The electrons bound to the surface of liquid dielectrics by image forces are described as a two-dimensional, classical, one-component plasma with inverse distance interactions. Exact expressions for the collective modes and the dynamic structure factor are obtained from first principles in the limit of long wavelengths. The differences and analogies with uncharged particle fluids and with the three-dimensional one-component plasma are explicitly displayed. The previously used mean-field approximation is shown not to describe weakly coupled systems and to be inadequate in the long-wavelength region.