We present ab initio calculations of the local current density j(r) as it arises in dc-transport measurements. We discover pronounced patterns in the local current density, ring currents (“eddies”), that go along with orbital magnetism. Importantly, the magnitude of the ring currents can exceed the (average) transport current by orders of magnitude. We find associated magnetic fields that exhibit drastic fluctuations with field gradients reaching 1  T nm−1 V−1. The relevance of our observations for spin relaxation in systems with very weak spin-orbit interaction, such as organic semiconductors, is discussed. In such systems, spin relaxation induced by bias driven orbital magnetism competes with relaxation induced by the hyperfine interaction and appears to be of similar strength. We propose a NMR-type experiment in the presence of dc-current flow to observe the spatial fluctuations of the induced magnetic fields.