# Deriving Kirchoffs voltage law using the lumped element model

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"

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 $$\nabla \times E=0$$?

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.