The present manuscript is a slightly expanded version of the article published in Foundations of Computational Mathematics, the main difference being a sharpened version of Theorem 4.1 with full proof thereof. We study inverse problems for the Poisson equation with source term the divergence of an R^3-valued measure; that is, the potential phi satisfies Laplacian(phi) = div mu, and mu is to be reconstructed knowing (a component of) the field grad(phi) on a set disjoint from the support of mu. Such problems arise in several electro-magnetic contexts in the quasi-static regime, for instance when recovering a remanent magnetization from measurements of its magnetic field. We develop methods for recovering mu based on total variation regularization. We provide sufficient conditions for the unique recovery of mu, asymptotically when the regularization parameter and the noise tend to zero in a combined fashion, when it is uni-directional or when the magnetization has a support which is sparse in the sense that it is purely 1-unrectifiable. Numerical examples are provided to illustrate the main theoretical results.