Can we use KVL in a circuit having non-conservative field. I mean if its true then it denies the Maxwell equations which says that closed loop integral of E.dl is not zero in non-conservative fields.


Yes, we can use KVL in circuits with non conservative E fields without violating Maxwell’s equations. To see how this is possible, it is important to understand the basis of KVL. KVL is part of circuit theory, and circuit theory rests on three assumptions:

1) The circuit is small relative to the speed of light and the time scales of interest

2) Inside any circuit element the net charge is negligible

3) Outside any circuit element the magnetic flux is negligible

In particular the last assumption limits what types of non conservative E fields are compatible with KVL. Specifically, the assumption about magnetic flux requires that any nonconservative E field must be contained entirely within a circuit element. Then from the perspective of the circuit as a whole the EMF is simply treated as any other voltage across the terminals of a circuit element.

For a detailed justification of why this works see section 11.3 here: http://web.mit.edu/6.013_book/www/book.html But the basic idea is that as long as the fields go to zero outside of some region (the boundary of the circuit element) then regardless of the fact that the fields inside are non conservative, the energy crossing the boundary is equal to the current times the voltage and so it can be treated as a standard circuit element.

EDIT: I note that there is some comment about Dr Lewin’s famous anti-KVL lecture. The circuit he draws deliberately violates the third assumption, and so it is unsurprising that KVL fails since circuit theory is not justified in that case. KVL works any time the assumptions of circuit theory are satisfied

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  • $\begingroup$ Are you telling about the assumptions of the Lumped Circuit abstraction of the MIT 6.002 course? $\endgroup$ – John Cena Nov 25 '18 at 20:28
  • $\begingroup$ My textbook was Nilsson and Reidel “Electric Circuits” where they cover the assumptions in Chapter 1, but I am working from memory so I didn’t cite them directly. I assume that the MIT course covers the same assumptions, perhaps with different wording. $\endgroup$ – Dale Nov 25 '18 at 20:39
  • $\begingroup$ Are you an author? Really great to talk to you. $\endgroup$ – John Cena Nov 25 '18 at 21:10
  • $\begingroup$ Oops, sorry to give a wrong impression. By “my textbook” I meant “the textbook that purchased many years ago in college and now sits on my office bookshelf” not “the textbook that I wrote”. $\endgroup$ – Dale Nov 25 '18 at 21:44

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