Fractal properties of equipotentials close to a rough conducting surface

The Koch curve is used in the problem of evaluating and characterizing the electric equipotential lines in the infinite semi-space limited by a rough conducting one-dimensional surface. The solution of Laplace's equation subject to a constant potential difference between the curve and a straight line placed at infinity is performed with the help of Liebmann's method, The fractal dimension, D-f, of the equipotentials is numerically evaluated with a box-counting method. It is found that D-f decays exponentially with distance, from the value D-f = 1.273 at the Koch curve to the D-f = 1.0 when the equipotentials become flat smooth lines. The method does not depend on the specific choice of the Koch curve to model the rough substrate.