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I am wondering if the fact that "the electric potential is continuous" should not be added to the set of axioms of physics.

I have never seen a problem where the potential is discontinuous, unless, possibly if the problem involved non physical assumptions (like infinite objects etc.). Are you aware of any real situation where the electric potential can be shown to be discontinous?

Qmechanic
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MikeTeX
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    Do you count point charges as "non-physical objects"? How about ideal dipoles? How about infinitesimally thin surface charges? One could argue that all three are equally reasonable; and if you grant me the last two I can give you a situation with a discontinuous potential (a thin sheet with a constant dipole moment per area.) – Michael Seifert Feb 21 '23 at 21:10
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    @MichaelSeifert In general, strictly speaking, nothing that is idealized is physical. – Steeven Feb 21 '23 at 21:16
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    Indeed, nothing that is idealised is physical, but then that means that uniform charge densities (and anything not made of point charges) is not physical either. But that is not a constructive way to work. Electromagnetism often uses various idealised sources (some homogeneous, some singular) for good reasons, and adding axioms into physics that forbid the use of any of those sources should only be done for even better reasons. – Emilio Pisanty Feb 21 '23 at 21:25
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    And speaking of which: the electric potential is discontinuous at surface dipole layers. – Emilio Pisanty Feb 21 '23 at 21:26
  • @Emilio Pisanty: I disagree with the fact that uniform charge densities are "as not physical" as point charges: the charge densities are defined by smoothing the microscopic charges by convolution with a Gaussian, leading to the macroscopic field which is smooth. Sure, the macroscopic field is a mathematical being, but it is real, just like the mean of your score in a competition is real. – MikeTeX Feb 22 '23 at 06:14
  • Possible duplicates: https://physics.stackexchange.com/q/131674/2451 , https://physics.stackexchange.com/q/212516/2451 , https://physics.stackexchange.com/q/500667/2451 and links therein. – Qmechanic Feb 22 '23 at 06:46

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It is discontinuous if there ever exists a sufficiently singular charge density, which is extremely common. For example, point-charges with charge density $\sim \delta^{(3)}(\vec x - \vec x_0)$.

You might argue that these singular charge-distributions are unphysical, and merely crude approximations to smooth underlying charge-distributions. The key is to realize that this doesn't really matter - Maxwell's equations work just fine with such singular charge distributions.

Arturo don Juan
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There is no "the set of axioms of physics". Assumptions in physics are often adopted or rejected flexibly, depending on their utility for the problem at hand.

An example of a discontinuous electric potential is electric potential due a point dipole, at the point where the point dipole is. Depending on the direction from which we approach the dipole, potential can remain always zero, or grow negative, or grow positive, and thus at the dipole, it has no universal limit, and thus potential is not continuous there.

But this "problem" is only at that single point, everywhere else it is continuous.

Point dipole is so useful a concept that the discontinuity did not prevent people to adopt it into EM theory.

Ján Lalinský
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Physics is based on phenomena, not axioms. Some mathematical models of physics have axioms, though. You are free to add your axiom to electromagnetic models since there is no experiment that can falsify it. I think most of us would rather not do so, as it would simply prohibit useful idealizations without adding any utility.

John Doty
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