A Conducting Checkerboard

Phase 1, with electrical conductivity σ1, occupies the “bowtie” region of angle ±α, while phase 2, with conductivity σ2 σ1, occupies the remaining region. Deduce the approximate form of lines of current density J when a background electric field is applied along the symmetry axis of phase 1. What is the effective conductivity σ of the system, defined by the relation I = σΔφ between the total current I and the potential difference Δφ across the system? It suffices to consider the case that the boundary arc (r = a, |θ| < α) is held at electric potential φ = 1, while the arc (r = a, π − α < |θ| < π) is held at electric potential φ = −1, and no current flows across the remainder of the boundary.