# Understanding Kirchhoff's first law in charged conductors

I wonder about Kirchhoff's first law in charged conductors. Consider:

$$j = \sigma E \implies \nabla \cdot j = \sigma (\nabla \cdot E) = \frac{\sigma \rho}{\epsilon_0}$$

This means that Kirchhoff's first law does not hold true if the conductor is charged ($$\rho \neq 0)$$

But is not that exactly the case with any conductor connected to a capacity?

Are the surface charges not exactly that excess in charge that create the electric field.

• Conductors in usual low frequency circuits have $\rho = 0$ inside, non-zero charge density can be only on their surface. But on the surface, the local Ohm law $\mathbf j = \sigma \mathbf E$ does not hold, because current component perpendicular to the surface can be zero, while electric field is not, so the two vectors are not parallel. Also, $\sigma$ is not position independent on the surface. One can have non-zero charge density on the surface while divergence of $\mathbf j$ is zero, this is e.g. when the surface charge is constant in time. Jan 25 at 17:46

You are correct, Kirchhoff's circuit laws only apply when the lumped circuit approximation is valid and one of the assumptions of the lumped circuit approximation is that no charge accumulates in the conductors connecting the circuit elements.

However, in case there is significant charge accumulation in the conductors, circuit theory can easily account for this by modeling parasitic capacitor(s) at the appropriate points in the circuit.

But is not that exactly the case with any conductor connected to a capacity?

Usually the charge accumulates significantly only on the "plates" of the capacitors. These are considered part of the capacitor element and not part of the interconnect wiring, so charge accumulation here presents no special issue in terms of using KCL to solve a circuit. The charge that accumulates on the capacitor plate is simply accounted for as a current in or out of the capacitor element.

There is a way to state KCL without stipulations regarding charge accumulation. Rewrite the continuity equation (or take the divergence of Ampere's law) to get $$\mathbf\nabla\cdot\left(\mathbf j + \frac{\partial \mathbf D}{\partial t}\right)=0.$$ $$\partial \mathbf D/\partial t$$ is the so-called displacement current density. If we take $$\mathbf j +\partial \mathbf D/\partial t$$ to be "the current density", thus including the displacement current in what we call "current", then the total current leaving any closed surface is always zero.

Most famously, when current is flowing into one plate of a capacitor and out the other plate, displacement current is present between the plates of the capacitor. Taking this into account partially justifies that language of "current flowing through a capacitor". On the other hand, displacement current can be considered negligible in conductors (and zero at steady-state), so there is no real ambiguity in which version of current we speak of in practice.

• Reference for stating KCL this way? It does not seem viable for circuits, as displacement current is not limited to conductor body and also because KCL would refer to additional unknown variables not appearing in KVL, not just $I$ which appears in KVL. Jan 26 at 2:08
• This is the correct answer. When you use KCL, you are implicitly promising to account for displacement current. And when you use KVL, you are implicitly promising to account for induction. Jan 26 at 2:09
• @JánLalinský The "extra unknown variables" come from the geometry. You can't understand the relation between charge and potential without them. If you do IC design, one of the last steps is to take the physical layout, extract a circuit model from the geometry, and simulate that to gain confidence in the design. The extracted circuit has a lot of tiny capacitors in it to address this problem. Jan 26 at 2:25
• @JánLalinský R. S. Eisenberg, "Kirchhoff's Law Can Be Exact," arXiv preprint arXiv:1905.13574, 2019. I agree that the subtlety of displacement currents is lost if you are just given a lumped circuit. It's still useful to keep them in the back of one's mind when converting a real circuit into an approximate lumped circuit for instance, where important paths of displacement current flow can be modeled by adding parasitic capacitors. $I$ doesn't appear explicitly in KVL except via Ohm's law etc. where the displacement current doesn't really matter anyway.
– Puk
Jan 26 at 2:25
• Thanks. I meant $I$ due to resistors, and $\dot{I}$ due to inductors, and $\int I dt$ due to capacitors; the fact KCL refers to $I$ only allows us to solve the circuital equations in these simple linear cases. When one introduces fictitious "parasitic" network elements, the same KCL with conduction current is still used in the resulting network. So I don't see how KCL with displacement current helps - one can introduce as many parasitic elements as needed to get a model to fit anything. Are you saying one can justify this fitting enterprise or limit its complexity using laws $\mathbf D$ obeys? Jan 26 at 2:39

When there is a charge in a conductor, then the net supplied voltage will be the algebraic sum of the voltage source and the voltage because of the charge in the conductor. The net voltage source will be equal to the net voltage drop and is the Kirchhoff’s Voltage Law (KVL).