We present a theory of the nature of plasmons in nanowires along with the response of such systems to a spatially uniform applied electric field in the plane perpendicular to the symmetry axis of the wire. We confine our analysis to the electrostatic description of these properties and to modes with infinite wavelength parallel to the wire's symmetry axis. Our theory thus focuses on the limit where the linear dimensions of the cross section are small compared to the wavelength of radiation that may illuminate the system. We derive integral equations that involve only the electrostatic potential on the boundary of the wire for a wire of arbitrary cross section. Once this is solved, a complete description of the potential may be generated. Companion equations are obtained for the function that provides one with the lines of electric field in the system. The homogeneous versions of these equations provide one with a means of finding the plasmon eigenfrequencies and eigenfunctions, whereas the inhomogeneous equations allow one to generate the response of the nanowire to a spatially uniform applied field. We present numerical studies of the plasmon normal modes and electromagnetic response of nanowires of rectangular cross section, and we compare our results to experiments and to a calculation done by other methods. © 2010 The American Physical Society.