The structure of the axisymmetric steady magnetosphere surrounding a rotating black hole is studied, allowing for electric potential variations along the lines of the poloidal magnetic field, which is idealized to have the geometry of a split-monopole. Maxwell's equations are written for the slow-rotation limit, using the Macdonald-Thorne 3+1 formalism for the Kerr geometry. A spherical source of plasma is supposed located beyond a certain radius, and a volume positron current flows inwards towards the horizon. A steady state is maintained by an inflowing net negative equatorial sheet current. It is shown that the boundary conditions at the horizon and the plasma source enforce a potential gradient along field lines, which can accelerate positrons to high energies. The eigenvalue for the total currents flowing into the hole is slightly less than that of the force-free case (≍ 82 per cent). In a fuller treatment the associated radiation drag should be included in the equation of motion, as in analogous pulsar models.