Dear Christian,

   Thank you very much.

   1. div B=o Maxwell equation tell us that the div of the magnetic induction field B is zero. This in principle doesn't tell us how the magnetic poles are created. For instance this equation doesn't tell us how the poles are only created in ferromagnetic materials below to a critical temperature.

   2. Magnets needs to have spinors and also the magnetic moment that they have. The question is if you can see how the spinors (or they magnetic moments) form the poles. For instance, how many spinors are necessary for having a magnetic pole? Can they be free?

   When you write the eigenvalues of the spinors are you telling that they are the eigenvalues of a Dirac equation? Do you need relativistic spinors?

   3. Sorry, your answer was wrong and I think that you don't understand it, if I understood you in the other thread, because you said that the force interaction is due to the gradient of the magnetostatic energy and the magnetostatic energy doesn't distinguish the kind of pole. E= 1/2 HB.

   4. Yes I know what is the Dirac monopole, Schwinger monopole, t'Hooft monopole...etc. My doctoral thesis was in the magnetic monopoles and its topological configurations. But how the topology is needed to be introduced and the quantum mechanics in a problem so classical in principle?

    Dear Christian,

    I cannot understand how we cannot put agree with so basic things.

    1. The definition of the vector potential is not related with the exitance of a pair of poles. That is only a vectorial condition that mathematically ir related with topology, B is a closed form that for simply connected manifolds is also exact. Thus it is possible to define globally A. This is far of the question and it might be related with the magnetic monopole that you asked me to look for it.

   2. Spinors are not related with the magnetic poles. Spinors are eigenfunctions of Dirac equation why magnetic poles are just magnetic sources created in macroscopic magnetism.

  3. I think that your answer was always wrong and I don't understand how is possible to discuss about this point. The magnetostatic density of energy is 1/2 BH, Isn't? Thus if you calculate its gradient you can have a force, that is true, but you don't distinguish between the interaction of NN, SS or NS, SN which was the central point of the question. This is the key which suggested me to put again the same question but a little bit more explicit.

  4. The magnets were calculated and quite well understood in XIX and even later by Gilbert and so on. Curiously it is possible to have a classical electromagnetic view of them without needing to describe a ferromagnetic phase. Obviously this has many non realistic outcomes and one of them is the magnetic poles as such.

    Dear Christian,

    No, again, my answer was correct. We speak about the (integral energy of the) superposition of two magnetic dipole fields. Of course the total field energy depends on their respective orientation. (I assume hard magnets where we can neglect any change of their magnetization). Don't know how to make this clearer.

Why does appear always two magnetic poles in a magnet and.... Available from: https://www.researchgate.net

    Thus you don't distinguish what the attraction or the repulsion force.

    Let me try to explain what was the reason to put this question separated of the other thread, which was full of other questions without answering the question.

    1. The question is simple but it has no answer, no body knows it in all that I know. In the same form that no body knows why the electron and the proton attracts, while two electrons or two protons repel. This is a fundamental issue that is not found calculating with energy or with other thing.

    2. I tought that this was very well known and I was very surprise to see that you said that you were doing the same as Ricardo, when I thought him that he was wasting his time is he tried to arrange the electronic spins. This is wrong!

   3. I have a good concept of you and it surprised me quite a lot. I hope that you can understand then my procedure. In any case, I hope that not so disorganized because Ricardo didn't say anything and this is not going my case, but this question I hope that can help to people who could be interested in going deeper on this issue.

    Dear Christian,

    Chikazumi, ref. that I have given you, starts in its first chapter:

The most direct manifestation of the magnetism is the force of attraction or repulsion between two magnets. The phenomenon can be described by assuming that there are "free" magnetic poles on the ends of each magnet which exert forces to one another. These are called "Coulombic forces" by analogy with Coulomb forces between electrostatically charged bodies. Consider two magnetic poles with strenghts of m1 ((Wb)Weber) and m2 Wb respectively separated by a distance r (m). The force F exerced (N (Newton) on one pole by other is given by

        F= m1 m2/ 4ᴫμ0 r2 

     This was the concept of magnetic pole in analogy with electric charge. Thus you can understand what where my analogies and the magnetic dipole is only one special case. Notice that if the magnet or a solonoid are very long you have not a magnetic dipole and you can consider the pole as if it were free less in a straight line. But no dipoles! Dipoles are only when the magnets are short.

