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

High energy cosmic ray self-confinement close to extragalactic sources