Dear Ricardo,

    The reason is the same as the charges of the same sign repel while the ones of the opposite sign attracts. Nobody knows the deep reason for such behaviour, this is a phenomenological result and don't waste time trying to obtain it (less from the classical electrodynamics due that the poles are got by spins ). Even you can associate a north pole with a positive charge and a south pole with a negative one. In such a case you have (Gilbert model) for the interaction between magnets just dipole interactions if their length is not too great with respet to the distance which separates them. The interaction is therefore a non linear force depending strongly between the angle that they form, and if you leave them in free interaction they would try to form a quadripole in 2-D etc...

Dear Ricardo,

You will not get coherent results with the Lorentz forces because they relate to the relative densities of ambient magnetic and electric fields that are used to orient charged particles, and do not address the mutual interaction between charged electromagnetic particles.

For example, equal densities ambient electric field and magnetic fields will cause an electron, which is an electromagnetic particle, to move in straight line. Velocity will increase with fields intensity. If the ambient magnetic field is more intense than the ambient electric field, the trajectory of the electron will be curved accordingly.

The magnetic fields of magnets are very well understood and this knowledge is used by manufacturers to magnetize magnets in all sorts of ways.

The macroscopic magnetic field of permanent magnets is caused by forced parallel spin alignment of unpaired electrons in the material.

In electromagnets, the moving electrons in the wire are de facto spin aligned parallel simply because they move in the same direction, so this forces unpaired electrons in the material of the magnet to align in opposite spin parallel alignment.

Mutual electric interaction between charged particles is addressed by the Coulomb equation, and obeys the inverse square law of interaction with distance, while mutual magnetic interaction between the same particles obeys the inverse cube interaction law.

You will find quite a bit of information in the CRC Handbook of Chemistry and Physics, for example, in Section  Properties of Magnetic Materials, starting at page 12-117 in the 2003-2004 edition.

This inverse cube interaction was directly measured in 2014 between 2 electrons belonging to two different atoms that were maintained in parallel spin alignment. See first paper below.

This inverse cube interaction can also be measured between permanent magnets. See second paper below.

http://www.nature.com/articles/nature13403.epdf?referrer_access_token=yoC6RXrPyxwvQviChYrG0tRgN0jAjWel9jnR3ZoTv0PdPJ4geER1fKVR1YXH8GThqECstdb6e48mZm0qQo2OMX_XYURkzBSUZCrxM8VipvnG8FofxB39P4lc-1UIKEO1

http://www.ijerd.com/paper/vol7-issue5/H0705050066.pdf

What strikes me is that for this seemingly simple question, there are already four completely different answers. This already proves the statement of Ronald Starke that this question definitely "deserves attention".

Dear Giulio,

You write "This already proves the statement of Ronald Starke that this questions definitely "deserves attention"."

Total agreement, and urgently at that.

Well established knowledge is not taught anymore in many institutions.

The main trend in teaching has apparently drifted to only studying popular orthodox theories, disregarding confirmed experimental facts, while really studying physics means studying physical reality grounded on actual experiments.

The up coming generation is not being taught to correlate theories with experimentally confirmed facts.

I think the best answer is presented by Richard Feynman in the following link:

https://www.youtube.com/watch?v=MO0r930Sn_8

   The question is:

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.

   The reason of the attractive (N-S, S-N) or repulsive (N-N, S-S) force between poles can be translated in what is the reason to have attractive (+-, -+) or repulsive (++,--) force between the electric charges. The reason is that there is a symmetry of electrodynamics known as duality rotations which can exchange the properties of the magnetic field and sources with the electric ones. Energetically nothing change and the Hamiltonians are kept invariant under such transformations. Obviously this is an interesting question but not solved even in QED and if you do it perhaps you are closed to the Nobel prize because there is a connection with Dirac see and the antiparticles C invertion through the CPT relativistic discrete symmetries, for example with the chirality in the weak interaction or the Weinberg-Salam unification model for electro-weak interaction.

   What is the reason in electrodynamics of this behaviour of the sources? Nobody knows (nobody knows), for example in gravitation doesn't happens the same.

   But what is clear is that this cannot to be related with minimize the energy of a configuration, which only are many different possible dipolar interactions due to the nonlinearity between the pairs or associated poles for always providing an external divergence of the fields to be zero, i.e. nabla B=0. This is a very different story.

    Dear Christian,

    Magnetic pole is pure magnetostatic concept that is associated to the number of magnetic force lines which go out ( N pole) or go into (S pole). The magnetic field is not radial as the electric one but it depends of an angle that the lines form between the two poles, which makes difficult to calculate the force as the gradient of their fields. In fact it depends of details as the form of the magnet (for instance, it is not the same an sphere than a cilinder or a torous), the long that the poles are separated each other and with respect to the ones of their are interacting.

