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Suppose we have a closed loop in a changing magnetic field. By Faraday's law this would induce an emf in the loop. However by the Kirchhoff's law the total emf around a closed loop is zero. It seems like we've reached a contradiction. What is going on here?

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Kirchoff's Law is an approximation of Faraday's Law in the specific situation whereby the change in magnetic flux is negligible.

Kirchoff's Law means that voltages are like heights in gravitational potential; at every point in space, there is only one voltage, just like how at every point on the surface of the Earth, there is only one height of the land's surface.

Faraday's Law implies that Kirchoff's Law is wrong, and that there is a new contribution to voltages, confusing that it may be, that you can measure. Of course, for convenience, we will continue to define the scalar electric potential as if this new contribution does not exist, i.e. still has the nice property of having just one value anywhere in space, but there is a new contribution coming from the time derivative of the magnetic vector potential.

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  • $\begingroup$ It's too strong to say "Kirchoff's Law is wrong" it just requires a slight modification, including an induced EMF term in the KVL expression (e.g. add an equivalent voltage source in the network). Circuit theory is supposed to be practical, just make the adjustment in this special case. IMO instead of "implies Kirchoff's Law is wrong" we should say "Kirchoff's Law is incomplete unless you include induced EMF" $\endgroup$ Dec 11, 2023 at 14:30
  • $\begingroup$ @SamGallagher After two earlier statements giving the same idea that you are suggesting, I do not see the point of holding back at that point. It is quite important to hammer in the idea that, contrary to expectations, there is no uniquely positionally defined voltage function whenever magnetic fields are changing. $\endgroup$ Dec 12, 2023 at 1:08

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