In a normal conductor under steady-state conditions, the drift velocity of charge carriers reaches a constant value when the electric field is zero. This occurs because in the presence of an electric field, the carriers experience a net force that accelerates them until they reach a terminal velocity. When the electric field is zero, the carriers do not experience any net force and therefore have no acceleration, resulting in a constant drift velocity.
However, in the case of a superconductor, the behavior is fundamentally different due to the phenomenon of zero resistance. In a superconductor, at temperatures below the critical temperature, electrical current can flow without any resistance. This means that there is no need for an electric field to be present to sustain a current. In fact, in a superconductor, the electric field is zero even when a current is flowing. The lack of resistance in a superconductor is a consequence of the formation of Cooper pairs, which are bound electron pairs that move through the material without scattering off impurities or lattice vibrations.
In the two-fluid model of superconductivity, the current is carried by two types of charge carriers: normal electrons and superconducting Cooper pairs. The behavior of these two fluids is described by different equations. The normal electrons follow Ohm's law, where the current density is proportional to the electric field, similar to what we observe in normal conductors. On the other hand, the superconducting Cooper pairs, which are responsible for the zero resistance, do not follow Ohm's law. The motion of Cooper pairs is described by the London equations, which relate the derivative of the supercurrent density to the vector potential (related to the magnetic field) rather than the electric field.
In summary, the absence of a steady-state condition or a unique solution for the electric field in a superconductor arises from its unique property of zero resistance and the presence of two different charge carriers with distinct behaviors. The behavior of superconductors is described by the two-fluid model, where the current density of the superconducting component is not directly proportional to the electric field.