Infinite Grid of Resistors

Summary

Infinite Grid of Resistors ����������������� Remain, remain thou here, While sense can keep it on. And, sweetest, fairest, As I my poor self did exchange for you, To your so infinite loss, so in our trifles I still win of you: for my sake wear this... ������������������������� ������������������������������� Shakespeare There is a well-known puzzle based on the premise of an �infinite� grid of resistors connecting adjacent nodes of a square lattice. A small portion of such a grid is illustrated below. Between every pair of adjacent nodes is a resistance R, and we�re told that this grid of resistors extends �to infinity� in all direction, and we�re asked to determine the effective resistance between two adjacent nodes, or, more generally, between any two specified nodes of the lattice. For adjacent nodes, the usual solution of this puzzle is to consider the current flow field as the sum of two components, one being the flow field of a grid with current injected into a single node, and the other being the flow field of a grid with current extracted from a single (adjacent) node. The symmetry of the two individual cases then enables us to infer the flow rates through the immediately adjacent resistors, and hence we can conclude (as explained in more detail below) that the effective resistance between two adjacent nodes is R/2. This solution has a certain intuitive plausibility, since it�s similar to how the potential field of an electric dipole can be expressed as the sum of the fields of a positive and a negative charge, each of which is spherically symmetrical about its respective charge. Just as the electric potential satisfies the Laplace equation, the voltages of the grid nodes satisfy the discrete from of the Laplace equation, which is to say, the voltage at each node is the average of the voltages of the four surrounding nodes. It�s also easy to see that solutions are additive, in the sense that the sum of any two solutions for given boundary conditions is a sol...