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When there is an induced emf, Kirchhoff's Loop Rule no longer is true, because electric fields are nonconservative when there is an induced current, as stated by Faraday's Law:

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However, I have seen explanations that incorporate inductors and induced emfs into circuit analysis by treating them like batteries. For example, for the following circuit, if V is the voltage of the battery, Vinduced is the induced emf from the inductor, R is the resistance of the resistor, and I is the current, then V - Vinduced = IR:

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To me, this seems to be treating the inductor like a battery with voltage Vinduced. I see why this is justified; the only difference between the electric field created by a battery and by an inductor is that the inductor's field is nonconservative, while the battery's field is conservative due to the electric field inside the battery. However, are there any cases where an inductor acts differently than a battery with the same voltage, at least for circuit analysis purposes?

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    $\begingroup$ "Are there any cases where an inductor acts differently than a battery with the same voltage, at least for circuit analysis purposes?" Most cases because it's an inductor not a battery. With no forward EMF, if you treated an inductor as a source it would be a current source, not a voltage source. And this would only be valid for an instant in time. Then of course, there are all the times where it's not a source. $\endgroup$
    – DKNguyen
    Commented Jul 19, 2022 at 13:31
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    $\begingroup$ Inductors have transient effects, batteries don't. $\endgroup$
    – Jon Custer
    Commented Jul 19, 2022 at 13:31

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It seems I was mistaken in my original answer.

You can treat (emphasis on the word treat) an inductor as a battery if you took into consideration the emf in the voltage formula and apply Kirchoff's law. But in technical details, since the electromagnetic field is not conservative here, you can't in general apply Kirchoff's law.

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    $\begingroup$ An external magnetic field affects a circuit without inductors, too. If you use KVL, you are implicitly promising to account for all induction. $\endgroup$
    – John Doty
    Commented Jul 19, 2022 at 13:40
  • $\begingroup$ If you added the voltages across the entire circuit and a changing external magnetic field is present, the voltage doesn't add up to 0. In the case of self induction, the formula of the voltage of the inductor accounts for the self induced magnetic field. $\endgroup$
    – Habouz
    Commented Jul 19, 2022 at 13:50
  • $\begingroup$ If you include the EMF due to the external magnetic field, the voltage sums to zero around a loop. $\endgroup$
    – John Doty
    Commented Jul 19, 2022 at 14:29
  • $\begingroup$ Yes I agree on this. $\endgroup$
    – Habouz
    Commented Jul 19, 2022 at 14:31
  • $\begingroup$ Huh? Of course you can apply KVL. All you have to do is account for induction (both self and external). $\endgroup$
    – John Doty
    Commented Jul 19, 2022 at 14:48

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