If there were a uniform current density, then there would be a tangential magnetic field within the wire by Ampere's Law. Electrons traveling through that field (opposite the direction of the current) would experience an inward radial magnetic force due to current at a smaller radius. With electron density nearer the center increased, an outward electric force develops to balance the inward magnetic force on the electrons. With this balance maintained, electric flux and magnetic flux remain constant in all directions. The fault lies in assuming the conductor is ideal. If this were so, then resistivity would be zero and conductivity would be infinite. If a constant current density is proportional to electric field, then resistance must exist. This in turn allows electric energy to dissipate as heat energy. Electromagnetic energy flows inward according to ExB, thus causing the wire temperature to increase. If the wire does not heat and has no resistance, then current can flow without a driving electric field.
