The Algebra of an Infinite Grid of Resistors

Summary

The Algebra of an Infinite Grid of Resistors In a previous note we discussed the well-known problem of determining the resistance between two nodes of an �infinite� square lattice of resistors. The most common approach is to superimpose two �monopole� solutions, one representing the field for one amp of current entering a given node and flowing �to infinity�, and the other representing the field for one amp of current being withdrawn from a given node flowing in from infinity. If the two nodes are adjacent, the solution is fairly unambiguous, but the resistance between two arbitrary nodes of an �infinite� resistor grid is actually indeterminate unless we impose restrictions on the voltage and current levels �at infinity� (such as stipulating that we seek the solution for a finite grid in the limit as the size of the grid increases to infinity). According to the na�ve approach, we imagine injecting 1 amp of current into a single node at the origin, and allow it to flow outward symmetrically to infinity. Let Vm,n denote the voltage drop from the source node with coordinates (0,0) to any given node with coordinates (m,n). Likewise we could imagine extracting one amp from the node at (m,n), flowing in symmetrically from infinity, and the voltage drop from the origin to (m,n) would also be Vm,n. Superimposing these two solutions, we have one amp entering node (0,0) and exiting node (m,n), and the voltage drop is 2Vm,n, so the net effective resistance between (0,0) and (m,n) is Rm,n = 2Vm,n/I, where I = 1 amp. Of course, as mentioned in the previous note, the voltage �at infinity� must go to infinity relative to the voltage at the source, because the resistance to infinity is infinite. Nevertheless, it might seem that with suitable care we can evaluate the limit to arrive at an unambiguous answer. However, the stated problem has a unique solution only if we stipulate some restriction on the boundary conditions. This is a delicate proposition, since the boundary is �at inf...