Effective resistances of two dimensional resistor networks

Abstract: We investigate the behavior of two dimensional resistor networks, with finite sizes and different kinds (rectangular, hexagonal, and triangular) of lattice geometry. We construct the network by having a network-element repeat itself $L_x$ times in $x$-direction and $L_y$ times in the $y$-direction. We study the relationship between the effective resistance ($R_\mathrm{eff}$) of the network on dimensions $L_x$ and $L_y$. The behavior is simple and intuitive for a network with rectangular geometry, however, it becomes non-trivial for other geometries which are solved numerically. We find that $R_\mathrm{eff}$ depends on the ratio $L_x/L_y$ in all the three studied networks. We also check the consistency of our numerical results experimentally for small network sizes.