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In his Introduction to Electrodynamics, D.J. Griffiths is warning against the idea that magnetic fields $\vec B$ and $\vec H$ are similar. Here is a passage:

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The equation 6.20 he is talking about is $\oint \vec{H}\cdot d\vec{r}=I_f$, where $I_f$ is the free current.

Fine. Then he presents a problem. Image below.

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When I look at the solution, what do I find? He concludes that $\vec H=0$ because there is no free current. Image below

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Well, now, isn't this precisely what he just warned the reader against doing??

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    $\begingroup$ this being a homework problem I can only give you a hint: a "short cylinder" and an "infinitely long cylinder" are very different things. $\endgroup$
    – hyportnex
    Commented Sep 19, 2021 at 18:56
  • $\begingroup$ @hyportnex it is not homework, I am just studying this book and had this question. How exactly does the length of the cylinder make a difference? Is it that $\vec{\nabla} \cdot \vec{M}\neq 0$ at the "top" and "bottom" of the cylinder? $\endgroup$
    – thedude
    Commented Sep 19, 2021 at 19:04
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    $\begingroup$ @hyportnex If a question like this is considered as homework or homework-like this site is going to die. Question and answers are about conceptual understanding of magnetism in materials. That is perfectly in agreement with this site policy. $\endgroup$
    – GiorgioP-DoomsdayClockIsAt-90
    Commented Sep 19, 2021 at 19:15
  • $\begingroup$ Of course, my comment applies also to the tag editing by @Frobenius. $\endgroup$
    – GiorgioP-DoomsdayClockIsAt-90
    Commented Sep 19, 2021 at 19:15
  • $\begingroup$ @GiorgioP I agree with your comment wholeheartedly, this is a fair question and in fact I upvoted the question itself; but unless I was hallucinating, as I recall now, there was a "homework" tag added to the question that since has disappeared, hence my "hint" without a detailed answer... $\endgroup$
    – hyportnex
    Commented Sep 19, 2021 at 19:56

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As suggested by @hyportnex, and correctly understood by the OP, there is an essential difference between Griffith's example in the warning and the Problem 6.12. In the former case, the cylindric magnet is finite, while in the latter is infinite. The two cases differ for a region of non-zero divergence of magnetization, precluding the possibility of getting the value of the field $\bf H$ by using only the information about zero circulation in the absence of free currents.

A picture, taken from Wikipedia enter image description here

should make evident that the vector field $\bf H$ lines do start and end at the left and right surfaces, making non-zero the local divergence.

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