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  • $\begingroup$ I'm sorry but I think you missed the entire reason why I asked this question, this explanation or ones similar to it I did find and are the reason as to why my confusion rose in the first place, when you evaluated the line integral you essentially wrote the work done per unit charge by the electric force, but you missed the induced non conservative electric field, I get the fact that potential is defined for conservative fields only but work can be defined for any force and consequently the line integral for the electric field. $\endgroup$
    – bm27
    Commented Jun 29 at 18:14
  • $\begingroup$ Also to your reply to my first doubt, it is fine that I imposed Faraday's law but don't I by invoking Faraday's prove the claims stated? Because the terms can be rearranged to get the form of KVL. $\endgroup$
    – bm27
    Commented Jun 29 at 18:18
  • $\begingroup$ Under strict KVL you should never have $L\frac{\mathrm dI}{\mathrm dt}$ term. KVL requires that voltages be the initial potential function definition, whereas time-varying magnetic fields require that such voltage "potentials" be multi-valued. $\endgroup$
    – naturallyInconsistent
    Commented Jun 29 at 18:25
  • $\begingroup$ As for the "missed the induced non-conservative electric field", you are sorely mistaken. The very fact that a loop integral is not identically zero is the non-conservation already right there for you to see. $\endgroup$
    – naturallyInconsistent
    Commented Jun 29 at 18:26
  • $\begingroup$ Ohk so, essentially the line integral doesn't contain the non conservative component which is accompanied with time varying magnetic fields. So if we were to write $ \frac{d\phi}{dt}$ as the line integral of non conservative component and shift it to the other side then it implies KVL still holds? (As it should because it essentially is conservation of energy) $\endgroup$
    – bm27
    Commented Jun 29 at 18:40