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I'm struggling to understand Gauss's law for magnetism, which states that the net magnetic flux through any closed surface is always zero. I understand why it holds true if you have a single magnet creating a magnetic field, because the field lines form a closed loop and therefore must both enter and exit the Gaussian surface.

But, I don't understand why it holds true for the magnetic field created by a pair of magnets. For example, in the image below there are field lines in the space between the north pole of the left magnet and the south pole of the right magnet... but those don't look like closed loops to me. Couldn't you construct a Gaussian surface that intersects those lines thus creating net flux? What am I missing here?

A magnetic field diagram for two magnets attracted to one another

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    $\begingroup$ In between the 2 magnets, the flux goes from the left magnet to the right magnet. If you take any volume in that region, it is easy to guess that whatever flux enters from the left of that volume escapes to the right of that volume, making the total flux in the volume element picked null. $\endgroup$
    – untreated_paramediensis_karnik
    Commented Apr 3, 2022 at 8:12
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    $\begingroup$ I guess there are also magnetic lines inside the material. Have you considered them? $\endgroup$
    – StefanH
    Commented Oct 13, 2022 at 12:32

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One answer might be: If you believe that it works for one magnet, then you have to believe it works for two magnets because the Maxwell equations are linear and the field of the two magnets is just a superposition.

If in the image above we take a closed surface that encloses one of the magnets, for example, the flux through it is the sum over two contributions: One created by the field of the enclosed magnet - which you believe to be zero (the flux, of course) - and one created from the field of the magnet outside the closed surface - which you believe to be zero. ;)

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  • $\begingroup$ Haha, so basically 0 + 0, don't think too hard about it? But still, where do those flux lines in the middle go? $\endgroup$
    – the_sky_is_pink
    Commented Apr 3, 2022 at 17:35
  • $\begingroup$ From one magnet to the other...? ^^ They are part of lines that go from one magnet to the next in the middle, through the magnet, round the outside, and back through the first magnet to form a closed loop. $\endgroup$
    – kricheli
    Commented Apr 3, 2022 at 18:45
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There are of course still closed loops of magnetic field. The magnetic (B) field lines run through the magnets from S to N but have not been drawn. This may be what you're missing?

I can't see any volume you could draw which does not have just as many field lines entering the volume as exiting the volume.

Note that the poles of a magnet are not sources and sinks of B-field. They are sources and sinks of H-field for which the flux out of a closed surface can be non-zero.

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