On the conductance of a lattice-like network

The conductance of a lattice-like network of thin uniform wires of cross section A and conductivity σ is calculated in an elementary manner. It is shown that the conductance S_μ of a network between two parallel metal plates of area A_p and separation d (with the neglect of the edge effect) is given by S_μ=A_pd^-1δ_μδ ∑ channelsδ_jcos²μ_j where cos μ_j are the directional cosines relative to the measurement direction μ (perpendicular to the plates) and the channel directions j (the directions of wires); σj are the channel conductivities and, in the present case, they are given by σ_j = n_jσA with n_j denoting the areal densities of wires for the channels j The above formulas (a) and (b) can be applied to a wide range of situations including the cases where the wires intersect with each other, the wires within each channel are not evenly spaced and the channels are not mutually orthogonal. The effective conductivity σ_μ, which in general depends on the measurement direction μ, can be regarded as the diagonal projection of a structure-material tensor characterizing the conduction property of the network. A few applications are discussed.