In geosciences and paleomagnetism, estimating the remanent magnetization in old rocks is an important issue in order to study the Earth magnetic field. However, the magnetization can not be directly measured, but it produces an external magnetic field that can be recorded. We consider the situation of very thin and weakly magnetized rocks samples with magnetization m (modeled as a 3-dimensional vector field defined on some planar set S). Scanning microscopes furnish measurements b_3[m] of the vertical component of the associated magnetic field on a planar region Q located at some fixed height h > 0 above the sample plane. We assume that both S and Q are Lipschitz-smooth bounded connected open planar sets and that the magnetization m belongs to [L^2(S)]^3, whence b_3[m] belongs to L^2(Q). Such magnetizations possess net moments defined to be their integral on S. Recovering the magnetization m or its net moment from the available measurements b_3[m] are inverse problems for Poisson-Laplace partial differential equations in the upper half-space with right hand side in divergence form. Neumann type data b_3[m] are available on Q while we aim at recovering m or its net moment on S. We thus face recovery issues on the boundary of the harmonicity domain from (very partial) data available inside. These inverse problems are typically ill-posed and require regularization in order to be solved. Indeed, magnetization recovery issues lack uniqueness, due to the existence of non trivial silent sources m (i.e., such that b_3[m] = 0). Although such sources have vanishing net moment, moment estimation turns out not to be stable with respect to measurements errors. The present work points at non-uniqueness and instability properties of the inverse magnetization problem. We characterize silent sources, (unique) equivalent magnetization of minimal L^2(S)-norm to some given m in [L^2(S)]^3 (two magnetizations are called equivalent if their difference is silent), and establish density / instability results.