The Gauss's law for magnetism states that the magnetic flux through a closed surface is always zero. So how can we determine a change in magnetic flux? While calculating the induced EMF by Faraday's law, we equate the induced EMF to a change in magnetic flux through the closed loop( in which we want to find the EMF). So how are the above two valid at the same time? Or is there something that i don't know about these laws?
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$\begingroup$ Note that a closed loop is not the same as a closed surface. $\endgroup$– Clara Díaz SanchezCommented Mar 23, 2020 at 9:15
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3 Answers
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The Gauss's Law applies to a CLOSED Surface. What does it means? Imagine a sphere. The sphere has a inside and a outside. If something is in the inside, it must cross the surface in order to get outside. It means that all closed surfaces divides the space in two regions and you need to cross the surface to go from one region to the other.
What Gauss's Law says is that the magnetic flux though a closed surface is always 0. If you have a magnet inside a sphere, the total flux in the surface will be 0. In analogy to the Gauss's Law for electric fields, you can interpret it as "there is no magnetic charge, or monopole". Poles always appear in pairs, and the net "magnetic charge" will always be zero.
In the case of the Faraday's Law of Induction the flux is not through a closed surface. It is through a OPEN surface. Imagine a sheet of paper. It does not divide the space in two regions. You can simply go around the paper and get to the other side. No need to cross the surface.
A property of this kind of surface is that they have a border. In the case of a conductor wire you can imagine a ring. This ring has a surface inside it. It is a circular surface and its border is the ring. The surface does not need to be physical, just imagine it exists.
When the surface is open, the magnetic flux is not always 0! We can simply find a place in the field where there is more lines going in one direction than in the other. And a open surface placed there will not has a 0 magnetic flux.
When the magnetic field is varying over time, the flux trough a open surface will vary too. If this surface is the one limited by the conductor ring, it will causes current to circulate around the ring.
Notice that the flux can be different if we change the location and size of the surface in the field.
In the case of a bar magnet, we can choose a tiny region where the flux is always positive. But if this region is "a slice of a sphere" as you mentioned in a comment, in this situation the flux will be 0 because the flux of lines going in one direction through the slice is the same as the flux going through the opposite direction.
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$\begingroup$ I need to make a correction to my answer. The flux in the "slice of sphere" IS NOT always 0. Otherwise it would contradict what I said before it. In a case of symmetry it could happen to be 0. But not always. This is what I forgot to say. This has to do about where you choose to put the surface. $\endgroup$ Commented Jun 28, 2020 at 5:59
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$\begingroup$ I got the point, magnetic flux for a closed loop depends on the actual magnetic field lines that pass through it, and flux could be zero or non zero depending upon the distribution of magnetic field lines. Thank You for the answer. $\endgroup$ Commented Jun 28, 2020 at 6:26
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If you look the direction magnetic field lines of a bar magnet , magnetic field lines with a direction of entering the magnet cancel magnetic field lines with a direction of leaving the magnet over a closed surface.
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An example of a closed surface is that of a surface of a sphere. The net magnetic flux through that surface would be zero. This is the consequence of Gauss's law. However a closed loop is something like a wire bent on itself. The surface enclosed by the loop is not closed so there can be a net magnetic flux through it.
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$\begingroup$ So, in Gauss's law , a closed surface means a 3 D surface, while when we apply Faraday's law, we assume a 2 D surface? $\endgroup$ Commented Mar 23, 2020 at 11:41
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$\begingroup$ Also, if i take a bar magnet and place it inside a sphere, the net magnetic flux will be zero. But if i cut a ring out of the sphere, will the flux be non-zero through the ring? $\endgroup$ Commented Mar 23, 2020 at 11:43