     Dear Christian,

     I'm afraid that you were thinking in a very different concept of magnetic pole. This is the standard concept and I have assumed that Ricardo was asking on this picture. Thus you can see that the SS,NN or SN interactions are easily calculated with this analogy because it is a kind of Coulomb force. What you cannot say is why the sign of S and N are not the same and they attract or SS repel. In the same form that the electric charge do.

     Dear Christian,

     This is a very old assumption which helps to make many magnetic calculations without entering in how the magnetism is produced. In spite that we nowadays know how the ferromagnetic phase is produced and how the magnets work on the canges of temperature or the chemical composition, the engeeners and many applied physicists still use these old concepts.

   Dear Chistian,

   The full question was:

   Why do magnets attract or repel?

   I am trying to understand with a deeper foundation the attraction (N-S) or repel (N-N, SS) between two magnets. 

   I have tried to reason with Lorentz forces pulling or pushing the electrons spins, but the resulting forces does not coincide with the direction of attraction-repel.

   Dear Christian,

   I'm not sure to understand what are you trying to say, but it is true that in materials you have many different possibilities. Surface insulator (bands, Anderson,..) with high or small gap (direct or indirect), semiconductor, semimetal, metal, ferromagnetic, antiferromagnetic, RKKY coupling with. itinerant magnetism .....etc... There are many subtleties if one go to face the reallity of a given material and it is crazzy even to think in only method for solving everything.

   But the question was only on the magnetic poles which have a very clear historical meaning which hasn't changed too much. In practice the main interest if for calculating the demagnetizing field of one sample where the magnetostatic interaction plays the fundamental physical role. The Chikazumi is a fantastic book for explaining this topic and develop it in different situations.

    Dear Christian.

   The text, including the formula of "Coulomb form", is just a copy of how the Chikazumi starts. I did it for showing that this is not my particular form of understanding the magnetic poles.

   This is a concept that belongs to pure classical electrodynamics as it stands and which copy the idea of electric charge interaction. The first question is if that formula can be assumed to be true given that nowdays we know that magnetism in matter has a very different source than the electricity. For instance, the electric charge always conserve while magnetization doesn't do it.

   Curiously there is classical symmetry of Maxwell equations and no of the electromagnetic action which allows to affirm it. I have devetod a chapter of my book to study this symmetry (duallity rotations) in my book. One of the things necessary for such symmetry is that the Hamiltonian is kept invariant and therefore their eigenvalues.

   This is a much deeper issue than to calculate the magnetic configuration between the interaction of several magnets, which in most of the cases can be reduced to the calculation of dipolar interactions (forgetting important details as form or strength of the poles).

  Dear Christian,

  Obviously a Coulomb law for the magnetic poles, as I copied from the Chikazumi, cannot be a realistic true law. The magnetic poles cannot be free and less cannot have the spheric symmetry that the electric charges have. For instance, notice the difficulties involved to try to apply a Gauss theorem on these sources.

  I don't know what charge and current sheet numerical method are you actually thinking?

   Dear Christian,

   I never used such a methods, but as you said there are many different kind of " boundary" conditions. Many years ago I was employing the slices and tubes for selecting electric for the magnetic conditions, but these in fact are not necessary if we think in the finite elements or the Garlekin method which is fully developed.

   The problem was not this kind of calculations but the concept of magnetic pole that , even now after so long discussion, I'm not sure that it was clear between us. But it never minds, because in fact it is not so important or fundamental. I only was very surprised with your answer in the other thread, answering a quetion which is fully wrong as I had said to Ricardo.

    Have a good day Christian!

   Dear Christian,

   Let us have electron (e) and proton (p) interactions. Does depend the interactions ee, ep, pp of their energies? No they just depend of a Coulomb force which is function of the chage and the distance between them. The attraction or repulsion only depends of the sign too.

   This is what I have told you for the magnetic poles, but perhaps you have some other ideas that I cannot understand. Sorry if this has extended the discussion without any need.

    Dear Christian,

    I'm tired too and it doesn't matter. Have a good day!