Ambiguity on the notion of potential in electrical circuits?

As everybody else I have been taught elementary electrical circuits from secondary school to engineering level in analog electronics at university. Invariably, the notion of potential used to described what is going on in these circuits is seemingly that of electric potential that would follow then Maxwell's equations.

Yet, when learning a bit more about fundamental elecrokinetics in solid state physics, a current arises whenever there is a difference/gradient of chemical potential of the charge carriers and not necessarily a gradient of electric potential.

Basically, the chemical potential can probably be expressed as the sum:

$\mu = \mu_{id} + \mu_{exc} + \mu_e$

where $\mu_{id}$ is an ideal (albeit quantum) contribution, $\mu_{exc}$ an excess part comprising the interactions with the lattice, interactions/correlations with other charge carriers, quantum effects etc... and $\mu_e \sim q \phi$ an electric part containing a mean field contribution from the rest of the charges and that follows Maxwell's equations.

If the piece of metal is always the same, retrieving elementary laws for electrical circuits is kind of easy according to the above definition.

The only problem I have is that in most circuits there is more than one type of material, there are in fact many different materials that contribute to the overall current flow.

At the junction between two pieces of materials for instance, current will flow for various reasons related to electric potential differences but also to differences in the overall chemical potential (where the ideal and the excess non electrostatic part would also differ).

A good example of that is of course NP junctions that develop an electric potential difference at the junction to make uniform the chemical potential of the charge carriers and as a consequence there is no flow of charges in a diode.

Also, from what I understand, chemical batteries based on oxydo-reduction reactions impose a chemical potential difference for electrons only (it is chemistry driven). Hence although it is easy to claim that the potential difference for such a chemical battery will be given by Nernst equation, this potential difference can only make electrons flow and not for instance ions in an electrolyte solution. I may be mistaken here but, from what I understand, you can't close a circuit using a chemical battery by bridging the two cells with a material that would not conduct electrons (even if it were to conduct other stuff). Note that I do know that there is a salt bridge in chemical batteries to ensure that the chemical potential difference is not balanced precisely by an electric potential difference.

So here is my question(s):

• Am I overthinking it by believing that there is some ambiguity going on in the use (and potentially misuse) of the notion potential as being always interpreted as an electrical potential in circuits?

• Why then would the concept of potential in circuits be so ubiquitous? Are all my worries simply solved by the gauge freedom of the electric potential?

• Are there cases where the notion of potential would indeed be misleading if understood as electrical potential? I think I have mentioned some but I may be misunderstanding something...

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Generally, one includes as few interactions as possible to describe your physical problem. So, for almost all EE problems you consider the applied electric potential and everything works great. Until your hit something like a thermocouple. Then you have to generalize your potential to include the additional effect that reared its ugly head. But you don't start applying that more general potential form to every single EE problem from then on - you save it for when you need it. We still use Newtonian mechanics to good effect, even knowing it fails at some point and relativity is needed then. – Jon Custer Aug 4 '14 at 13:47
The same question for a semiconductor P–N junction: physics.stackexchange.com/questions/108314/… – Incnis Mrsi Oct 24 '14 at 9:22

• There is an ambiguity. Although I did not understand your analysis of the problem completely, charge carriers certainly can run against the (averaged) electric force due to difference in available bands and other particle statistics effects.
• The gauge freedom is irrelevant. There are two cases for the “ubiquity”. First, these non-Maxwellian deviations usually have the order of 1 volt or even less, so for such technologies as the mains electricity or electrical transport they are negligible. Second, even where they are not negligible (such as in semiconductors and heterogeneous metallic circuits) they usually almost cancel themselves when form a closed circuit. Thermocouples form a notable exception.
• Inside a piece of usual atomic matter, even if it is an electricity conductor, there is no such thing as value of the electric potential. Nuclei emits their Coulomb fields, electronic shells emits theirs, electrons in conduction bands are also changed, and even worse, all this things (especially electrons) obey the laws of quantum mechanics and hence have not any certain spatial position. Do you imagine a Maxwellian field in these circumstances? I do not. Maxwell’s theory is hardly usable inside conductors. Insulators provide some emulation of it on the macroscopic scale, although not unperturbed vacuum Maxwellian fields (I refer to such things as electrical permeability and optical density), but also have aforementioned hell inside them on the molecular level.
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