Outside the lumped circuit model, is it possible to define voltage satisfying the known relations for inductors, capacitors and resistors? The 3 most common devices in electric circuits are inductors, resistors and capacitors. The relation between current and (terminal) voltage across these elements is often used in circuit analysis, namely:
\begin{align}
U = RI   \\
U = L \frac{d}{dt} I \\
\frac{d}{dt} U = \frac{I}{C}
\end{align}
When the assumptions of the lumped element model (no charges in inductors connecting the elements, no change of magnetic flux outside elements) don't hold anymore, people mostly worry about wether Kirchhoff's laws are still applicable. I'd like to ask another question, namely wether the relations written above can still hold, even with the most ideal resistors, inductors and capacitors possible.
My own answer would be that this is impossible, but I'm not sure wether I made a mistake:
Let's assume there is an electric field $\vec{E}_0$ throughout the whole circuit, created by charges inside and (possibly) outside the circuit. If I define voltage as the negative line integral of the overall electric field, then the voltage-drop across a capacitor is affected by $\vec{E}_0$, and doesn't depend solely on $Q$, the charge inside the capacitor, anymore.
However, defining voltage as the negative line integral of the overall electric field is the only definition by which $U=RI$ holds, because this is just a reformulation of $$\vec{j} = \sigma \vec{E}$$
Is my reasoning valid, and this is another caveat when talking about "voltage" outside the realm of the lumped-circuit model?
 A: In microwave circuit analysis, it is standard practice to introduce voltages, currents, impedances, etc., such that Kirchhoff's laws (1&2) hold. This is based on the observation that waveguides sustain a well-defined discrete set of propagating modes. Take, for example, a junction to which one or several waveguides are connected and launch a wave in one of the guides far away from the junction so that all the evanescent modes between the launcher or loads and the junction can be ignored, and the frequency is such that only a single propagating mode may exist in each waveguide.
Let $V_k, I_k$ denote the amplitudes of the $E$ and $H$ propagating field, resp., in the $k^{th}$ guide. Then it is possible to show that there is a matrix $\bf{Z}$ or $\bf{Y}=\bf{Z}^{-1}$ that depends only on the geometry of the conductors and $\bf{V}=\bf{Z}\bf{I}$ or $\bf{I}=\bf{Y}\bf{V}$. (This is almost always true except for a handful of singular cases, such as open or short circuits, etc., but these can still be handled with scattering matrices that always exist.) For an excellent description see, [1]
[[1]: https://archive.org/details/in.ernet.dli.2015.16056/page/n145/mode/2up Montgomery, Dicke, Purcell: PRINCIPLES OF MICROWAVE CIRCUITS, chapter 5]
