# Explaining how a magnetic field affects currents in a conductor, in the limit of high conductance

Recently, I gave an answer to this question, explaining my intuition on how much electric field impinges into a good conductor under certain conditions, in order to reason about the limit of perfect conductors. Partly because I criticized how the accepted answer explained Gauss's law, and also perhaps because I accidentally interpreted the (ambiguously worded) question as about a changing field, rather than a rotating conductor, I received downvotes.

But I still want to understand whether my explanation has intuitive merit, so I'm asking the trio of questions:

• How does electric field look within a loop of conductive material when it is not moving itself but is exposed to a changing external magnetic field?
• How does the electric field look in the limit of infinite conductance? For a superconducting loop?
• Does my explanation (in terms of eddy currents) lead to the right intuition? If not, how should I revise my intuition?

A final thing I am curious about is learning more about the physics of superconductors, as I chose not to be explicit in that case to avoid being incorrect.

There are several related questions asking about this issue, but the point of this question is to critique this intuition, not to answer the question without reference to it.

I apologize for its length, but my is explanation below:

The Maxwell–Faraday equation says $$\nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t}.$$

Put roughly into English, this says: A changing magnetic field produces a swirling electric field that can accelerate charges around a loop.

Let's look at how a changing external magnetic field affects a conductor in the cases of a normal conductor, a "perfect conductor" (I mean this loosely, meaning in the limit of a normal conductor approaching infinite conductance), and a superconductor.

## Normal conductor

In a conductor with finite conductance (a "normal conductor"), we already know the story that the a little bit of the electric field can penetrate inside and also induce a drift current in the bulk electrons. (I've made multiple answers on this sort of topic, so you can see some of my recent answers for intuition in this area.)

## "Perfect" conductor

In general, we like to say a perfect conductor doesn't allow electric field lines in. This is a little inaccurate, first of all, because there are no perfect "normal" conductors, and second of all, because this catch-all explanation misses the important physics of why conductors typically cancel much of the incoming field, and why better conductors cancel more of the field.

Let's first take a step back.

### What cancels (either all or most of) the electric field in conductors?

Main Process: Charge Build-Up

For most circuits, it all has to do with charge build-up somewhere in the material. In a conductor, charges move in response to whatever electric field penetrates, which usually often leads to an electrostatic case of essentially all the electric field being canceled in the conductor, even for the case of a normal conductor.

But a characteristic of this situation is that there is "an electromotive force," (EMF) which is fancy word for saying "we are consistently changing something about our situation that is not letting these charges settle in the way they normally do." Whenever we hear the term electromotive force, we typically should understand it as a summary of the actual forces and effects that continuously push charges around a circuit.

For example, in a circuit, we have a battery. Charges move to attempt to cancel the electric field the battery produces, but each the charges accumulate on the battery to try to alter the voltage on the terminals, the battery simply plucks that charge and puts it on the other terminal, sustaining the motion. (This act of restoring the charge to maintain voltage is due to a chemical process, which we describe as producing an "EMF".) Thus in this case, there ends up being a steady-state of nonzero current in a conductor, and nonzero electric field in the conductor due to needing to overcoming the resistance of the conductor to sustain the current.

For the case of a battery, in the limit that that conductor is perfect, in order to completely cancel out the field, the charges still need to accumulate somewhere in the circuit, e.g. on a resistor. Now the electric field in the conductor will be essentially zero, but charges will still flow, keeping the momentum they initially had from being accelerated the instant the field wasn't zero in the near-perfect conductor.

In the case of electromagnetic induction, a changing magnetic field also produces an EMF due to Faraday's law. Charge confined to a loop would pushed around and around in a circle due to the swirling electric field. But asking how much of the magnetic field gets into the conductor in the first place brings me to my next point.

### What cancels (part of) a changing magnetic field in conductors?

Main Process: Eddy currents:

For this particular case, for a conductive loop exposed to a changing magnetic field, there will actually turn out to be another important effect: eddy currents. These occur because of Faraday's law: charges in a changing magnetic field want to swirl. They swirl locally in eddy currents then tend to counteract the magnetic field, and thus prevent some field lines from entering.

However, the case is precisely the same as before, in the sense that any resistance present prevents electrons from swirling fast enough to stop all entering field lines. So the external, changing magnetic field penetrates some distance inside, in any case; however, this penetration depth may become very small as the conductance approaches infinity.

### Applying to the original scenario

@Orpheus was asking about a coil formed by a perfect conductor in a changing magnetic field. Let us simplify to discuss a simple loop.

The answer is simply: eddy currents in response to the changing magnetic field keep the changing external magnetic field relatively close to the surface, but it still penetrates some distance that decreases as conductance increases. The impinging, changing magnetic field still has a total EMF around the loop, and there is a swirling electric field (confined more and more tightly to the surface in the limit of high conductance) that accelerates charges around the loop. This acceleration around the loop ends when the external EMF balances with the "reverse EMF" produced by accelerating electrons producing their own flux, but for finite conductance, the acceleration will end sooner because the resistance will help slow the electrons down.

You can also think of the net flow around the loop as one big eddy current, if you want. Now, let me explain some finer points about the previous paragraph in a few more words.

