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In most practical cases I have seen, voltage was regarded as a difference in electric potential, save the odd change of magnetic flux over time which, as a form of change, also conveys the notion of a difference.

However, from what I gather, we can ask ourselves what a sensible basal level for an electric potential might be (for example, the potential an infinite distance away from the electric charge producing it) and treat any voltage expressed in relation to it as an absolute voltage. But then we need to agree that the potential at infinity is zero. I do not know whether there are other sensible choices, but I am willing to accept that there might be.

My question is whether there are any other senses, not involving a prior agreement on a reference value, in which we could speak of an absolute voltage and, if there are, what senses would those be?

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Yes: if one is attempting to do electromagnetism in 1D then the force becomes constant with distance and it makes sense that $U = k~x$ for some $k.$ This may seem artificial because we are located in 3D space, but it corresponds to an infinite wall of uniform charge, which can be used to describe what things would look like if one were shrunk down and found one's self inside a parallel-plate capacitor. The natural zero would be at the plane of the uniform charge.

Similarly if one is thinking about a point charge next to an infinite conductor, the fact that an infinite conductor must be at constant voltage leads to a natural desire to set this voltage equal to zero. One can then fully enforce this using the method of image charges.

In 2D rather than 3D, one finds a $1/r$ force law which must admit a potential $U_0~\ln(r/r_0)$ and this cannot be normalized to be zero at infinity or at the place where the charge is located. This corresponds to a geometry of a charged infinite wire, and one just has to arbitrarily choose a distance $r_0$ away from the infinite wire where the charge will be zero.

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