# Does a current carrying conductor carry a net total charge or only a net bulk charge?

I would like to start a discussion in which we have an ongoing debate elsewhere with no convincing solution in sight. So I decide to ask here:

The question is mainly academic: Is a current carrying conductor charged? I do not mean any charges imposed by electrostatic boundary conditions, because it is clear that a voltage drop between two conductors in a loop , which are connected with a resistor at their ends are making up a capacitor and therefore carries charge. I talk about a free conductor without any other boundary conditions.

There are many papers on this topic, in principle they state, that the bulk of a current carrying conductor gets slightly negatively/positively charged when a current of negatively/positively charged carriers flows through.

It took me a while to understand this, but in the meantime I think this result is beyond discussion. As an example there is a slab which carries current in z-Direction. The image depicts the cross section of this slab.

A Lorenz force acts on each moving carrier and it would get deflected in y-direction. Because the conductor is infinite in flow-direction, there must be an equilibrium state, where the carrier flows parallel to the conductor. So an additional force is needed. A simple analysis shows, that the Field B is

$$B = \mu_0 J \cdot y$$

and the Lorenz force (ignore sign now) on an electron, moving moving with speed v is

$$F_L = \mu_0 J \cdot y \cdot e \cdot v$$

So there must be a compensating electric field of amount

$$E = \mu_0 J \cdot y \cdot v$$

Because of

$$\vec \nabla \cdot \vec E = \rho/\epsilon_0$$

We must have a bulk charge density

$$\rho = \mu_0 \epsilon_0 J \cdot v = \frac{v^2}{c^2} \cdot e \cdot n_e$$

This is just a basis for further discussion now. One paper describing the scenario is

https://www.researchgate.net/publication/338736210_Self_induced_Hall_Effect_in_current_carrying_bar

but only chapters I, II are relevant in my context, because it goes much further and does not address my problem.

The question that arose here is whether the excess charge inside is balanced out by a surface charge density of opposite sign, or the conductor is charged as a whole.

Personally, I prefer the assumption of surface charge density, but the others disagree. However, none of the opinions has convincing evidence.

This is solution one with a charge density on the surface built up:

While the other possibility is, that the conductor carries a total charge:

Personally, I find case-1 more logically, because the moving carriers are first deflected into the center so that, within a short time, they are finally missing at the surface and distributed homogenously within the bulk, thereby leaving a positive surface charge behind. I would consider this analogous to the charge separation in a metal by means of influence, where surface charge is build up in a way to exactly satisfying boundary conditions.

Case-2 would mean, that the source must provide additional charge to load the whole conductor negatively. However, this means, that the source itself must charge-up positively, because where do the additional negative charges come from? Although the effect is actually very very small, I would still find that pretty absurd.

In my opinion, both variants meet the boundary conditions at the interfaces between air and metal. However, due to the external field, variant 2 has a much higher energy as compared to variant 1 and is therefore energetically less favorable. It would be like a capacitor charging itself as a whole instead of charging both plates with opposite charges in order to keep it neutral. Why should a total charge be built up, when an energetically much cheaper scenario can satisfy the imposed boundary condition (have a fixed potential difference between its plates) as good?

I think over this problem already for more than three days and cannot come to a solution...any ideas?