Sensing the magnetic field has been established as an essential part of navigation and orientation of various animals for many years. Only recently has the first detailed receptor concept for magnetoreception been published based on histological and physical results. The considered mechanism involves two types of iron minerals (magnetite and maghemite) that were found in subcellular compartments within sensory dendrites of the upper beak of several bird species. But so far a quantitative evaluation of the proposed receptor is missing. In this article, we develop a theoretical model to quantitatively and qualitatively describe the magnetic field effects among particles containing iron minerals. The analysis of forces acting between these subcellular compartments shows a particular dependence on the orientation of the external magnetic field. The iron minerals in the beak are found in the form of crystalline maghemite platelets and assemblies of magnetite nanoparticles. We demonstrate that the pull or push to the magnetite assemblies, which are connected to the cell membrane, may reach a value of 0.2 pN -- sufficient to excite specific mechanoreceptive membrane channels in the nerve cell. The theoretical analysis of the assumed magnetoreceptor system in the avian beak skin clearly shows that it might indeed be a sensitive biological magnetometer providing an essential part of the magnetic map for navigation.