Conductivity of a Medium Containing a Dense Array of Perfectly Conducting Spheres or Cylinders or Nonconducting Cylinders

The effective electrical conductivity σ is computed for a composite medium consisting of a dense cubic array of identical, perfectly conducting spheres imbedded in a medium of conductivity σ₀. When f, the fractional volume occupied by the spheres, is near its maximum value π/6, the result is $${\mathrm{\sigma /\sigma }}_0\text{=−(π/2)}\log \text{[(π/6)−f]+…,(π/6)−f≪1}$$ . This result exhibits the singularity of σ at f=π/6, when the spheres touch each other. The previous results of Maxwell, of Rayleigh and of Meredith and Tobias are not valid near the singularity and they fail to reveal it. For f=0.5161 our result yields σ/σ₀=7.65, while the measurement of Meredith and Tobias yielded σ/σ₀=7.6. For a medium containing a square array of perfectly conducting circular cylinders we obtain $${\mathrm{\sigma /\sigma }}_0{\mathrm{=\pi }}^{\frac{3}{2}}{\text{/2[(π/4−f)]}}^{\frac{1}{2}}\text{+…,(π/4)−f≪1}$$ . This result agrees well with the numerical results of H. B. Keller and D. Sachs. We also prove that for any value of f, σ/σ₀ for a medium containing a square array of nonconducting cylinders is the reciprocal of σ/σ₀ for the same array of perfectly conducting cylinders.