When is Kirchhoff’s Current Law True?
Recently I had the opportunity to reflect on the famous Kirchhoff Current Law (KCL). I was reviewing the reasons behind this law as well as its cousin, the Kirchhoff Voltage Law, and how to prove them, in preparation for a lesson with a student when I realised to my own great surprise that I didn’t really know why they were true!
In fact, I suddenly realised that these laws were very nontrivial and, having further thought about them for a while, realised (at least if I haven’t made a mistake) that Kirchhoff’s Current Law is not even always true — they only hold under certain conditions which I was not previously aware of.
The state of my understanding right now is this: KCL is not always true, so it is not a consequence of Maxwell’s equations, but KCL is true in most cases we care about in practice. So there exists a set of reasonable assumptions from which KCL can be proven. But what assumptions? I don’t know.
It is a bit weird to write a blog post where I basically admit I don’t understand something and haven’t found the answer. Nevertheless, I will proceed — partly for the sake of consolidating my own understanding and partly because it may be interesting to someone to see how I thought through these issues. Anyway, I believe it is important to admit imperfections in understanding, to oneself as well as to others. :)
(Note: In case there is any misperception, this is definitely not new research or any kind of groundbreaking stuff. Probably, this topic is completely satisfactorily discussed in various introductory circuit analysis or electrical engineering textbooks which I haven’t bothered to read. I am writing this from the perspective of a confused student working through something hard and recording his progress, not the perspective of some researcher.)
In a simple and naive understanding, KCL says: the current flowing into a junction in a circuit is always the same as the current flowing out. Reasonable enough, right? I had always thought this law was just a trivial consequence of charge conservation, since charge buildups are not allowed at any junction. In some sense, this is true — but in another, if we think about it carefully, it is not trivial at all.
It is reasonable enough to think "at any point along any circuit in any situation, there cannot be a charge buildup”. From this claim, it’s true that KCL follows trivially. I do not know at what point this claim entered into my intuitive understanding of circuits — perhaps it is from primary school where we understand the amount of electric current through a circuit as a constant quantity independent of position along the circuit (which I guess is itself a simple form of KCL). In any case, it is possible to notice that this claim is not a fundamental law of physics! Therefore, if we want to really rigorously justify it, we have to find some way to prove it based on an understanding of classical EM through Maxwell’s equations.
Suppose that there were a charge buildup somewhere along the circuit. My first thought was that, if so, the current state of the circuit has led to a positive and finite rate of change of charge at that point; due to time-invariance of physics, this means that at any point in time the circuit at that position will also have a finite rate of change of charge. This implies the total charge built up at that point tends to infinity as time increases to infinity. Although this is not a mathematical contradiction, I felt this is enough of a physical contradiction with intuition to end the argument.
But wait! This argument makes no reference to what the point was; in particular, the point may be a point on a capacitor in which case my argument seems to imply that capacitors can never charge up or discharge! That’s clearly wrong. So there must be something wrong with the argument. Yes, there is: the trouble is that the time-invariance of Maxwell’s equations does not imply that the circuit will be in the same state at all future times, and therefore it does not imply $dQ/dt$ at that point will be constant. In particular the $Q$ at that point can and must change, and this is a change in the state of the circuit, and the argument falls apart. I mean, to take a mechanical analogy, this is like saying that the time-invariance of Newtonian mechanics implies that all things are always in inertial motion. That’s clearly wrong.
How else to proceed? Whatever argument we produce must take into account the notion of a junction: KCL only holds at a junction, whatever that means. (How to precisely define junction?) In particular, as discussed above, KCL obviously doesn’t hold for a capacitor plate. (Although somehow if you think of the whole capacitor together, it’s still true?! What is this magic? There’s a gap between the plates and current doesn’t even flow directly between them and yet the current in is still equal to the current out! I guess this is a consequence of the charges on both plates of a capacitor being equal, but why is that true?! Okay it is ‘obvious’ and we may take it for granted but how do we really prove it?)
And what’s wrong with a charge buildup anyway? Intuitively, I think it is fine enough to say something like: if there is a charge buildup somewhere, these charges would repel each other, and so it would be more energetically favourable for the charges to be spread out evenly and smoothly across the circuit to ensure uniform current. Well, that’s fine, but it’s not rigorous. Usually such things can be proven rigorously — if I want to argue there’s no charge buildup in the interior of a conductor in electrostatics, for example, we can do so rigorously by an application of Gauss’ Law. But that same argument doesn’t work now, because, for one thing, the charge buildup may very well be on the surface of the wire and not the interior.
Well, here I am stuck, honestly. I do not see a clear mathematical reason why charge buildups are not allowed! In fact if we make an analogy of the charges as a sequence of balls in series connected by springs (analogy for the Coulomb repulsion), if we imagine suddenly giving a push to the first ball, that would cause a ‘density wave’ to propagate down the springs in such a way that causes the number of balls per unit length to not be uniform, precisely a ‘charge buildup’ at different locations at different times through the springs! One can imagine, perhaps, that in the circuit case, this is analogous to taking a wire loop and suddenly connecting it to a voltage source. Perhaps it would take time for the current to propagate (well, we know this is true since we can take some kind of distributed element model where each element has RLC and compute the time constant and so on…) and therefore we can temporarily have some charge buildup. But if this is correct, then that means we can’t have any mathematical proof that charge buildup is impossible, since actually it is possible! (I don’t know, by the way, how good the ball-and-springs analogy is… such things are confusing. Even the hydraulic analogy for circuits, which I found totally convincing for a long time, fails in some cases for some reasons I still don’t fully understand.)
So what assumptions, actually, do we need for KCL to be true?
It is enough to assume that the circuit is in steady-state, interpreted strongly to mean something like: the state of the circuit is constant with time. If so, then the argument based on time-invariance I provided before really does work. But it is really depressing if we have to take this strong assumption: it would mean we cannot use KCL in the analysis of the behaviour of even a simple RLC series circuit in DC. If we want to know, for example, how quickly a capacitor charges up when connected in series to a resistor and battery, we need to be able to consider the process of charging, in which the circuit is obvously not in steady-state.
I have seen online some sources that claim KCL is true if we assume that the frequency of the circuit, in the AC case, is not too high. I guess the implication is that if the frequency were very high, then inductance of the wires themselves becomes very significant (and maybe also tied together with that effects due to displacement current?) and so the lumped-element assumption doesn’t work? But I’m not sure, and I don’t understand this at all, to be honest. Or, I guess intuitively this kind of makes sense because in the balls-and-springs example it’s like assuming the density wave propagates so fast through the whole loop that its effects may be neglected because they ‘average out’ or maybe ‘dissipate out’ due to resistive forces or something? In that case then the low-frequency assumption is to make sure that extra density waves aren’t being generated so quickly that they overwhelm the circuit’s ability to even itself out. Like the time scale of density wave generation needs to be much greater than the time scale of the density wave dissipation. (By the way what would be the latter even be in the circuits case? I am confused about this also. Is it the length of the circuit divided by the speed of light? That would sound reasonable to me but I am not sure. I do not even know if the balls-and-springs analogy is reasonable or not.)
In the end, I am a simple man: I would simply very much like to use KCL when analysing DC RLC series circuits. If that’s all I get, I am happy. But I still haven’t found any reasonable combination of assumptions which would allow me to rigorously prove that KCL is true in this case. I am sad. :(
Brief remark: I also had similar troubles with KVL, due to certain subtleties in the definition of the notion of voltage. But I think I have figured it out that assuming the lumped-element model holds, which is a good enough assumption for me, we can get a good and simple proof of KVL. I don’t see, though, how this extends to KCL.
