The approximability problem of electromagnetic field

The relationship between the geometrical properties of a surface and the properties of the electromagnetic field generated by a current arbitrarily distributed on the surface is discussed. There is a continual cardinality of surfaces for which this field cannot even approximately describe any randomly chosen pattern or any field in the near region. Study of these surfaces is based on the fact that they are zero surfaces of some auxiliary electromagnetic field which obeys the Maxwell equation. The mere proximity of the surface to any surface having these properties gives rise to nontrivial properties in the fields generated by the currents inducted on the surface.