Determination of the fractal dimension of equipotential surfaces in a region confined by rough conductors

We consider a region bounded by two conductors held to a constant voltage bias, one of them with an irregular rough shape and the other being a flat one. The irregular profile can be either a curve with a formation rule or the result of a deposition process. The rough shape of the profile influences the equipotential lines, which we have characterized by numerically evaluating their roughness exponent α and fractal dimension Df. For a fixed finite size system, the less corrugated lines, far away from the rough profile, have higher α. For a line corresponding to a fixed value of the potential, the roughness exponent decreases with the size of the profile, suggesting that a single constant value characterizes all lines for an infinite system.