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When a magnetic field is time-varying, then Faraday's Law can be used to define a conservative field with which a scalar potential can be associated. 1

When a circuit loop deforms, the flux through it changes, and an electric field is induced. Much like in the case of a changing B field, this field is nonconservative. Can Faraday's Law be applied here to define a scalar potential as well? If not, is there any way to define an electric scalar potential in this case?

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I don't think a scalar potential can be defined when the magnetic field is time varying. When the magnetic field is varying, electric field and magnetic field couple through Faraday's and Ampere-Maxwell's law. In these cases, the scalar potential isn't enough to describe the electric field. You need to include the time derivative of the Magnetic vector potential. So the electric field will be the minus gradient of the scalar field minus the time derivative of magnetic vector potential.

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  • $\begingroup$ Yes, I should have been more clear on that; it's mentioned in the source I linked, but that inclusion of the time derivative of the magnetic vector potential allows you to define a conservative field with a scalar potential associated with it; my question was, does this still apply when the magnetic field is not changing in time, but the boundary surface is, since in that case, there still is a changing flux inducing a field, see e.g. the second half of the proof here: en.wikipedia.org/wiki/Faraday%27s_law_of_induction#Proof $\endgroup$ Jul 16, 2021 at 13:30
  • $\begingroup$ Hmm I see. I think you can replace the problem with an analogous one where the loop is stationary and the magnetic field is time and position dependent. This way you can find the potentials as well. Also, in this case, the loop integral of electric field will be equal to magnetic field dotted with the time derivative of the surface area. Using this, you can redefine the potentials. I hope this helps ! $\endgroup$
    – emir sezik
    Jul 16, 2021 at 13:58

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