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The most common reason we get for why the same current flows through all resistors in series is that because electrons cannot accumulate at any point. But why is that? Why can't some free electrons move from their free state to positive ions and similarly why can't new valence electrons join free electrons? Would it not change the current flowing in the circuit? Can someone clarify what I am missing?

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  • $\begingroup$ As the band structure of solids is due to quantum mechanics, it would be helpful if you tell us on which educational level you need an answer. $\endgroup$
    – Semoi
    Commented Nov 4, 2019 at 18:39
  • $\begingroup$ Actually I am still studying in my class 12th $\endgroup$
    – Sharad
    Commented Nov 4, 2019 at 18:48
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    $\begingroup$ Possible duplicate of Kirchoff law of circuit confusing me $\endgroup$
    – Bob D
    Commented Nov 4, 2019 at 18:51
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    $\begingroup$ Your instructor may have mentioned the asummption of steady state that is part of the "proof" of Kirchoff's laws. My experience is that students often miss it, and then if they start to think hard are confused. $\endgroup$ Commented Nov 4, 2019 at 18:56
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    $\begingroup$ @dmckee No, steady-state is not among the assumptions in the derivation of Kirchhoff's laws. What is assumed is a quasi-static regime (no they are not synonims). For a full derivation, have a look at Fano, Chu and Adler, Electromagnetic Fields, Energy, and Forces. As far as I know, this is the only book that presents a complete derivation, which is much more complicated than many people think. There's also a derivation published in the Bell's Lab Journal, but I cannot search it now. $\endgroup$ Commented Nov 5, 2019 at 0:56

4 Answers 4

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The reason is quite simple: Elecrons repel each other, and they do so quite strongly. The number of electrons is bound to be approximately equal to the number of protons in the solid (out of which only the free electrons can move).

A fun exercise is to calculate the coulomb force resulting from displacement of all electrons in a penny by one meter.

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  • $\begingroup$ that penny must be bigger on the inside :) $\endgroup$
    – dlatikay
    Commented Nov 5, 2019 at 22:12
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This question is a bit confusing. Electrons CAN accumulate in a point in a circuit, that's how capacitors charge up.

And the second part of your question concerning free electrons and ions is not very clear.

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  • $\begingroup$ Why is this an answer? $\endgroup$
    – Semoi
    Commented Nov 4, 2019 at 18:39
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    $\begingroup$ Q: Why can't electrons accumulate at a point in a circuit? A: They can. $\endgroup$ Commented Nov 4, 2019 at 18:42
  • $\begingroup$ IF THEY CAN THEN HOW WE CAN OBTAIN A STEADY CURRENT THEN? $\endgroup$
    – Sharad
    Commented Nov 4, 2019 at 18:48
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    $\begingroup$ Hmm, the question is about resistors in series. So in my opinion this is not an answer to the question. I agree that the question is not perfectly stated. However, the topic is Kirchhoff's law. So I expect that the questioner is rather young. We should try to answer the question he/she has and just stating that the question is wrong. Do you agree? $\endgroup$
    – Semoi
    Commented Nov 4, 2019 at 18:50
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    $\begingroup$ I believe a node is defined in electrical engineering as any point on a circuit where the terminals of two or more circuit elements meet. I don’t think the single capacitor plate meets that definition. I raise this distinction because the OP in a previous post questioned the relationship between KCL and conservation of charge. Don’t want the OP to think the accumulation of electrons on the capacitor plate would violate that law since KCL applies to an electrical node. $\endgroup$
    – Bob D
    Commented Nov 4, 2019 at 20:35
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With real wires, there will be some parasitic capacitance between any given wire and whatever other wires are nearby. Because of this some charge will accumulate in the wire (and an equal charge will be repelled from some other conductor, leading to no net charge accumulation in the system as a whole). However often (and if we design carefully) this capacitance is small enough that we can ignore it and get a "good enough" model to predict how our circuit will behave.

In comments you asked,

IF THEY CAN THEN HOW WE CAN OBTAIN A STEADY CURRENT THEN?

Because once charge has accumulated in the wire, it will oppose further accumulation. We eventually end up with a steady state charge (attributed to the parasitic capacitance) and a steady state current through the wire, which doesn't depend (at all in many cases) on the accumulated charge on the wire.

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There are multiple quantum mechanical ideas involved:

  1. Due to the interaction between the charged particles, the solid obtains a band structure. As you already stated some electron occupy states "close to the ion". These electrons are bound to the ions and can not travel freely inside the conductor. In contrast, some electrons form the so called free electron gas inside the conductor. They are the charge carriers (i.e. current carrier).
  2. The band structure is also due to Pauli's exclusion principle (which is a symmetry principle and not an interaction). Pauli's principle states that two electrons can't occupy the same state (simplified version). Thus, if a bound state ("near an ion") is occupied, no electron from the free electron gas is allowed to occupy this state. The state is filled.

Now the idea of a solid is to fill up the allowed electron states starting with the states which posses the lowest energy. These are the bound states (they are "closer" to the positively charged ions). If the solid possesses only "few" electrons, such that all of them find a bound state, then we call the solid a non-conductor. In contrast, if the solid possesses "many" electrons then some of the electrons must occupy free electron states and we call the solid a conductor (I won't go into semi-conductors, here) -- note that I define "few" and "many" with respect to the available number of bound states.

I believe that the simplest picture is that the "bounded electrons" shield the ions from the "free electrons". Just as every single atom can occupy only a certain amount of electrons before it is electrical neutral, this is also true for a solid. However, since two ions can share a "bounded electron", they don't need all their electrons to compensate their electrically positiv charge.

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