In this talk, we present a new integral representation for outgoing solutions to the time harmonic Maxwell equations in an unbounded domain in R^3 . This representation leads to numerical methods for solving the problem of scattering from a perfect conductor which do not suffer from spurious resonances or low frequency breakdown. In the course of our analysis we prove the existence of new families of time harmonic solutions, which we call k-harmonic fields. These generalize, to non-zero wave numbers, the classical harmonic Neumann fields known to exist when the boundary of the region is not simply connected.