# 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?

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]
• Imagine a circular pipe and a rectangular pipe, and between them some kind of box (maybe filled with dielectric, ferrite, etc.). The pipes are connected to the box. The box is the junction and it can be represented as a circuit of a theoretically infinite number of $i=Cdv/dt$ capacitors and $v=Ldi/dt$ inductors. Commented Jun 19, 2020 at 22:00