We study a two-dimensional (2D) fermionic quantum-field-theoretical model used to model 1D strongly correlated electrons in the presence of a time-dependent impurity that drives the system out of equilibrium. In contrast to previous investigations, we consider a dynamic barrier switched on at a finite time. We compute the total energy density of the system and establish two well defined regimes in terms of the relationship between the frequency of the time-dependent perturbation Ω and the electron energy ω . Finally, we derive a relaxation time t R such that for times shorter than tR the finite-time switching process is relevant.Instituto de Física La Plata