I'm having a fundamental problem with Gauss's law for magnetic fields. I understand that what goes in must come out. What I'm having difficulty with is that we are dealing with vector fields measured with respect to the normal of the surface. Since the fields entry and exit have different normals(So the dot product will be different) the flux will be different.
I'm obviously not looking at this correctly. Can anyone point out where my reasoning has gone haywire!!
I do understand these surfaces in the diagram are supposed to be closed 3d surfaces.
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$\begingroup$ try this: play.google.com/store/apps/… $\endgroup$– Samuel OwinoCommented Oct 15, 2019 at 23:11
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What enters must exit. The normals can be different, the intensity of the fields can be different, the surface areas can be different, but at the end of the story all these different values combine exactly to give a zero net magnetic flux.
OK, I am not good at drawing so I'll describe a situation.
Imagine a place where the magnetic field is vertical. Say, just at the magnetic pole. Suppose it varies slowly, so it can be assumed constant on the size of a little cabin I build. The cabin has a square, horizontal floor. So the flux than enters is the product of the field intensity by the area of the floor. The walls are vertical, the field is parallel to them, no flux at all. Now look at the roof. It is not horizontal, but slanted at some angle $\alpha$ from the horizontal. So its area is larger than that of the floor, by a factor $1/cos \alpha$ . You can see that !
But what is the flux out of the roof ? The product of the field intensity (same as at the floor) times the area (larger by $1/cos \alpha$ ) times the cosine of the angle between the field and the normal. But this angle is just $\alpha$, so this exactly cancels the effect of the larger area.
And this will always work, even in the field is not constant everywhere It is guaranteed by the zero flux rule !
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$\begingroup$ Yes that does help thank you. My biggest problem is that this applies regardless of the complexity of the surface. If for instance a "shaft" of magnetic field enters a surface, that is normal to the field, of area dA and exits via a complex surface that is not normal to the field then since the exit area will be some multiple of dA that compensates for the loss due to the exit surface not being normal. How would you prove that? If it was just stuff in=stuff out then it would be very intuitive, but the addition of vectors complicates it in my small mind!!! $\endgroup$– BiskwitCommented Oct 16, 2019 at 0:17
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$\begingroup$ As I said, in all cases the loss due to the exit surface not being normal will indeed be compensated by a larger area. And if the field varies, that will also be compensated. This must be the case, because the magnetic flux has zero divergence and the net flux through any closed surface is zero. I don't see how to be clearer. $\endgroup$– AlfredCommented Oct 16, 2019 at 0:28