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My understanding of how the electric field produced by a battery gets transferred along the length of the wire is summarised in the following picture:

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Is it correct to say that the force that the electrons in the wire experience is actually an indirect force from the battery which is transferred along the length of the wire by several planes of electrons? Can we say that an electric field is only present in the vicinity of the battery but due to this transfer of force along the wire by the sheets of electrons, the electrons a long way away from the battery still experience a force in such way that it's as if they are the first layer of electrons in contact with the terminal of the battery? Can we describe this as an extension of the electric field produced by the battery (due to the transfer of the forces)?

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Is it correct to say that the force that the electrons in the wire experience is actually an indirect force from the battery which is transferred along the length of the wire by several planes of electrons?

Yes, it is correct, but 'indirect' is also an appropriate description. The battery, instead of being a voltage source, functions here as a source of current, and the current contained in the narrow channel of the wire is generating, by Ohm's law, the local voltage gradient. The electric field inside the wire can most easily be derived from the battery current, not the battery voltage.

The full solution of the problem does require an appreciation of the battery electric force, but the Thevenin equivalent of a battery is voltage source in series with resistive impedance, and the Norton equivalent is current source in parallel with resistive impedance, and they are both (as labeled) equivalent.

Because the two pictures are equivalent, the voltage picture is possible. But, because 'a wire' is commonly a long thin channel, a model treatment of electric field inside the wire is rather more complicated than necessary: the current picture is simpler to apply, and... is equivalent. There's theorems that say so.

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  • $\begingroup$ But anywhere in the wire, according to Ohm's law $$\vec J=\sigma \vec E $$ a current only flows if there is an electrical field. And the question is, what produces the electrical field along the wire causing the current? $\endgroup$ – freecharly Apr 6 '18 at 16:44
  • $\begingroup$ @freecharly That Ohm's law equation doesn't say which is the cause and which the effect; they aren't separate in time, but simultaneous. So, "current if electric field" and "electric field if current" are both implied. $\endgroup$ – Whit3rd Apr 6 '18 at 20:43
  • $\begingroup$ Describing a battery as a dipole (two points of charge) is easy; describing the extension of those points to a shape of wire is an electric field problem requiring the solution of that wire's shape as a boundary for second-order differential equation solution. So, we wouldn't do the solution that way (it's difficult). $\endgroup$ – Whit3rd Apr 6 '18 at 20:51
  • $\begingroup$ Ohm's law $$\vec J=\sigma \vec E$$ with the specific conductivity $$\sigma=\frac {ne^2 \tau}{m^*}$$ where $n$ is the electron concentration, $e$ the elementary charge, $\tau$ the electron scattering time, and $m^*$ the effective electron mass, is determined on a microscopic level by the effect of an electric field $\vec E$ on the drift velocity $v=\frac {e \tau}{m^*}E$ of the electrons in the conductor. Thus the electric field is definitely the cause and the current is the effect in this relation. $\endgroup$ – freecharly Apr 6 '18 at 21:15
  • $\begingroup$ You are right. The solution of this problem is not easy. See also my answer here: physics.stackexchange.com/questions/390270/… $\endgroup$ – freecharly Apr 6 '18 at 21:22
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When you study electrical engineering, there are two main topics: (1) fields and waves, and (2) circuits.

2 is simply a very simplified version of 1 - doing things and making things happen in ways you can 'ignore' the fields and waves (Maxwell's equations) and focus on the circuit elements, and corresponding laws (e.g., Kirchoff's current and voltage laws).

With a bit of arm waving, there is indeed an electric dipole at a battery in free space. But wire has the ability to allow ready conduction/motion of electrons. So, in the end, the voltage gradient gets 'realized' in the wire, causing electrons - which are mobile - to move, giving rise to a currrent.

You then use circuit theory. But that is not absolute: a current flowing through a wire gives rise to a magnetic field around it which we usually ignore. But when you suddenly disconnect the circuit, the back-emf will induce a voltage from the collapsing magnetic field.

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