimates, revealing the main properties of a system and deriving exact expressions. Symmetric bracelet resistor networks are constructed by connecting n identical resistors in a circle, and then connecting two such circles by another set of n identical resistors. First, using van Steenwijk’s method, we establish that the equivalent resistance or two-point resistance (TPR) between any two nodes is derived when the layer-to-layer resistance R 0n is known. We then determine R 0n by an elementary recurrence relation which converges rapidly to its large n limit. Using this reference value of R 0n , accurate estimates of other TPRs follow for all values of n, characterised by a leading 1/n variation. In addition, exact explicit expressions of the TPRs can be calculated for any value of n. These networks, prototypes of three-dimensional networks considered in research, can be used to illustrate the diversity of the physical approach, the power of elementary methods, and to learn to be comfortable with approximations. Easy to make and use for experimental tests, they can support hands-on activities and conceptual changes.