The addition of small quantities of an absorbing dye can modify the optical properties of liquid crystals, in particular by producing a photo-induced torque reorienting the director. Such an effect is modelled by including in the free energy the contribution of dye-solvent electrostatic interactions between the molecular charges and the solvent polarisation. The system evolution is described through the torque acting on the director and the diffusion equations for the orientational distributions of the dye in the ground and the excited states. In the presence of a time scale separation between molecular variables (orientation) and collective properties (director), an explicit relation is derived for the dye photo-induced torque, which appears to be compatible with the experimental findings.