Analytical solution for DC field, transparency condition, and estimation of resistivity for radially anisotropic sphere of conductivity rm

We investigate the direct current (DC) field distribution for a sphere of special conductivity. While a general case requires the introduction of either special functions or numerical approach, we show that in cases where specific conductivity of the sphere increases as rm function, where m ≥ 0, the analytical solution can be found. This analytical solution is then used to analyze three concrete examples: a DC field distribution of a sphere submerged into a electric current field, the conditions required for the sphere to become transparent, and the resistivity of the sphere for a symmetric pair of electrodes of a variable size.