Question on the constant current in electric circuits. [ Not homework ]

[Nanyang]

Hi I'm new here and have this question I would like to ask. Sorry if I posted in the wrong section! My level of education is not high so please forgive me for this simple problem, though not simple to me.

Q: Why is it that in, for example, a series electric circuit that the electric current is constant?

I thought about this for very long (a year) when I am free and now, I kind of gave up trying to think of a solution independently.

note: this is not a homework

I asked my teacher about this and he said:

In a closed circuit with potential difference. There will be a net flow of charges in a specific direction. The collision of the electrons with the atoms of the conductor is represented by the resistance of the wire.

You can see current as a sea of electrons moving at a certain direction in general. According to Kirchhoff law, there should not be any charge accumulation anywhere in the circuit. The charges should move together. So the flow of electrons per unit time (current) in a series circuit is always constant.

I thought of it and replied him saying:


Note: I used div and grad to put my points forward, but I mostly self learned that and thus is not an excellent user of this mathematics. So do forgive any ugly mistakes made and if possible, correct it so that I can learn.

So Kirchhoff nodal rule is div J = 0 which means that there is not net current through any closed surface formed on the wires. The closed surface can be infinitesimally small so that it's a point. So for any point on the circuit the net current flow is zero, thus the current is constant throughout the circuit for those charges that are moving, if there is no resistance.

I think this is something like a longitudinal wave and this wave should propagate at around c, assuming the apparatus is in a vacuum. Which means at an arbitrary time t, not all points on the circuit has charges moving.

But if I look at ohm's law, V=IR, if R is constant,

grad V = R grad I

E = - R grad I

So I is constant at all points in the circuit if E=0, where E is the electric field parallel to an infinitesimal displacement on the circuit. But electric fields extend to infinitely large distances, thus the parallel component of the electric field of the potential source is definitely not always zero on the circuit, as it is a closed loop. In other words, the charges, when going through the points where there is an electric field, accelerates and thus it's velocity changes.

Therefore, the current is not a constant as it goes around the circuit.

Now moving to the problem with the collisions of the electrons with the atoms of the conductor... More specifically the positive ions, since the metal loses negative electrons for conduction... etc... etc... etc... Anyway, the collisions occur because of the repulsion of the electrons of the atom and the free electrons that are flowing. To conserve momentum, the velocity gained by the electrons in orbit is slightly less than the velocity lost by the flowing electrons, and the orbiting electrons now taking a longer orbit.

Unless the previously flowing electron somehow enters the shells of the metal ion and the previously furthest electron is no longer orbiting the nucleus and now becomes a flowing electron, I cannot see how div J = 0.

Also, the positive nucleus gets attracted to the previously flowing negative electron and moves a little. Hence the 'missing' momentum from the previous consideration of the conservation of momentum is in the positive nucleus and therefore resistance is a property of the material, as the speed of the 'ejected' electron (that was previously orbiting before the flowing electron arrives near the ion) depends on the number of electrons orbiting as well as the mass of the nucleus. I don't know if this is true, but it's an idea.

If the flowing charges don't pass through, I think it will be something like a capacitor...where charges accumulates and the charge density changes with time, implying a non zero value for div J. Then the current is not the same at all points in the circuit.

Sorry if this seems a little long, but these are some of the considerations I had over about one year thinking about this.

So do correct me where I'm wrong and hopefully help me understand why the current is constant in a series circuit.

 

[HallsofIvy]

Think, first, of direct current- electrons moving in one direction through the wire. If one section of current were greater than in the next, there would be a "build up" of eletrons where the faster moving electrons ran into the slower electrons: a "traffic jam".
That can't happen in a resistance only circuit. It can happen, I believe, on a capacitor where charge can build up.

 

[Dadface]

In terms of "free electron theory"each electron accelerates between collisions and the current carried by each separate electron will indeed vary as it makes its way around the circuit.Remember,however ,that the same theory deals with the huge numbers of electrons we get with typical currents and the effects average out to a constant current.Free electron theory is useful up to a point but at a more advanced level we have to start using quantum theory.
May I say that although I have not absorbed the points made in your analysis you are clearly a very deep thinker. Well done.

 

[Nanyang]

Thanks for the replies!

Think, first, of direct current- electrons moving in one direction through the wire. If one section of current were greater than in the next, there would be a "build up" of eletrons where the faster moving electrons ran into the slower electrons: a "traffic jam".
That can't happen in a resistance only circuit. It can happen, I believe, on a capacitor where charge can build up.

