# Isn't Kirchhoff's junction law a violation of conservation of charge?

As a beginner of classical electrodynamics I am quite confused in understanding Kirchhoff's junction law or it may be the reason that I misunderstood the law of conservation of charge.Please correct me if I am wrong:-

I understood the law of conservation of charge as follows:-

The net charge of an isolated system will always remain constant.

By isolated system I think that it is a system in which we can't take out any charge from it as well as we can't add any charge to it.

Now considering Kirchhoff's junction law, Imagine a junction of wires at $$J$$ which is a part of an electric circuit. When the switch is open then no current passes through $$J$$. I consider $$J$$ as my system. My teacher told me that the junction law is just the other form of law of conservation of charge taking junction as the system. When the switch is closed then current flows through it and when we apply the law of conservation of charge at $$J$$ then we arrive at the junction law.

Now my question is how can we apply law of conservation of charge at $$J$$ because $$J$$ is not an isolated system?

I don't think that it is an isolated system as when switch is turned on then we can't write the equation using law of conservation of charge. So I think that Kirchhoff's junction law violates the law of conservation of charge! Am I correct or totally wrong?

I understood the law of conservation of charge as follows:-

The net charge of an isolated system will always remain constant.

You're missing a crucial distinction between local conservation of charge and global conservation of charge. The statement as you have given it is a correct encapsulation of the global version of the principle, but electromagnetism is also subject to the local conservation of electric charge. Local conservation can hold for any arbitrary volume, whether it is isolated or not, and it says that for the total charge within the volume to change, there needs to be a flux of charge into or out of that volume.

As you correctly note, Kirchhoff's junction law does not follow from the global conservation of electric charge, which would be completely untouched if electric charge could instantaneously teleport from one part of the circuit to another. Instead, it is an expression of the local version, i.e., it embodies the principle that to get from A to B, electric charge actually needs to move along a path that joins the two.

For some more detailed explanations, see this section in Wikipedia.

And finally, it seems that you're strongly misusing the word "violation" here. There's a difference between "A does not follow from B" and "A is in direct conflict from B": the fact that B does not imply A does not mean that they're incompatible. The fact that Kirchhoff's junction law does not follow from global charge conservation doesn't mean that it represents a violation of it. (Similarly, Kirchhoff's junction law cannot be derived from relativistic invariance ─ but that doesn't mean that it is a violation of the latter!)

Conservation of charge is better understood as a special case of the continuity equation: $$\frac{\partial Q_{system}}{\partial t} = \dot{Q}_{incoming} - \dot{Q}_{outgoing}$$ This formula becomes a conversation law if $$Q_{incoming}=Q_{outgoing}=0$$ (i.e. the system is isolated), $$\frac{\partial Q_{system}}{\partial t} = 0.$$ As to not confuse the students who are new to these concepts, introduction level materials doesn't name the continuity equation, but rather talk about its results as the conversation laws. So, in this example, $$J$$ is indeed is not an isolated system. But, it still must obey the continuity equation.

Since there is no element capable of storing charge (i.e. a capacitive element) in $$J$$, $$Q_{system}=0,\ \therefore\ \frac{\partial Q_{system}}{\partial t} = 0.$$

From this, substituting into the original equation we get the Kirchhoff's junction law: $$I_{incoming}=I_{outgoing}$$ (since $$\dot{Q} = \frac{dQ}{dt}=I$$)

Your idea that this junction is not isolated is correct, but this system cannot violate conservation of charge since it is not isolated in the first place.

TL;DR: Conservation laws apply only to isolated systems. There are other laws for general (isolated or non-isolated) systems from which these can be deduced.

The junction is not an isolated system. It is connected to a circuit. Your professor may have been a little sloppy with the choice of words if they said isolated. If we assume that no charge accumulates in the junction then what goes in must come out. Conservation of charge is $$\text{div}(\vec J) = -\frac{\partial \rho}{\partial t}$$. We are assuming that the r.h.s = 0 in the absence of any evidence to the contrary so the l.h.s = 0 and this is the junction rule. The junction is still a "system", it's just a system that is interacting with the rest of the circuit. The point is that it is an idealization of nature.

My small contribution to this comes in the form of addressing the assumption. It is reasonable considering the idealizations at work.

In the 80s there was a lot of interest in a few areas of "Exotic Systems in QM". I was working on some of these problems in graduate school with a few professors. Two of these are of interest here. Solving Schrodinger's equation on a graph, and solving Schrodinger's equation on curved 1-dim and 2-dim spaces where the electron is effectively trapped by a boundary condition by a limiting process. In these cases you find effective potential wells can emerge due to the geometry of the bends and the graph loops. I'm not trying to say that this contradicts the assumption that charge in not trapped in the junction as that is a fair idealization. And large scale circuits probably do not exhibit this type of trapping. I just this this question is a nice place to bring it up. In reality some defect in the circuitry could create an effective capacitance in a junction. Charge would probably grow a bit then break away. This would be observable as a small instability in the measured current on small (I'm assuming it would happen fast) time scales.

Rarely does nature follow these ideal system models. But we study them to understand the basic principles in the laws of physics and they are useful for building up an idea of what is happening in real systems.

• Hi Sir, would you please elaborate in terms of classical electrodynamics aas I have no idea about quantum mechanics. – Gaurav May 17 '19 at 16:48
• I have, you can ignore the last paragraph. My primary comments are in the first. I believe you might be confusing static charge conservation within a closed system versus the continuity equation that describes how the change of charge at a point relates to the flow of charge into or out of that point. – ggcg May 17 '19 at 16:51

Kirchhoff's current law can be stated that the algebraic sum of the currents flowing into a node (junction) is zero, or

$$\sum i_{in}=\sum i_{out}$$

Electric current $$i(t)$$ through a surface of a conductor is defined as the rate of charge transport through that surface of the conductor, or

$$i(t)=\frac {dq(t)}{d(t)}$$

Where $$dq(t)$$ denotes instantaneous charge.

Therefore Kirchhoff's current law is a statement of conservation of charge since it is saying that charge cannot increase or decrease at a node.

Hope this helps.

Actually you are somewhat wrong in taking your system.

You have just taken the junction but you forgot the battery that is supplying power.

So when the battery is supplying power to the junction your idea of just considering 'Junction' as a isolated system is flawed. Because there is an external intervention in your system via the battery

If you want a fully isolated system you will have to consider the whole electric circuit. *Including your source.

After answering a comment I got another way to explain this.

In case you still want to consider junction as a system then you must look at basics of conduction.

When a metal is conducting the number of electrons it has is still the same (it is just the motion of electrons that you recognise as current).

Hence I would like to say that as number of electrons in the conductor is constant the charge is still constant, it is just moving.

As the conservation of charge specifies that there is no creation or destruction of charge hence the law still holds.

• Doesn't really address the question. Kirchoff's Law is specifically about junctions, the presence of the battery is immaterial. I think the issue here is that the given statement of conservation of charge is too restrictive. A better definition would be something like "If the charge of a system is constant, then the net charge entering is zero" – garyp May 17 '19 at 15:27
• Actually net charge is still constant. The metal that is conducting has constant electrons all through the process. The metal that is conducting does not lose any electrons while conducting or when switch is opened. We are forgetting that fundamentally current is just movement of charge not creation of charge. – Berserker May 17 '19 at 15:30