Symmetries, recurrence, and explicit expressions of two-point resistances in 2 × n globe resistor networks

Abstract

Resistor networks are practical electric circuits and provide meaningful models to represent natural or artificial conductive structures. They can also be used to demonstrate how, in physics, general properties can be derived elegantly by combining general principles with symmetry properties. Here, this typical approach of physics is illustrated in the case of two-point resistances in the 2 × n globe network. It is a particular case of resistor networks on a sphere, where the n nodes of an equatorial frame of n identical resistors are connected by equal resistors to two axial poles. Recurrence relations are obtained using only Kirchhoff’s laws and Kennelly’s theorem. When complementing with relations derived using van Steenwijk’s method, explicit relations are obtained for all two-point resistances to any order n. Such analytical exact results are useful to test the results of numerical or integral methods. This complete treatment of the 2 × n globe network can be used to illustrate for students the efficient ways of physics to derive analytical results and understand their origin.