This paper addresses fully space-time adaptive magnetic field computations. We de-scribe an adaptive Whitney finite element method for solving the magnetoquasistatic for-mulation of Maxwell’s equations on unstructured 3D tetrahedral grids. Spatial mesh re-finement and coarsening are based on hierarchical error estimators especially designed for combining tetrahedral H (curl)-conforming edge elements in space with linearly implicit Rosenbrock methods in time. An embedding technique is applied to get efficiency in time through variable time steps. Finally, we present numerical results for the magnetic recording write head benchmark problem proposed by the Storage Research Consortium in Japan.