A new method is developed for expanding the electromagnetic field of radiating charges and currents in multipole components. Outside a sphere enclosing all sources, the field is represented in terms of Debye potentials which are shown to be closely related to the radial components of the electric and magnetic vectors. Attention is drawn to remarkably simple source representation of these radial components. A discussion of vector-potential versus Debye-potential representation of multipole fields is included. Only familiarity with ordinary vector calculus is required.