  The ultra-high energy cosmic rays observed at the Earth are most likely
accelerated in extra-galactic sources. For the typical luminosities invoked for
such sources, the electric current associated to the flux of cosmic rays that
leave them is large. The associated plasma instabilities create magnetic
fluctuations that can efficiently scatter particles. We argue that this
phenomenon forces cosmic rays to be self-confined in the source proximity for
energies $E<E_{\rm cut}$, where $E_{\rm cut}\approx 10^{7} L_{44}^{2/3}$ GeV
for low background magnetic fields ($B_{0}\ll nG$). For larger values of
$B_{0}$, cosmic rays are confined close to their sources for energies $E<E_{\rm
cut}\approx 2\times 10^{8} \lambda_{10} L_{44}^{1/4} B_{-10}^{1/2}$ GeV, where
$B_{-10}$ is the field in units of $0.1$ nG, $\lambda_{10}$ is its coherence
lengths in units of 10 Mpc and $L_{44}$ is the source luminosity in units of
$10^{44}$ erg/s.
