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What's wrong with the following diagram?

enter image description here

Image source: Page 183, NCERT Physics Textbook for Class XII Part I

The reason stated in my textbook is as follows:

Magnetic field lines between two pole pieces cannot be precisely straight at the ends. Some fringing of lines is inevitable. Otherwise, Ampere’s law is violated. This is also true for electric field lines.

I don't understand how is Ampere's law violated when fringe fields are absent.
Can anyone please explain how Ampere's law is violated?

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Consider the two paths $ABCDA$ and $EFGHE$.

enter image description here

Path $AB$ contributes a positive value to the $\vec B\cdot d\vec l$ integral but the other parts of the loop contribute nothing, so overall there is a finite value for the $\vec B\cdot d\vec l$ integral but no enclosed current which violates Ampere's law.

Again path $EF$ contributes a positive value to the $\vec B\cdot d\vec l$ integral but now all the other parts of the loop contribute a negative value to the $\vec B\cdot d\vec l$ integral with the result that the total integral is zero as is the enclosed current.

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    $\begingroup$ Very interesting argument. ... just wonderin' if you'd have a textbook (or other) reference to this. $\endgroup$ – ZeroTheHero May 19 '18 at 17:01
  • $\begingroup$ @Zero This really is the qualitative argument in total (a quantitative treatment would be need to get the shape of the lines). I've seen it in several text books, but I don't have mine in front of me today. $\endgroup$ – dmckee --- ex-moderator kitten May 19 '18 at 17:16
  • $\begingroup$ @dmckee Think about me please. I realize the quantitative aspect would be overly complicated but I've never seen this argument before so I wouldn't mind something beyond PSE I could reference later. $\endgroup$ – ZeroTheHero May 19 '18 at 17:52
  • $\begingroup$ @ZeroTheHero I have not seen a quantitative treatment. Perhaps it would be easier to do the computation for the fringe electric field and hence the line integral for a parallel plate arrangement? $\endgroup$ – Farcher May 19 '18 at 18:18
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    $\begingroup$ I think a quantitative approach would completely miss the physics at play here. The whole point is that Ampère's Law requires that the line integral around any loop that passes through this field must be zero due to the lack of current anywhere. The example above shows that, without fringing, any loop that passes outside of the poles would have a net value if there are no field lines there, hence the need for fringing. $\endgroup$ – Dancrumb May 19 '18 at 21:29

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