I don't know much about capacitors or inductors so my question is based on having a potential source (maybe some charges at some point) and a closed metal wire loop.

I thought about what you said and thought that:

When the faster ones run into the slower ones, the previously slower ones become the now faster ones and the previously faster ones become the now slower ones... so it's like a longitudinal wave? Hence for some points the electrons get closer together (contraction) and some points further apart (rarefaction), so the charge density is not constant, therefore the current is not a constant.

Hmm... could it be possible that texts are referring to the total charge flow as a whole being constant... and not talking about the current flowing through two or more arbitrary chosen points on the circuit being the same?

In terms of "free electron theory"each electron accelerates between collisions and the current carried by each separate electron will indeed vary as it makes its way around the circuit.Remember,however ,that the same theory deals with the huge numbers of electrons we get with typical currents and the effects average out to a constant current.Free electron theory is useful up to a point but at a more advanced level we have to start using quantum theory.
May I say that although I have not absorbed the points made in your analysis you are clearly a very deep thinker. Well done.

Hmm...I thought of that before, but I'm not sure if there are experiments to confirm, at a very precise level, that the current is not a constant for two arbitrary points on the wire.

But texts (and teachers and people) always say that the current is constant like a 'miracle', no mention of averaging (at least for those I read) So I'm looking for an explanation for this 'miracle'...

Again, thanks for the replies! Would be great if there were more opinions from other people on this matter...

 

[Dale]

Hi Nanyang,

Consider conservation of charge. If you look at a small segment of a wire than any current that flows into the segment must either flow out or increase the charge of that segment. One of the implications of the "small circuit" assumption of circuit theory is that charge does not accumulate in any circuit element, therefore if the charge of the segment is not changing then the conservation of charge implies that the current entering is exactly equal to the current leaving.

 

[Nanyang]

Hi Nanyang,

Consider conservation of charge. If you look at a small segment of a wire than any current that flows into the segment must either flow out or increase the charge of that segment. One of the implications of the "small circuit" assumption of circuit theory is that charge does not accumulate in any circuit element, therefore if the charge of the segment is not changing then the conservation of charge implies that the current entering is exactly equal to the current leaving.

Does this assumption also hold for practical circuits in 'real life'? I mean like a battery and a wire.

If it isn't true, then is the 'averaging' effect mentioned by Dadface and also that the electric field of the voltage source is very small hence making it seem almost like a constant electric current?

In other words, is the ammeter (those used in ordinary school labs) not accurate enough to measure the tiny variations in the current, if the assumption that charge cannot accumulate at a point is not true for practical, 'real life' circuits.

 

[Dale]

Does this assumption also hold for practical circuits in 'real life'? I mean like a battery and a wire.

Yes, it holds very well which is why your electronic appliances work. In practice it works well for any circuit that is smaller than about 1/8 of a wavelength of the highest frequency involved, which is why it is called the small circuit approximation.

 

[alxm]

Also, the positive nucleus gets attracted to the previously flowing negative electron and moves a little. Hence the 'missing' momentum from the previous consideration of the conservation of momentum is in the positive nucleus and therefore resistance is a property of the material, as the speed of the 'ejected' electron (that was previously orbiting before the flowing electron arrives near the ion) depends on the number of electrons orbiting as well as the mass of the nucleus. I don't know if this is true, but it's an idea.

Well, if I remember my physics right, it's true in some cases, but only for highly energetic electrons. In general, it's assumed electron transitions occur so fast that the nuclei don't have time to 'react' and change their position/momentum. This is the basis of the Franck-Condon principle.

 

[Dadface]

There will be tiny current fluctuations at each point in a wire but an ammeter cannot measure these because it measures the current through the whole cross section of the wire.
alxm,it is wonderful what you can pick up in these threads and I have never heard of the Frank Condon principle.I have just googled and I am going back for another look.

 

[cabraham]

A helpful reference you can explore is to search for "Johnson noise" and "shot noise". Electrons moving through conductive media do incur collisions, and the dc value is only an statistical average. Noise components are always present. The magnitude of said noise increases with bandwidth. The higher bandwidth the circuit, the less these variations average out. Hence, a very low BW network has low Johnson & shot noise.

Claude

 

[Nanyang]

Thanks a lot everyone :)