What happens microscopically when an electrical current starts to flow? I'd like to understand microscopically what happens in detail when electrons start moving (quasi-classically).
Electrons can have different velocity, they can produce electromagnetic fields, leads have free electrons and rigid atom cores and there exist electromagnetic fields. That's all the ingredients you should need?
Electrons only move due to EM fields, so basically this question boils down to what the EM fields look like and how they build up?! In steady state, what is the electric and magnetic field distribution in/around the lead? And what about the transient state?
What happens when you attack a battery to a lead? Are there EM fields between battery poles or why are electrons pushed? How do the EM field start to push electrons along an arbitraritly shaped long lead?
[EDIT: Ideally an explanation with the Drude model (which partly derives from Fermi model) or an explanation why that model isn't sufficient. Also stating the EM fields consistent with the electron density distribution would be important (i.e. $\vec{E}(r,\theta,z)$ and $\vec{B}(r,\theta,z)$) because otherwise it's hand-wavy arguments.]
(Please consider all remarks in this question. I know common arguments for parts of the question, but I've never seen a full microscopic in detail explanation.)
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