Deriving Kirchoffs voltage law using the lumped element model

Question

I have one question about Kirchoffs laws using the lumped element model: https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws

The proof is on the Wikipedia page. They say "Approximate the circuit with lumped elements, so that (time-varying) magnetic fields are contained to each component and the field in the region exterior to the circuit is negligible"

On the Wikipedia page on lumped elements: https://en.wikipedia.org/wiki/Lumped-element_model

It is stated that: "1. The change of the magnetic flux in time outside a conductor is zero."

This is what I assume gives us:

$\nabla \times E= -\frac{\partial B}{\partial t}=0$,

so the curl of the field is zero.

But it says for lumped elements that "the time change of the magnetic flux outside the conductor is zero. But what about the magnetic field inside the conductors and resistors? Do we have that $-\frac{\partial B}{\partial t}=0$ here? How is exactly Kirchoffs law derived to take account for this effect? What are we assuming?, and why does not the magnetic field inside the resistors and conductors affect the fact that we can still use $\nabla \times E=0$?

Answer 1

Lumped model is there to make circuit analysis easier for electrical engineers. Imagine if you have a circuit, and you have to take into account every little detail effecting the current and voltage, which would take a lot of time and is usually unnecessary, like calculating how long it takes for a ball to return to your hand once thrown vertically, you solve basic Newtonian equations, and it gives a "good" approximation, depending on how accurate one needs to be.

To specifically address why $\frac{d\phi}{dt}$ is 0 outside the conductors and not inside them, is to be able to have less complication when analyzing these circuits. This means that, say if you have an inductor and a resistor connected to each other with a conductive material, the only factors effecting the voltage/current in your system is what happens inside of it, so you can ignore the magnetic flux change in the space caused by the inductor, which ofc would affect the magnetic field inside the resistor, but we assume that it is not significant.

Now you might ask, what about Radio Frequency (RF) or Wireless systems? because those systems need to be effected by something out of their system. From my experience, electrical engineers are not too bothered with the definition rather getting the system to work! but if you wanna get technical, I would assume that an antenna in a wireless communication system is not obeying the fact that the rate of change of magnetic flux outside of the circuit is 0 since we know that this is not definitely the case. I guess you can look at it as an external part of the circuit, where it can be looked at as a lump, but one with different charactrsitcs when compared with other parts of the inter-connected circuitry.