In this paper, some optical properties of a two-walled carbon nanotube are studied. The system is represented by two infinitely long and infinitesimally thin coaxial cylindrical shells, each containing a two-dimensional electron nano-fluid. The quasi-static approximation and the quantum hydrodynamic model are adopted in order to obtain the optical response of the system. General expressions for polarizability, dispersion of dipole surface plasmons and induced surface charges of the system are derived, by solving the Laplace and fluid equations with appropriate boundary conditions. Also, by using the polarizability formula, scattering, absorption and extinction spectra are computed and expressed in terms of widths, which are defined as the cross sections per unit length of the system. Furthermore, the Maxwell-Garnett theory for the effective medium approximation of composite materials is developed to the effective dielectric function of a composite of aligned two-walled carbon nanotubes.
