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I have some doubts related to electric fields and flow of current. So, let us assume an electric circuit, which contains a battery and a wire connecting positive and negative terminal of the battery. I read that at its negative terminal, there is a lot of negative charge/electrons, and because of repulsive force, electrons try to move away from each other, and so they try to move to the positive terminal. So, my first question:

When electrons start moving through the wire to the positive terminal, do they all move at once? Because otherwise, while they are moving, they will still exert repulsive forces on each other? Does this repulsive force affect their movement?

So, let us assume they do not move one by one. So, some of them flow through the wire, and because they are moving apart from each other, repulsive force also reduces. So my second question:

Shouldn't some of the electrons stay in the wire itself? If, at some point of the wire, there is not enough repulsive force present, will they stop at all, or will they reach the positive terminal?

Now, suppose there are many circuits, in which the shape of wire is different in each case(some straight, some circular). So, my third question:

Will the shape effect the movement of current? Does it have any effect on the electric field?

And, my final question. Suppose we have a very long wire. Now, I read that as we have concentration of negative charges on the negative terminal, we have concentration of positive charges on the positive terminal. So, if the wire is very long. So, the force between negative charge and positive charge becomes less:

Will the length of the wire effect the speed of the flow of charges? If we have an infinite length of wire, will charges flow at all?

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You've asked some really good questions here. Before starting, I want to first mention that the traditional picture of particles moving through a wire in electostatics is missing some physics; for instance, it ignores the quantum mechanical nature of electrons. The reason we still teach this model is because it captures the main effects (the phenomenon of current) without dealing with microscopic details, but I wanted to warn you that some of the answers will involve physics that is probably not contained in your readings in electrostatics.

To put things in perspective, we now know Newtonian physics is "wrong" (or perhaps more accurately, incomplete), and doesn't give the right answers if, for instance, an object is very small or moving very fast. But we still teach Newtonian physics because it's "good enough" for describing macroscopic objects like cars and baseballs.

Now, to answer your questions,

When electrons start moving through the wire to the positive terminal, do they all move at once? Because otherwise, while they are moving, they will still exert repulsive forces on each other? Does this repulsive force affect their movement?

The microscopic picture of a metal is (crudely) a collection of negative charges, aka electrons, moving through a lattice of positive ions. Indeed, there will be an attraction between these ions and the electrons, and repulsion between any two electrons. Surprisingly, there is also an attractive force between the electrons. The origin of this attractive force is that the electrons attract positive charges around them, and can in some cases lead to the formation of a bound state called a Cooper pair, which are relevant for explaining the phenomenon of super-conductivity, a phase of metals where the resistance is exactly zero. Note, this requires quantum mechanics to do properly, and is extremely subtle.

Shouldn't some of the electrons stay in the wire itself? If, at some point of the wire, there is not enough repulsive force present, will they stop at all, or will they reach the positive terminal?

Again, we need a more refined model, in this case statistical mechanics. Before connecting the terminals, the electrons all have a random distribution of energy which manifests itself as temperature. The presence of an electrostatic field causes a net flow of charge, but at the micro level, electrons are colliding and moving in a variety of directions. Often times you will see electrostatics books speak of drift velocity of the electrons, which is a statistical representation of the net flow. A single electron is probably moving much faster than the drift velocity, even perhaps in the opposite direction of the current flow, due to the random thermal energy and the collisions between particles.

Will the shape effect the movement of current? Does it have any effect on the electric field?

In electrostatics, no, but in reality, yes. In mechanics, one has statics and dynamics. In electromagnetism, one has electrostatics and electrodynamics. If you keep learning about electromagnetism, you will soon encounter another field, the magnetic field, and you will learn that the electric fields and magnetic fields are intertwined in such a way that lead you to reconsider the two fields as components of a single entity (hence, "electromagnetism"). In particular, you will learn that current carrying wires produce magnetic fields (Ampère's Law) and that changing magnetic fields can produce EMFs (Faraday's Law). This is a legitimate concern for building real world circuits, and the quantity associated with this effect is called impedance. Impedance is measured in Ohms, like resistance, and depends on the geometry of the circuit.

Will the length of the wire effect the speed of the flow of charges? If we have an infinite length of wire, will charges flow at all?

You're definitely on to something here. The resistance of the wire is proportional to the length of the wire. By Ohm's Law, the current is inversely proportional. The current is proportional to the drift velocity, so the current is inversely proportional to the length of the wire. See http://hyperphysics.phy-astr.gsu.edu/hbase/electric/ohmmic.html#c1 for a derivation.

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1.) When you first close a circuit, there is a very brief period of time in which the electrons push each other forward "one by one", so to speak. This very brief period of time is probably on the order of nanoseconds, so we usually ignore it. After that, a steady state condition is established, and the electrons move all at once ... more or less. Don't forget that the electrons are constantly colliding with impurities and lattice vibrations.

2.) So your second question is moot.

3.) Shape can have an effect. Resistance is greater where there are bends in the wire, and all circuits have stray capacitances and inductances which can affect operation. Often, (usually?) these effects are small and negligible. Sometimes they are introduced intentionally.

4.) Length will affect the speed. For a given applied voltage, a longer wire will have a lower voltage per meter (electric field), and the resistance of the wire increases. Eventually, a length will be reached where the current generated is unmeasurably small, lower than the current produced by thermal fluctuations. At that point one the current has effectively stopped.

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  • $\begingroup$ I don't think that the physical image in your first point is a very good one. The electromagnetic field outside of a conductor travels at the speed of light in the surrounding medium. It's this field that moves charge carriers at "the far end" of the circuit. Indeed, one can measure the "pushing of charge carriers" in ionic conductors with time resolved impedance spectroscopy because it happens on microsecond and slower time scales. In metals the "pushing" leads to skin-depth dominated losses on twisted pairs. $\endgroup$ – CuriousOne Jan 29 '16 at 22:36
  • $\begingroup$ @CuriousOne Yes, it's crude. The fields the drive the charges are due to surface charge density gradients in the wire. I don't think those fields propagate at the speed of light. $\endgroup$ – garyp Jan 30 '16 at 0:39
  • $\begingroup$ The point is that "electricity" i.e. the fields on wires doesn't move at the speed of charge carriers, charge carriers follow the fields, which means that the electrons at the other end won't care for a long time what the first ones in the cue are doing. One can measure that response both in the time and frequency domain on mismatched electrical conductors, but thankfully for us it doesn't matter. If it did, there would be no digital communications. $\endgroup$ – CuriousOne Jan 30 '16 at 1:10

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