  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, $E_J^{\rm crit}(E_C,q_x)$, 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.
