We have studied a quantum Hamiltonian that models an array of ultrasmall Josephson junctions with short-range Josephson couplings E J and charging energies E C due to the small capacitance of the junctions. We derive a new effective quantum spherical model for the array Hamiltonian. As an application we start by approximating the capacitance matrix by its self-capacitive limit and in the presence of an external uniform background of charges q x. In this limit we obtain the zero-temperature superconductor-insulator phase diagram E J crit (E C, q x) that improves upon previous theoretical results that used a mean-field theory approximation. Next we obtain a closed-form expression for the conductivity of a square array, and derive a universal scaling relation valid about the zero-temperature quantum critical point. In the latter regime the energy scale is determined by temperature and we establish universal scaling forms for the frequency dependence of the conductivity.