# Voltage and current spacetime propagation in a circuit

It is often stated that for lumped circuits the signal propagation can be considered instantaneous, so the the circuit parameters do not depend on space coordinates.

But how to actually derive this fact from Maxwell's equations?

First I would like to know how to derive the general space-dependent voltage and current expression in a circuit. Then the lumped case can be obtained by letting the space dimensions go to $$0$$.

Edit: I have seen this question. The problem is that there is no "small-sized" circuit $$d << \lambda$$ assumption made anywhere in the derivation. I cannot even see, why that derivation cannot apply to large circuits and I want the answer to this question to explain this as well. Also, I wanted to see how to go about deriving the distributed circuit equation first, and then using the approximation, instead of using the approximation along the way - and let alone using it somehow implicitly, unrigorously, without even writing it down, as was in the answer I linked.

• Possible duplicate physics.stackexchange.com/questions/102458/…? Commented Apr 17, 2023 at 0:09
• I made an edit and explained why it is not
– Sgg8
Commented Apr 17, 2023 at 8:25
• Just because the answer there does not use the specific condition $d \ll \lambda$ does not mean that it didn't answer that part. The part about discretisation is the relevant part, and can be translated into the lumped circuit case. Commented Apr 17, 2023 at 9:12
• @naturallyInconsistent I don't see how to translate this. In the answer the discretization was actually mentioned only after the laws were derived.
– Sgg8
Commented Apr 17, 2023 at 9:39
• That is because Maxwell's equations are the correct equations; its classical use is valid until quantum corrections are needed. So the form of the equations in deriving the lumped circuit approximations do not come with the lumped circuit limitations. It is only when you take the discretisation step do you insert the lumped circuit simplifications. Commented Apr 17, 2023 at 9:42