# Can there exist an arrangement of charge densities that creates a loop in the electric field lines of force?

Can there exist an arrangement of charge densities that creates a loop in the electric field lines of force?

In this answer, a poster writes:

Note 2 5/8: superconducting loop

A uniformly perfectly conducting loop poses some extra problems because inside the perfectly conducting material there cannot be any resultant electric field Etot. This means that the surface charge will redistribute in such a way as to produce a coulombian field Ecoul that completely obliterates Eind everywhere inside the ring.

If no net electric field could exist within a superconductor, then the London Equations, which reference the electric field within a superconductor would make no sense.

However, equally important is the fact that a fixed distribution of charge density cannot, no matter how arranged, bring about an electric field which contains a line of force which is a loop.

The electric field resulting from a distribution of charge density is conservative, and a conservative vector field cannot contain loops.

Since the electric field resulting from a distribution of charge density (what the poster refers to as the Coulombian field) cannot contain any loops, it cannot "completely obliterate" the electric field loops created by a time varying magnetic field.

Am I correct? Or incorrect?

Edit:

In a reply, @tobalt wrote:

The London equations posit that there is no current in the bulk of the superconducting body. Therefore, it has to be free of both E field (otherwise current would build up) and B field

I agree that an E field in a superconductor will cause a time varying current. However, I would like to see a reference for the claim that "there is no current in the bulk of the superconducting body[, t]herefore, it has to be free of [an] E field"

This paper, for example contradicts the notion that there cannot be an E field in the body of a superconductor.

We show that a London superconductor in a steady uniform external magnetic field must support an electric field in its interior. The existence of an electric field implies that a superconductor has a nonvanishing charge in its interior, a fact consistent with measurements of charge imbalance in steady-state superconductivity.

See also: this paper, and this paper

As a consequence the superconductor in its ground state is predicted to have a non-homogeneous charge distribution and an outward pointing electric field in its interior.

@tobalt responded that:

I just repeated what the London equations say. For magnetic field free interior, it would also be free of E field. It is also briefly mentioned in wiki below the introduction of the single London equation: A is zero in the bulk. Maybe that paper is referring to a nonideal situation, where there are non-SC volumes in the bulk, or where the magnetic field is present during the NC->SC transition and becomes eternally trapped.

I am not sufficiently knowledgeable at this point to determine whether the claimed E field within the body of a superconductors found in various papers violates or is consistent with the London Equations. So, I withdraw the claim that

If no net electric field could exist within a superconductor, then the London Equations, which reference the electric field within a superconductor would make no sense.

However, the main issue that I wish to resolve is the claim made in the referenced answer that

that the surface charge will redistribute in such a way as to produce a coulombian field Ecoul that completely obliterates Eind everywhere inside the ring.

I cannot see how a Coulomb field (an irrotational Electric field) could cancel a solenoidal/rotational Electric field, even if we restrict our attention to a single loop in the solenoid field.

Can there exist an arrangement of charge densities that creates a loop in the electric field lines of force?

# No.

This is easily seen as a direct consequence of Maxwell's laws, specifically Faraday's law of induction,

$$\nabla × \mathrm {\bf E} = -\frac{\partial \mathrm {\bf B}}{\partial t}$$

This states that the curl of the electric field must be zero in the absence of time-varying magnetic fields. If there is a loop in any field line, the curl must necessarily be nonzero near that loop.

As you are asking specifically about the case where there are only static charges, nothing can be time-varying, and so $$\\nabla × \mathrm {\bf E} = \frac{\partial \mathrm {\bf B}}{\partial t} = 0\$$ (not to mention the fact that only static charges implies that $$\\mathrm{\bf B}\$$ itself must also be zero).

• Hope you don't mind. Repeated the question so it is clear what the "No" is referring to. Dec 18, 2021 at 20:28

Maxwell says that you need a changing magnetic field to produce rot E. No static charge distribution will do this.

The London equations posit that there is no current in the bulk of the superconducting body. Therefore, it has to be free of both E field (otherwise current would build up) and B field

When subjecting a superconducting ring to a changing magnetic field (as per your linked question & answer), the Meissner Effect will prevent that changing magnetic field from reaching the interior of the ring. I.e. there is no rot E to be compensated by charge redistribution in the first place. All induced current flows on the surface of the superconductor in the London penetration depth zone. The interior of the SC is completely oblivious to anything happening outside.

• I agree with your first statement. No static charge will produce a rotational E field. Would like to see a citation for second claim. See my edit, but stack exchange and quora answers aside, there are academic papers that contradict the idea that an E field cannot exist in the body of a superconductor. Dec 18, 2021 at 19:58
• I just repeated what the London equations say. For magnetic field free interior, it would also be free of E field. It is also briefly mentioned in wiki below the introduction of the single London equation: A is zero in the bulk. Maybe that paper is referring to a nonideal situation, where there are non-SC volumes in the bulk, or where the magnetic field is present during the NC->SC transition and becomes eternally trapped. Have you checked if that paper is being refuted by others citing it ? Dec 18, 2021 at 21:50
• A is not the electric field, but the vector potential. Dec 18, 2021 at 22:01
• @MathKeepsMeBusy and A being zero mandates jS to be zero. if no current can exist, it also mandates that either no electric field can exist or that resistivity would be infinite (which is obviously false for a SC) Dec 18, 2021 at 23:14
• So, it seems the correct version of sredni's answer should say, w.r.t. superconducting ring, something like "the current (which is changing due to E_induced) near the surface of the SC will produce a changing counter induced field B_reaction, which in turn will produce an E_reaction field that nearly completely obliterates Eind everywhere inside the ring." Dec 19, 2021 at 0:54