Two-point resistance of a cobweb network with a $2r$ boundary

TL;DR

This paper derives a general formula for calculating the two-point resistance in an m by n cobweb network with a 2r boundary, extending previous solutions to more complex boundary conditions.

Contribution

It introduces a simplified direct summation method for resistance calculation in cobweb networks with a 2r boundary, surpassing traditional Greens function and Laplacian approaches.

Findings

Derived a general resistance formula for finite and infinite cobweb networks.

Presented examples illustrating the application of the formula.

Compared the new method with existing techniques, highlighting its simplicity.

Abstract

We consider the problem of two point resistance on an $m times n$ cobweb network with a 2r boundary which has never been solved before. Past efforts prior to 2014 researchers just only solved the cases with free boundary or null resistor boundary. This paper gives the general formulae of the resistance between any two nodes in both finite and infinite cases using a method of direct summation pioneered by Tan [Z.Z.Tan, et al, J. Phys. A 46, 195202 (2013)], which is simpler and can be easier to use in practice. This method contrasts the Greens function technique and the Laplacian matrix approach, which is difficult to apply to the geometry of a cobweb with a 2r boundary. We deduced several interesting results according to our general formula. In the end we compare and illuminate our formulae with two examples. Our analysis gives the result directly as a single summation, and the result is mainly composed of the characteristic roots.