Certain magnetic bearing designs, both radial and axial, contain a cylindrical radial flux return path. The magnetic flux in this path produces a radial negative stiffness and, if the journal is displaced, a side-pull force. In this paper, closed-form solutions are found for both of these properties by determining the magnetomotive force in the eccentric gap. This is achieved by solving the Dirichlet boundary value problem in the eccentric annulus. A conformal transformation to bipolar coordinates is utilized which results in a much simpler boundary value problem than if the physical coordinates are used. An example problem is presented which indicates the significance of these two properties.