What's wrong in this diagram?

diagram of magnet

The reason is this: magnetic field lines between the two poles of magnet are perfectly straight in the given diagram, which is not possible.

Fringing effect of the lines is inevitable

I was investigating why there is fringing effect, and then I came across a line in NCERT physics text book that if there was no fringing effect it would violate Ampere's law. Can anyone please explain how Ampere's law is violated?


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.

  • 1
    $\begingroup$ 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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