A normalized two-dimensional band-limited Brown fractal function is presented for modelling the rough surface. An analytic solution of electromagnetic scattering field from this conducting band-limited Brown fractal surface is given based on Kirchhoff approximation. The average scattering field, the average scattering coefficient and the average intensity of scattering field are derived with emphasis on examining the relation of fractal dimension with the scattering pattern. The important conclusion is obtained that the diffracted envelope slopes of scattering pattern can be approximated to a slope of linear equation.