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I have been introduced to circuits and told that an electric field, along with a emf provided by an external source (which can't be electrostatic), causes electrons to move. They follow the direction of lower potential,and then this emf "pushes" them back to the original potential, like going down a slide and then having someone take you to the top again.

However, I don't really have an intuition as to how this electric field would be generated . How can the source create a field that magically follows an arbitrary direction of wire? Or how does it adapt if we twist the wire or move it, for instance? Should I consider the field created by the rest of electrons as well in order to account for it?

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  • $\begingroup$ Too many questions. You need to focus. $\endgroup$
    – Bob D
    Commented Jan 27, 2023 at 19:59
  • $\begingroup$ Would you make different questions? For me, they are related in the sense that they are all aspects of the very basics of why a circuit works. $\endgroup$
    – Jaime Yepes de Paz
    Commented Jan 27, 2023 at 20:02
  • $\begingroup$ @JaimeYepesdePaz Then make an attempt to sum it up with one, overarching, question. That makes it easier for folks to provide an answer which may cover all your other points. And if it doesn't, you can always provide comments/questions on the answers given. $\endgroup$
    – Bob D
    Commented Jan 27, 2023 at 20:08

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How can the source create a field that magically follows an arbitrary direction of wire? Or how does it adapt if we twist the wire or move it, for instance?

The important thing in this regard is the distribution of charge on the surface of the wire. Inside the wire the E field is mostly along the wire, and outside the wire the E field can vary substantially in direction relative to the wire. The surface charge is responsible for the often sudden change in the E field inside vs outside the wire.

My favorite paper on this topic is this one: https://www.tu-braunschweig.de/index.php?eID=dumpFile&t=f&f=138440&token=2cc8a71e4fdbf159121c6b8ef8348952a2e0c197 (R Muller. Am. J. Phys., Vol. 80, No. 9, September 2012)

It describes in a graphical semi-quantitative manner how to determine easily where surface charges will accumulate to provide the necessary E fields for the current to take the correct path.

They follow the direction of lower potential

As a side note, it is important to remember that this is a statement that is only true in Ohmic materials and circuit elements, like resistors and conductors. It is not true in general, and it is frequently violated. Do not hold on to this concept too rigidly.

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  • $\begingroup$ Great, it makes sense, although one has to guess that the distributions are the final result of a complex web of forces and such. While the type II surface charges seem out of reach, maybe knowing the microscopic mechanism behind type I could be done. Do you have any paper that addresses that, or know it yourself? I tried to see Jackson's cited paper but it's behind a paywall. $\endgroup$
    – Jaime Yepes de Paz
    Commented Jan 28, 2023 at 9:02
  • $\begingroup$ The only answer in this page that tries to explain it is physics.stackexchange.com/questions/516656/…. It almost makes sense, but if charges did accumulate at one end of the resistance (say left), wouldn't they repel the incoming charges, thus creating another "zone of resistance"? $\endgroup$
    – Jaime Yepes de Paz
    Commented Jan 28, 2023 at 9:19
  • $\begingroup$ @JaimeYepesdePaz it is indeed possible to explicitly calculate the “complex web of forces” using a numerical Maxwell’s equations solver (e.g. with a finite element model of the circuit). Unfortunately, I don’t know of a paper on that topic. During the initial transient (e.g. after closing a switch) the E field is not directed purely along the wire. The resulting non-parallel component of the current flows to the surface of the wire. At that point it stops and becomes an accumulation of surface charge. This redirects the E field to be more parallel to the wire $\endgroup$
    – Dale
    Commented Jan 28, 2023 at 12:47

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