Since there is no resistor in the loop, there is no place for charges to build up to counteract the loop EMF from Faraday's law in the charge-build-up way. So the charges accelerate. This produces an magnetic field that curls around the loop, a back-action on the applied flux. But this flux also obeys Faraday's law, and produces a reverse EMF. Thus, electrons will continue to accelerate until the reverse EMF from this produced electric field equals the applied EMF.

The question is, does that equality ever occur? I might try to go back and do some calculations, but, in any case, this is already a bit of a trick question, because perfect conductors don't exist, so in fact charges don't continue accelerating forever, and instead the drag force from the non-zero resistance adds with the self EMF to balances out the applied EMF. However, in this case, we are using a drag force to balance out the EMF. This drag force is a phenomenological model we use because we don't want to model to microscropic electric fields in a material. If we still want to use our space-averaged electric field $$\mathbf{E}$$, we better keep this drag force separate from Maxwell's laws, and just include it as another force.

Thus, the electric field penetrates even a very good conductor with finite conductance, because before the self EMF stops the charge accelerations, self EMF + resistance does. But, as stated before, this penetration is confined tightly to the surface due to eddy currents, so the current is almost all on the surface as well.

Now we get to the fun part: superconductors.

# Superconductors

Now, what is interesting to me is understanding how this works in the closest thing to a real-world perfect conductor, which are superconductors. I don't know much about superconductors, which have other properties like the trapping of magnetic flux lines inside of itself. Thankfully, how a superconducting loop responds to a applied magnetic flux has already been answered here on Physics StackExchange, for the case of a superconducting loop.

To paraphrase @Alfred Centauri's answer, the magnetic flux through a superconducting loop never changes; but to sustain this constant magnetic flux requires the current inside the loop to perfectly counteract any flux you try to push though. Because superconductors can only sustain a certain amount of current before becoming a normal conductor, this implies a high enough magnetic field will rupture superconductivity in the loop.

Note I make no mention of where in the superconductor this current is flowing, because I don't know much about superconductors. However, you should note that the case of superconductors already matches the intuition we gained from thinking about a limit of a perfect conductor: in order to stop accelerating, charges have to move fast enough to cancel the magnetic field.

### Summary

Thus, to summarize:

• In a normal conductor with fixed loop geometry, current in a loop flows only while the magnetic flux through the loop (from an external magnetic field) is changing, and dies after this applied magnetic flux stops changing, because the electrons will dissipate their energy due to the resistance of a loop. Eddy currents prevent this some of the external magnetic field from entering the conductor.
• In a normal conductor with very, very high conductance, a penetrating electric field accelerates the charges, but the self EMF of the electrons is almost enough to cancel all of the electric field to stop the acceleration. Surface eddy currents prevent almost all magnetic field from entering the material. Because of this, the the loop current remains tightly confined to the surface, and can be thought of as a surface current. But because we rely on resistance a little bit to slow the electrons, some electric field must penetrate a shallow penetration depth.
• In a superconductor, such flowing charges perfectly counteract the applied magnetic flux, and thus the flux through the superconducting loop is constant.

In at least in the first two cases, the electric field penetrates the loop a little bit; and we can use the second case to reason about the third. There is also a similar notion of penetration depth for superconductors, which may or may not match this intuitive notion of penetration. Also, note for a lot of the time when I talking about "stopping the acceleration," I mean for the case of a constant derivative of the magnitude of uniform applied global magnetic field.

Two relevant Maxwell Equations for isotropic materials are:

$$\mu \epsilon \frac{\partial \mathbf E}{\partial t} + \mu \mathbf J = \nabla \times B$$ $$\frac{\partial \mathbf B}{\partial t} = -\nabla \times E$$

For a normal conductor $$\mathbf J = \sigma \mathbf E$$, where the conductance $$\sigma$$ has some finite value. If it is a continuous current, there is a static electric and magnetic field inside the conductor.

For a superconductor, $$\sigma = \infty$$, and the electric field must be zero for the same continuous current. That means: a loop of a superconducting material has current, magnetic field, but no voltage.

For a superconductor loop, where some alternating current is generated by an external changing magnetic field, the second equation tell us that there is an electric field inside the material. This is different from the previous case because now it is not a static magnetic field.

In the first equation all terms are also not static.

The reason because it is not possible an electric field inside a superconductor, with a constant current, is that the current would be infinite. But if the fields are not static, the Maxwell equations show that both E and B are indeed present.

Using the analogy electro - mechanics, mass is inductance. Velocity is current and force is electric field. An oscillatory movement of the mass requires a changing force, even if there no damping (equivalent to no resistance).

And as for a mechanic oscillator, the maximum current corresponds to zero E-field, and the maximum E-field corresponds to zero current.

• It seems my analysis also led to the intuition that charges in a perfect conductor continue to accelerate in a changing magnetic field until they cancel the changing flux. Although this did not really answer the question of critiquing the explanation I gave, thank you for sharing the mechanical analogy. I also wonder about any lag in this cancellation that comes from electrons being slow to accelerate. Feb 7 at 23:07