# Is the superconducting wire in a equilibrium DC circuit free of surface charge?

Suppose we have a closed circuit composed of a time constant voltage battery, a resistor and superconducting wires connecting them in series. In the equilibrium state, both the free charge the electric field in the interior of the superconducting wire should vanish, lest the free charge accelerates and there be no equilibrium/steady current. The electric field generated by the charges in the battery should be counteracted by nonzero surface charges on the superconducting wire in order to generate the zero electric field inside the wire. At the locations of the wire that are far from the battery and with small curvature which should approximate a straight wire, the surface charge density should be close to uniform. However, I have often read people claiming a superconducting wire with a constant equilibrium (direct) current should be free of surface charge. I would like to know what the bases is for their argument. Are there some references on this topic?

• If there is a battery in the circuit made of superconducting wire, there can be no equilibrium while the wire is in superconducting state, the battery will work on electrons inside the battery against electrostatic field of the free charges all around and charge up the superconducting circuit magnetic field energy until magnetic field is strong enough to break down the superconducting state. Then dynamic equilibrium with non-zero current and non-zero free charge density can happen while the wire is in ohmic state. Nov 23, 2023 at 21:22
• @JánLalinský: I just realized I misstated the circuit. It should include a resistor. I have edited the question. I also added the statement that the variation of the surface charge density was close to zero but not the surface charge density itself. The circuit should reach a steady/equilibrium state and current. Do you agree?
– Hans
Nov 28, 2023 at 7:19
• With resistor in the circuit, there can be equilibrium. Surface charge density on the superconductor in such a circuit need not be zero. It will be close to zero if the whole circuit is superconducting, e.g. current flowing in superconducting ring. Nov 28, 2023 at 15:15

The claim that a superconducting wire should be free of surface charge is based on the unique properties of superconductors, particularly the behavior of the superconducting state.

In a superconducting state, the electrons form Cooper pairs and move without resistance, leading to zero electrical resistance. This means that the electric field inside the superconductor is precisely zero in the absence of any external magnetic fields. This is known as the Meissner effect.

Regarding the absence of surface charge, one key aspect is the expulsion of magnetic fields from the interior of a superconductor. When a superconductor is in its superconducting state, it expels any magnetic field within its interior, creating a perfect diamagnetic response. This expulsion of magnetic flux is closely related to the absence of electric fields within the superconductor.

Now, when it comes to a closed circuit composed of a time constant voltage battery and a superconducting wire, in the equilibrium state, there is indeed a transient current as the superconductor initially switches to its superconducting state. However, once the equilibrium is reached, the persistent current flows without any resistance, and the electric field is zero inside the superconducting wire.

The claim that a superconducting wire should be free of surface charge is likely related to the fact that, in the absence of an electric field inside the superconductor, there is no need for surface charges to maintain this equilibrium. Surface charges might be present during the transient period as the superconductor transitions to its superconducting state, but they should eventually disappear as the system reaches equilibrium.

References on superconductivity and the Meissner effect can be found in standard textbooks on solid-state physics and condensed matter physics. Some classic references include:

"Introduction to Solid State Physics" by Charles Kittel. "Principles of Condensed Matter Physics" by P.M. Chaikin and T.C. Lubensky. "Superconductivity: An Introduction" by Reinhold Kleiner and Werner Buckel.

• You are restating my question instead of answering it, especially answering to my rationale for the existence of the surface charge.
– Hans
Nov 18, 2023 at 19:36