  Electromagnetic radiation can be emitted not only by particle charges but
also by magnetic moments and higher electric and magnetic multipoles. However
experimental proofs of this fundamental fact are extremely scarce. In
particular, the magnetic moment contribution has never been observed in any
form of polarization radiation. Here, we propose to detect it using vortex
electrons carrying large orbital angular momentum (OAM) \ell. The relative
contribution of the OAM-induced magnetic moment, \ell \hbar \omega/E_e, becomes
much larger than the spin-induced contribution \hbar \omega/E_e, and it can be
observed experimentally. As a particular example, we consider transition
radiation from vortex electrons obliquely incident on an interface between a
vacuum and a dispersive medium, in which the magnetic moment contribution
manifests itself via a left-right angular asymmetry. For electrons with E_e =
300 keV and \ell = 100-1000, we predict an asymmetry of the order of 0.1%-1%,
which could be measured with existing technology. Thus, vortex electrons emerge
as a new tool in the physics of electromagnetic radiation.
