Timeline for Understanding Kirchhoff's first law in charged conductors
Current License: CC BY-SA 4.0
14 events
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| Jan 28 at 17:05 | comment | added | Puk | @JánLalinský [...] An isolated spherical capacitor can be represented in a lumped circuit by a capacitor between the spherical conductor and "infinity" (ground). If we are not including displacement currents, we are forced to say no current actually flows through the region this capacitor represents (no charge flow takes place in air). Including displacement currents allows us to speak of actual current flow through the capacitor. In any case, the theory is unambiguous and whether we accept displacement currents as "real" in this context is merely a matter of interpretation. | |
| Jan 28 at 17:05 | comment | added | Puk | @JánLalinský The distinction arises when we apply the two flavors of the continuity equation to a real circuit, as it appears OP is trying to do. One approach says the current flowing out of any closed surface is zero, the other says "it depends". There is no real distinction in the context of a lumped circuit graph (except what capacitor currents really represent): we have to assume KCL because it's a fundamental assumption of circuit theory. Yes, displacement currents provide rationalization of parasitic capacitances in a lumped circuit. [...] | |
| Jan 26 at 17:48 | comment | added | John Doty | @JánLalinský It is fully justified by classical electrodynamics. But of course, that's mere mathematical justification, not physics. | |
| Jan 26 at 17:30 | comment | added | Ján Lalinský | I meant not justified by the physical theory. It may be justified by its results, that's another kind of justification. | |
| Jan 26 at 17:12 | comment | added | John Doty | @JánLalinský I don't know why you think it unjustified. There's nothing mathematically shaky here, but more importantly, it works. That's the justification a physical idea needs. | |
| Jan 26 at 13:26 | comment | added | Ján Lalinský | Well finite element method is different. I meant introducing parasitic elements without justification in deeper theory than circuital equations, is a fitting enterprise, without justification. I agree that there may be a physics justification for parasitic elements in deeper theory involving $\mathbf D$. | |
| Jan 26 at 13:23 | comment | added | John Doty | @JánLalinský There's nothing fictitious about parasitic elements. When we intentionally concentrate displacement current in an object, we call the object a capacitor. Parasitic capacitors are the same thing, just unintentional and not so concentrated. As with intentional capacitors, their capacitance may be computed from geometry. This is not a "fitting enterprise". It's a special case of the finite element method. | |
| Jan 26 at 13:14 | comment | added | John Doty | @JánLalinský Nothing restricts the utility of Kirchhoff's laws to linear circuits. | |
| Jan 26 at 2:39 | comment | added | Ján Lalinský | 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 2:25 | comment | added | Puk | @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. | |
| Jan 26 at 2:25 | comment | added | John Doty | @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 2:09 | comment | added | John Doty | 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 2:08 | comment | added | Ján Lalinský | 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 1:42 | history | answered | Puk | CC BY-SA 4.0 |