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Consider the above question. I have been able to solve the question understanding area vector of A and B are opposite in direction.


However I have some conceptual doubts.

  1. In Faraday Law, when we say "area enclosed by a closed loop", does it coherently include all type of loops -- with twists and turns as given in the above question.

  2. Suppose I take only one loop say A and I try to apply Faraday in it. Can I do so simply while ignoring B?

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  1. Yes, the loops in question can be quite exotic. As long as the loop does not intersect itself, you can find an oriented surface which has that loop as its boundary$^\dagger$. The integral form of Faraday's law (which obtained from the differential form by application of Stokes' Theorem) can then be applied from there.
  2. Yes, you can get away with that here. Strictly speaking, we should only consider non-intersecting loops. However, if we allow a finite number of intersections and then decompose the result into a "sum" of non-intersecting loops, then we will get the same result when we add things back together. Of course, we'll have to pay attention to the orientations of each piece.

$^\dagger$The general procedure for constructing such a surface is called the Seifert algorithm.

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  • $\begingroup$ Didn’t Faraday’s law come many years before Stokes theorem? So it was discovered without assistance of Stokes. $\endgroup$ Commented Sep 15, 2020 at 4:03
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    $\begingroup$ @relayman357 Stokes' theorem was published in 1854. Faraday's observations were made in the 1830's, but were not cast into precise mathematical form until Maxwell in the 1860's. In any case, what I meant was that the integral form of Faraday's law, and all the specific rules governing fluxes and loops, is obtained by the application of the Stokes' theorem. I'll edit my phrasing. $\endgroup$
    – J. Murray
    Commented Sep 15, 2020 at 4:16
  • $\begingroup$ Good deal, i assumed that was your intent. Very nice answer! $\endgroup$ Commented Sep 15, 2020 at 12:39

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