On the theory of the electric field and current density in a superconductor carrying transport current

A theory is given to explain the physics behind the flow of low-frequency ac transport current around a closed superconducting circuit, where the circuit consists of two long, straight, parallel, uniform conductors, connected to each other at one end and to an applied emf at the other end. Thus one conductor is the return path for the other. A question of interest is what drives the current at any given point in the circuit. The answer given here is a surface charge, where the purpose of the surface charge is to spread the local emf around the circuit, so that at each point in the conductor it produces, together with the electric field of the vector potential, the electric field necessary for the current to flow. But it is then necessary to explain how the surface charge gets there, which is the central problem of the present analysis. The conclusion is that the total current density consists of the superposition of a large transport current and a very much smaller current system of a different symmetry. The transport current density is defined as a two-dimensional current density with no divergence. It flows uniformly along the conductor length, but can vary over the cross-section. The small additional current density has a much different symmetry, being three-dimensional and diverging at the surface of the conductor. Based on a slightly modified Bean model the transport current is treated as supercurrent having the value ±Jc, while the small additional system of current is like normal current, with a density given by the electric field divided by a resistivity. The electric field is computed from the sum of the negative time derivative of the vector potential and the negative gradient of the scalar potential due to the surface charge. It has components parallel and perpendicular to the long axis of the conductor. Thus the small normal current density has a perpendicular component which flows into or out of the surface thereby creating the surface charge. Since the circuit has charge neutrality one can picture the small system of normal current density at a given point along the conductor length, during a given half-cycle, as flowing in a perpendicular direction out of the surface and then turning to flow around the circuit to a similar point in the return path, where it turns and flows into the surface. The perpendicular component varies in strength along the length of the conductor, producing a surface charge density which varies approximately linearly along the length. Such a surface charge produces a uniform electric field inside the conductor which adds to the electric field of the vector potential and determines the flow of transport current. It is shown that one can arrive at the above results by a careful solution of the Maxwell equations, where the solution also satisfies the demands of the Bean model.