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We consider theoretically the dynamics of electric currents optically injected in carbon nanotubes. Although the plasma oscillations are not seen in these systems, the main effect on the carrier's motion is due to strongly nonuniform space-charge Coulomb forces produced by time-dependent separation of injected electron and hole densities. We calculate the evolution of the dipole moment characterizing the time- and coordinate-dependent charge-density distributions and analyze different regimes of the dynamics. The developed time-dependent dipole moment leads to a dipole radiation in the THz frequency range for typical parameters of injected currents.