Abstract
The radiation field of toroidal-like time-dependent current configurations is investigated. The infinitesimal time-dependent configurations are found outside which the electromagnetic strengths disappear but the potentials survive. For a number of time dependences, their finite radiationless counterparts can be found. In these cases topologically non-trivial (unremovable by a gauge transformation) electromagnetic potentials exist outside sources. The well-defined rule obtained for constructing time-dependent infinitesimal sources suggests the existence of finite non-trivial radiationless sources with an arbitrary time dependence. The latter can be used to carry out time-dependent Aharonov-Bohm-like experiments and to transfer the information. Using the Neumann-Helmholtz parametrization of the current density we represent the time-dependent electromagnetic field in a form convenient for applications.