Abstract
Searching for the explicit solutions of the potential function in an arbitrary resistor network is important but difficult in physics. We investigate the problem of potential formula in an arbitrary m × n globe network of resistors, which has not been resolved before (the previous study only calculated the resistance). In this paper, an exact potential formula of an arbitrary m × n globe network is discovered by means of the Recursion-Transform method with current parameters (RT-I). The key process of RT method is to set up matrix equation and to transform two-dimensional matrix equation into one-dimensional matrix equation. In order to facilitate practical application, we deduced a series of interesting results of potential by means of the general formula, and the effective resistance between two nodes in the m × n globe network is derived naturally by making use of potential formula.
Highlights
Searching for the explicit solutions of the potential function in an arbitrary resistor network is important but difficult in physics
Andre Geim and Konstantin Novoselov found the graphene network which is a real-plane resistor network existed in nature, who won the 2010 Nobel Prize in Physics for their investigations
Poisson equation and Laplace equation are the important equations of potential function[16,17], and the potential theory is about the general theory of solutions to Laplace equation
Summary
Searching for the explicit solutions of the potential function in an arbitrary resistor network is important but difficult in physics. We investigate the problem of potential formula in an arbitrary m × n globe network of resistors, which has not been resolved before (the previous study only calculated the resistance). Several effective methods for calculating the resistance of resistor networks have been found, such as Cserti[10] set up the Green function technique to evaluate the resistance of infinite lattices; Wu12 formulated a Laplacian matrix method and derived the resistance in arbitrary finite and infinite lattices by means of the eigenvalues and eigenvectors. The researchers have made new progress in the multi-functional N-order resistance network[38,39], and the result derived by the RT method has been applied to the impedance network[40]
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