We derive the equations governing the dipolophoretic motion of an electrically inhomogeneous Janus particle composed of two hemispheres with differing permittivities. The general formulation is valid for any electric forcing, including alternating current (AC) and makes no assumptions regarding the size of the electric double layer (EDL). The solution is thus valid even for nanoparticles where the particle radius can be of the same order as the EDL thickness. Semi-analytic and numerical solutions for the linear phoretic velocity and angular rotation of a single Janus particle suspended in an infinite medium are given in the limit of uniform direct current (DC) electric forcing. It is determined that particle mobility is a function of the permittivity in each hemisphere and the contrast between them as well as the EDL length. For a particle in which both hemispheres are characterized by a finite permittivity, we discover that maximum mobility and rotation is not obtained in the Helmholtz-Smoluchowski thin EDL limit but is rather a function of the permittivity and EDL properties.