   The most closer microscopic point of view is taking the magnetic dipolar interaction which is quite far of the spins associated to the electrons. Actually these models of magnetic poles belong to the age where such quantum concepts were not known at all. Usually this concepts starts in books of magnetostatics applied to magnetism to show the units of magnetim, force and energy. But knowing that the electronic structure of the material or how the magnetic domains are reached is far of the student at such moment. In fact the main analogy is with electrostatics.

    A very good book which devotes the first chapter to this issue and after applies it when a macroscopic picture is enough, is in

Soshin Chikazumi, Physics of Ferromagnetism, Oxford University Press, 2nd edition, 1997.

   Obviously, I don't need to say that this is not for understanding how the poles attract or repel which is granted as it happens the same with the singns of the electric charge and what I have understood that was the question of Ricardo was asking and suggesting that these interactions were related with the electronic (free) spins configurations, not withing a ferromagnetic phase having magnetic domains with walls as Neel or Bloch etc.

  Dear Christian,

  Yes, indeed, the calculation is only simple if the configuration is simple.

  But the question was why magnets repel or attract.

  The poles formation in a magnet is independent of the energy, it depends of the magnetic material features, the shape and crystalline anistropies fundamentally. Thus the attraction or the repulsion is something previous to do any energy configuration between magnets. 

    Dear Christian,

    I'm surprised with this question and I have put an specific question on it. Notice that if you have had the gradient on the magnetostatic energy for giving you the force, this would mean that it would be exactly the same for the north pole than the south pole, and obviously this is not true. In fact that is the question here in what I have understood and perhaps Ricardo can tell us if I'm wrong or not.

https://www.researchgate.net/post/Why_does_appear_always_two_magnetic_poles_in_a_magnet_and_what_has_created_them_physically

     Dear Ronald,

     In the magnets there are no currents and the energy is always constant.

Dear Daniel, at this point, unfortunately, I have to disagree with you. If we have a (static) magnetization M, then this definitely corresponds to a current j, which is precisely determined by the equation: rot M = j. For an idealized, homogeneously magnetized cylinder, this current is localized at the lateral surface of the cylinder. For details, may I also refer you to Appendix A of the following article?

https://www.researchgate.net/publication/303755035_Ab_initio_materials_physics_and_microscopic_electrodynamics_of_media

Article Ab initio materials physics and microscopic electrodynamics of media

    Dear Giulio,

    don't be worry to disagree with me! This is the important of a fruitful discussion. Who told you that the magnetization is has curl of M different of zero? Do you think that the magnets have Ampere's currents?

@Daniel: this current is usually referred to as "magnetization current". For example, this equation can be found here: https://en.wikipedia.org/wiki/Magnetization#Magnetization_current

@Christian: the magnetization current is transverse, meaning that div j = 0. Therefore (by the continuity equation) it does not imply any charge transport. Thus, in this case, we have to distinguish between "longitudinal" and "transverse" currents instead of "physical" and "virtual" currents.

    Dear Giulio,

    Do you have magnetic currents associated to the magnetic poles of a magnet? I think that we are speaking about different things and by the way, where was you draft published?

Dear Christian,

Giulio has it right.

There is need to become aware of the permanent orthogonal orientation by electromagnetic structure of the magnetic field with respect to the orientation the translational momentum energy of the electrons. Transverse orientation of the magnetic field with respect to this adiabatically induced momentum energy, even if the latter is translationally impeded while captive in its resonance state.

This is something I may already have mentioned to you, (or maybe to somebody else, not certain to recall correctly), which is not covered by the traditional concept of momentum/Lagrangian/Hamiltonian since in such circumstances the electron velocity, thus its motion, is impeded and reduced to zero.

So, there is no need for unpaired electrons to physically move in translation for their magnetic fields to be forced into parallel alignment and add up to contribute the macroscopic magnetic field of permanent magnets.

Their individual magnetic fields are oriented by structure perpendicularly to the direction of application of their momentum energy, whether it is expressed as an actual velocity or not.

Dear Christian,

1) A closed current loop is a highly singular current distribution j(x) which does indeed fulfill div j(x) = 0, provided the current is constant and the loop is really exactly closed (in the mathematical sense). Whether this ultimately applies as a realistic description to a typical closed current circuit may be a matter of debate. Anyway, this example would at best show that there can be purely transverse (i.e., divergence-free) currents which "heat up the conductor". However, it does not follow that any transverse current would necessarily produce heat.

