Will an electric field be still unable to penetrate a metal if the metal carries an electric current?

Question

In relation to this question I have another question. If a metal wire (or piece of metal) is placed in an electric field, the field can't penetrate the metal due to charges in the metal redistributing themselves, after which the field is canceled. So, initially, the field can enter, after which it is canceled. But what if the wire (piece of metal) carries an electric current? How can the charges redistribute to cancel the applied (external) field?

Answer 1

Will an electric field be still unable to penetrate a metal if the metal carries an electric current?

The answer depends upon what you mean by "unable to penetrate". In a very real sense, an electric field can "penetrate" anywhere. However, electric fields work according to the principle of superposition. If we divide all the charges that are contributing to an electric field into two groups, then the field resulting from the sum of all the charges is the same as the field resulting from adding the fields one would get from each of the groups of charges separately.

In the case where a conductor is involved, one of the groups of charges that needs to be considered is the electrons in the conductor, and the stationary nuclei of the conductor. Unlike insulators, conductors have an energy band where the outermost electrons are not tied to any one nucleus, but are shared across the whole conductor. These outermost electrons move freely within the conductor as if in an "electron gas". Under electrically neutral conditions, these electrons move randomly. However, in the presence of an electric field, these electrons have a drift motion in addition to their random motion. (Contrary to certain claims made on the internet, there is a qualitative difference between conductors and insulators. Insulators are not just poor conductors, they are missing this conduction band.)

There is a relationship between the drift current and the electric field that is causing that drift current. That relationship is known as the microscopic version of Ohm's law.

$$\vec{J} = \sigma\vec{E}$$

$\vec{J}$ is the current density. $\sigma$ is the conductivity of the conductor. $\vec{E}$ is the electric field.

From this, it is clear that there is an electric field in a conductor, whenever there is a current in that conductor.

So, where does the idea that electric fields cannot penetrate a conductor come from? It comes from the fact that when an external electric field is applied to a conductor, the electrons in the conductor re-arrange in a way that counters that electric field. Regardless of what field may be applied to a conductor, the resultant field, after re-arrangement of charges, is governed by the microscopic version of Ohm's law, and not by the applied field.

If the current density in the conductor is zero, (and the electric and magnetic fields are constant) then we have a simple case of electrostatics. By the microscopic Ohm's law, if the current density is 0, the electric field must also be 0. So in the electrostatic case, there is a sense in which we can say "the electric field is unable to penetrate the conductor".

What about in the case where there is a current flowing through the conductor? We have seen in this case, that there is an electric field in the conductor, proportional to the current density and inversely proportional to the conductivity. In order for there to be a steady state current, there needs to be a circuit. When one first applies an electric field to a conductor, electrons may flow from one end of the conductor to the other. But if the electric field remains constant once applied, that current will very soon stop. In steady state, the flow of current through a circuit must be equal everywhere in the circuit. It cannot "stop" somewhere.

From this, we can conclude that if the source of a time-invariant electric field is not electrically connected to a circuit, it cannot cause a current to flow through that circuit.

However, by the principle of superposition, if a time-invariant electric field cannot induce a current in a circuit to which it's source is not electrically connected, it likewise cannot alter a current which is flowing through a circuit for some other reason.

Since an external time-invariant electric field, cannot alter the current density, in the conductor, by Ohm, it cannot alter the electric field in the conductor either.

So, to finally give an answer to the question at hand, the electric field in the interior of a conductor is not altered by the presence of an externally applied, time-invariant electric field, even if there is current flowing in the conductor.

Answer 2

A current in a conductor requires a parallel E field to maintain the flow (and a power source which provides a separation of charge to maintain the field and the flow). An external field (not part of a closed circuit) will cause a separation of charge within the conductor which cancels the external field. That does not depend on whether there is a current in the conductor from some other source. The flowing free electrons (and the current density) can shift to cancel the external field. (Consider the shift which occurs in the presence of an external magnetic field.)