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I dont understand why the magnetic field does not curl outside the cable, according to this example, since I would expect the magnetic field lines to go back on themselves around the cable like shown in the picture and the curl would then be non-zero, right? Can someone please clarify why the curl is $0$ outside and $\neq 0$ inside?

enter image description here Source of the example.

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  • $\begingroup$ The diagram shows the equation you need. Do you need understand what $\mathbf J$ is here? $\endgroup$
    – BioPhysicist
    Commented Jan 13, 2021 at 14:17
  • $\begingroup$ @BioPhysicist, I get that equation but unfortunately, it does not make it intuitive for me. An explanation that there is no current and hence no curl outside the wire does not do it for me unfortunately. $\endgroup$
    – Clone
    Commented Jan 13, 2021 at 14:21
  • $\begingroup$ Also covered e.g. in my answer to I don't understand Ampere's circuital law. $\endgroup$
    – CR Drost
    Commented Jan 13, 2021 at 15:54

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Mathematically you can surely see that $\vec{J}=0$ outside the wire and therefore the differential form of Ampere's law demands that $\nabla \times \vec{B} = 0$.

I guess what is confusing is that curl is often described in terms of "curling field lines", but that isn't entirely accurate - straight field lines can have a curl and curved field lines can have zero curl (as in this case).

A better analogy is to consider the insertion of a small paddle wheel into the problem and imagine the field lines represent a fluid flow. The question you have to ask is whether the paddle wheel will rotate.

So in this case, what would happen is the paddle wheel would not rotate because the field is curl-free. See https://physics.stackexchange.com/a/302883/43351

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  • $\begingroup$ Thank you for the reply! That is exactly my problem, I dont understand why the paddle wheel does not rotate if I put it outside of the conductor. Thank you for the link to the other answer, what I didnt understand from that was why the paddle does not rotate even though the arrow on the bottom is the "biggest". The field is stronger closer to the wire so I would expect that the paddle would rotate clockwise. $\endgroup$
    – Clone
    Commented Jan 13, 2021 at 14:39
  • $\begingroup$ Okay, I figured it out. The net force is zero on the paddle, I suppose. Thanks for the answer. $\endgroup$
    – Clone
    Commented Jan 13, 2021 at 14:45
  • $\begingroup$ @Clone, probably because it isn't drawn accurately enough. In the end it is just a conceptual device, the reason it doesn't turn is that both $\nabla \times \vec{B} = 0$, which by Stokes theorem means that $\oint \vec{B}\cdot d\vec{l}=0$. $\endgroup$
    – ProfRob
    Commented Jan 13, 2021 at 15:27
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Field lines forming closed loops does not mean the curl is non-zero at all points in space. Curl is a local property defined through derivatives, so the curl at a point just depends on the field around that point. The curl operation doesn't depend on what the field is doing elsewhere.

In this example the current density $\mathbf J$ is $0$ outside of the wire, so by $\nabla\times\mathbf B=\mathbf J$ it must be that the curl of $\mathbf B$ is $0$ outside of the wire.

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  • $\begingroup$ Thank you for your reply! I was thought that curl is non-zero if you place a water paddle at a point (using fluid flow analogy) and if that paddle would rotate. So if I place a water paddle at the point where the wire would be it should rotate in the same direction as the B field in the picture, right? But that paddle would also rotate if I would put it outside the wire, wouldn't it? $\endgroup$
    – Clone
    Commented Jan 13, 2021 at 14:28
  • $\begingroup$ It will rotate with the general circular flow but this is not what the curl describes. You should fix the center of the paddle in one point and see if it spins around that point. The curl is a local quantity. Your intuition is more related to the line integral around a macroscopic loop rather than the curl. $\endgroup$
    – nasu
    Commented Jan 13, 2021 at 17:58
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Here is a diagram of a vector field like the one you're talking about:

enter image description here

If you insert a paddle wheel ("curl meter") in this field, then the stronger field on the left compared to the right would tend to produce a clockwise torque, but the angles of the top and bottom would tend to produce a counterclockwise torque. In the actual field, the radial fall-off of the field is such that these two effects cancel, and the curl is zero.

I have a longer description of this as an example in section 11.2.1 of my book Fields and Circuits: http://lightandmatter.com/fac/

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