  We study the electronic structure of gated graphene sheets. We consider both
infinite graphene and finite width ribbons. The effect of Coulomb interactions
between the electrically injected carriers and the coupling to the external
gate are computed self-consistently in the Hartree approximation. We compute
the average density of extra carriers, $n_{2D}$, the number of occupied
subbands and the density profiles as a function of the gate potential $V_g$. We
discuss quantum corrections to the classical capacitance and we calculate the
threshold $V_g$ above which semiconducting armchair ribbons conduct. We find
that the ideal conductance of perfectly transmitting wide ribbons is
proportional to the square root of the gate voltage.
