# Couldn't the electric field in a conductor be non-zero at points of zero charge?

In electrostatics, it's often claimed that we can assume the electric field inside a conductor is zero. The way I interpret this claim is:

If we have a region containing charge which can move around freely, then the only situations in which no charge is moving are situations in which the electric field is zero inside the region.

The reasoning is that if $E$ were non-zero at some point, then the charge at that point would start moving. But what if there's no charge at that point? It seems to me that the only thing we can really say is that the support of $E$ and the support of the charge density are disjoint.

Is this not correct?

• Coulomb interaction is rather long-range. If there was no charge at all, the electric field would have no effect. But then again, there would probably not be an electric field in the first place if there were no charges ;) – lmr May 2 '18 at 9:33
• @lmr I'm not sure what you're saying. Do you mean that if there were a point of zero charge inside the conductor, the electric field would automatically be zero at that point anyway? – Jack M May 2 '18 at 9:34
• If there is a non-zero electric field at a certain point, there have to be charges in the vicinity of this point. How else do you obtain an electric field? – lmr May 2 '18 at 9:43
• @lhr In the vicinity, yes, but not necessarily at the point. Couldn't you imagine a region of space in which the only places where there's a non-zero field, there's zero charge, and vice versa? – Jack M May 2 '18 at 9:47
• No, not really... I have no idea how that would be possible since you need charges to establish those fields. I'll think about it though. This is really hypothetical though - you always have electrons everywhere in a good conductor. It's simply the net charge that is zero. – lmr May 2 '18 at 10:11

## 4 Answers

By the definition of a conductor, all points of a conductor contain charges that can move. A region where no charges can move is called an insulator, and of course the electrostatic field can be nonzero there. For example, a dielectric is an insulator but dielectrics in capacitors contain fields.

• But in e.g. Feynman's book, the very next step is to show that, from the assumption of zero $E$, it follows that all of the charge must be concentrated on the surface of the conductor. So there are regions of zero charge in the conductor. – Jack M May 2 '18 at 9:55
• @JackM There are regions of zero charge density, i.e. the charges of electrons and protons are balanced out. But there are always electrons that can move. – knzhou May 2 '18 at 10:00
• And a region can still have zero charge density over a finite period of time while still having non-zero current, as long as much moves into the region as moves out? – Jack M May 3 '18 at 5:30
• @JackM What Feynman will prove is that any excess charge will be concentrated on the surface of the conductor. If a block of conducting material has no net electric charge, then there will be zero charge on the surface. – Mark H May 3 '18 at 6:41

Inside a good conductor (metal) there are always conducting charges, even if total charge density is zero; the part that distributes itself on surface of body is a minor fraction of all conducting charges.

Inside a semiconductor of very low temperature, density of conducting charges may be very low, so there is not enough of them to counteract the external static electric field. Then, hypothetically, there may be static electric field inside.

The electric field is only zero at the scale of the Fermi electron wavelength or larger. At shorter length scales the field is not zero, for example inside the positive ions.

In the absence of a current, the electric field in a conductor is zero on a macroscopic level.

In practice, it means that, in the absence of a current (electrostatic case), the potential difference between any two points inside a conductor, separated by a distance much greater than the distance between two atoms, is zero.

At a microscopic level, the electric field is constantly fluctuating because the protons and electrons, contributing to that field, are constantly moving, but, on average, the field along any path between our two points will be zero, leading to the zero potential difference.

We could note that, when we are talking about the absence of a current, we are talking about a macrocurrent, because, one could argue, there are gazillions of microcurrents inside a conductor due to the movement of charges.

So, we can say that, at any given moment, there are plenty of points inside a conductor where the momentary electric field is non-zero, but it does not contradict the notion that, at a macroscopic level, in the absence of a macrocurrent, the electric field remains zero.