We solve the time-dependent Maxwell's equations in all generality for an infinite medium in which the electric and magnetic permittivities and the conductivity are constant. In addition to the Ohmic current density, we include the effects of the presence of a charge density distribution and an external current density submerged in the medium. The longitudinal component of the magnetic field is shown to be zero, and the longitudinal component of the electric field is the gradient of the Coulomb potential, as in the case of zero conductivity. The longitudinal component of the external current density is given in terms of the charge density, as a generalization of the equation of continuity. The transverse electromagnetic fields are solved in terms of the initial values of the transverse fields and the transverse part of the external current density. Riemann functions are introduced for the problem. A critical wavelength is found such that fields associated with wavelengths shorter than this critical wavelength attenuate more rapidly in time than fields associated with wavelengths longer than the critical wavelength. Moreover, the fields with the shorter wavelengths are superpositions of standing waves that have a damping factor independent of the wavelength, whereas the fields whose wavelengths are longer than the critical wavelength are superpositions of `static' helical fields, except for the damping factor which does depend on the wavelength. Stratton's travelling and `diffusive' plane wave solutions appear as special cases of these fields. There appear to be implications in biology and naval communications, which are discussed briefly. To obtain our results we use eigenfunctions of the curl operator, which we have used earlier to solve a variety of problems involving vector fields.