2) Concretely, for a magnetized rod, there is of course no "flow of electrons" (in the sense of charge transport) at the surface of the rod, because (i) the current fulfills div j(x) = 0; (ii) the constantly magnetized rod is only an extreme idealization, which defines a surface current, whose magnetic field roughly equals the real magnetic field (outside the rod). In reality, the magnetization M(x) is more complicated and so is the current density j(x). In this sense, the surface current is indeed only "effective" (i.e., actually not there). Finally, (iii) the idea of a "real flow of electrons" is classical anyway. Rather one has to think about this phenomenon in terms of a conductor being in a many-electron quantum state with non-vanishing expectation value of the microscopic current operator.

Dear Giulio.

Your mention of the magnetized rod brings to my mind the Barnett effect with a demagnetized ferromagnetic rod that becomes magnetized by setting it in mechanical rotation.

My interpretation is that in the rod at rest, the momentum energy of each unpaired electron, which is by structure perpendicular to their related individual electric and magnetic fields, can only average out to mutual momentum cancellation due to their best fit least action antiparallel spin orientations.

When the rod is mechanically set in rotation, these electrons find themselves on circular orbits about the rotation axis and share this new momentum energy, unidirectional at their level, to then align parallel into this now lesser action orientation, forcing by structure their related individual perpendicular magnetic fields also in mutual parallel orientation perpendicularly to their local momentum energy orientation, thus adding up to the now measurable macroscopic magnetic field.

To be related to the fact that the magnetic field increases with the velocity as more kinetic energy is mechanically provided to the rod (in fact, to each participating electron).

This confirms that the electrons do not have to be in motion within the rod to generate the magnetic field. they only need to be forced into mutual parallel spin orientation within the rod. This can apparently be accomplished by forcing their unidirectional momentum energy (by structure perpendicular to each their local related discrete magnetic field) to align into a mutual least action orientation.

References to the Einstein-de Haas experiment and  the Barnett experiment are in the Reference section of this analysis:

http://ijerd.com/paper/vol6-issue12/B06120711.pdf

I feel more comfortable talking about magnetic field gradients, that can separate

a beam of particles into separate spin components, a well known experiment in modern physics, stern gerlach.

In a magnet the atomic spins point mostly in the same direction. Therefore the force of

the inhomogeneous magnetic field of one magnet on the spin or magnetization of the other. If we reverse the spins of the second magnet, the force will also be reversed.

You have mainly a torque effect to cause the second magnet to swerve around, if the poles are not aligned.

thinking of spins as small current loops , therefore oriented in a certain way to the magnetic field is also valid.

Dear Mr. Juan,

I agree that the modern quantum physics is the perfect way to explain the asked 

One could compare with a mechanical gyro, to get the spin direction which is constant following the body.

Then, for a 'simple' gyro, with one rotating disc inside a bag, the actions and reaction is on a sublevel to the rotating angular velocity. Analysing a little more, it is not entirely clear if the reaction of an incremental angular velocity is a torque in orthogonal direction, or more of an angular velocity too.

It depends on how it is constrained, and that can be done to collect the energy to the upper level maintaining the gyroscopic motion, but then it is not the first simple gyro, more a spherical constraining, (possibly sometimes called wobbling). This was more to explain how a magnet is created, and googling, there are several ways, (a current in a helix wire around, one said mechanical with a hammer). For the first 'simple' gyro, I made a sketch, if I find I can post it.

Magnets have electrons that are moving and positive ionised atoms that are not. Permanent magnets have electrons spinning in around atoms. When two electrons are moving in the same direction, and the positive charges are not moving, relativity changes the attractive and repulsive forces between the charges. The moving electrons feel the same repulsion from each other as they would when not moving, because they don't move relative to each other. However, the positive charges are moving relative the the electrons and the attractive electric force is distorted by length contraction so it is stronger in the sideways direction and weaker in the direction of movement. This means that when electrons move in the same direction and the positive charges stay still there is a total attractive force to each side.

When the electrons are going in opposite directions the relativistic contraction means that each sees the same increased attraction from the stationary positive charges, but an even bigger increase in repulsion from each other, so there is a total repulsive force. This is what causes the repulsion between two north poles, there are spinning electrons which result in circular currents that are in opposite directions in each magnet (the same direction when both north poles are pointing the same way) that repel each other. Similarly, the circular currents in north and south poles are in the same direction and attract each other.

Dear Ronald,

Quite interesting paper.

With reference to your equation (1), you might be interested in a relation that can be drawn for this equation between the wavelength (thus indirectly with the frequency) of the energy making up magnetic field B.

See equation (32) in this paper:

http://www.gsjournal.net/Science-Journals/Research%20Papers-Mechanics%20/%20Electrodynamics/Download/2256

I see that you also relate your derivation to the Biot-Savart equation.

The derivation that allows the equation (32) representation directly stems from a discovery by Paul Marmet of the relation between the progressive increase of the magnetic field of an accelerating electron and it simultaneous relativistic mass increase that he discovered after quantizing the charge of the electron to its invariant value in the Biot-Savart equation.

All references are given.

Best Regards

André