How can one practically measure electrostatic potential differences in a circuit when there's a varying magnetic flux?

Question

Suppose we have an electrical circuit in the presence of a varying magnetic field. As in https://physics.stackexchange.com/a/847326/404476, we define the electrostatic potential at a point $\mathbf{x}$ at time $t$ by $$ V_\text{static}(\mathbf{x},t) = \frac{1}{4\pi\varepsilon_0}\iiint_{\text{3D space}} \frac{\rho(\boldsymbol{\xi},t)}{|\mathbf{x}-\boldsymbol{\xi}|} \, \mathrm{d}^3\boldsymbol{\xi} $$ where $\rho(\boldsymbol{\xi},t)$ denotes the charge density at point $\boldsymbol{\xi}$ at time $t$, and $\varepsilon_0$ is the permittivity of free space.

Given two points $\mathbf{x}$ and $\mathbf{y}$ along the circuit, is there a practical way to measure $\,V_\text{static}(\mathbf{x},t) - V_\text{static}(\mathbf{y},t)$? I'm happy to make idealised approximations such as negligible self-induction, if that helps.

(I once saw a YouTube comment that seemed to indicate that a particular person had devised an experiment for measuring this - but although the comment gave the name of the person [which I don't remember], it did not provide any reference.)

Answer 1

The quantity that you call $V_\text{static}$ is more conventionally referred to as "the electric potential in the Coulomb gauge". It's impossible to measure exactly, even in principle (and so is the difference $V_\text{static}({\bf x}) - V_\text{static}({\bf y})$ between two different points). That's because the value of $V_\text{static}$ at a given point depends on the instantaneous charge configuration even far outside of the light cone of that point - but light doesn't have time to travel from that faraway charge configuration to the point in question, so there's no way to know what that charge configuration is without taking a measurement across all of space. The electric potential in Coulomb gauge is a useful theoretical construct, but it is physically unmeasurable.

See this answer for details on what a conventional voltmeter actually measures. The short version is that it directly measures the current traveling across a very short wire at a given point in space, and it indirectly measures the electric field at (approximately) that point.

You can measure $V_\text{static}$ in the electrostatic limit where the charge configuration (along with any currents) are assumed to be essentially stationary in time. In this case, a voltmeter does measure this quantity to a good approximation. But if there are high-frequency EM fields passing through (or being generated by) your system, then it won't. I don't think there's any feasible way to even approximate that quantity in the case where there are significant high-frequency EM waves.