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particlezoo

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**same**direction as the applied magnetic field.

Since the total magnetic field energy is based on the square of the magnetic field strength, it would appear that the mean square of this field would increase as these magnets approached each other (or at least rotated to have mutual magnetic alignment), maximizing the average alignment between the magnetic fields over the whole spatial domain. Since this process releases kinetic energy, I would have to assume that magnetic potential energy should have gone down in this process. But from what I see according to the definition of the magnetic energy density based on (1/2)B^2/mu_0, which is not preceded at all by a negative sign, the energy density in the magnetic field, according to this, would have increased, not decreased. If both the magnetic field energy and the kinetic energy are increasing as two permanent magnets approach each other, it would beg the question of where the compensating energy change lies.

It is quite clear though if I have two electromagnets that in order to generate a magnetic field I have to expend electrical energy to develop the field in the first place, and then when I allow the solenoids to attract each other, the increase in the magnetic field energy energy due to (1/2)B^2/mu_0 is not without the introduction of back-emf which is clearly necessary to preserve the energy conservation. In this case, it is very clear that (1/2)B^2/m_0 (as well as kinetic energy) increased at the expense of my input electrical energy.

With the permanent magnets, we do not have this simple explanation as to why both these occur. Potential energy is apparently expended in order to bring the magnets to acceleration, but how could both the magnetic potential energy and kinetic energy increase in this process without something else in the picture? Something that is somehow expended as the permanent magnets are brought together?

I still think it makes sense that magnetic potential energy decreases as the permanent magnets come close together under attraction, just as we might find with approaching solenoids whose back-emf reduces the magnetic field of each solenoid, reducing the power consumption, but not the total energy already invested into creating the magnetic field. The problem is that there does not appear to be any mechanism by which the attracting and approaching permanent magnets could generate emfs into each other that could in any way act against the intrinsic electron magnetic moments. So I am left with a suspicion that the magnetic potential energy change would be negative.

Now, if I look at the potentials formulation of electromagnetism, it is quite evident to see where we might get a negative change in potential energy, say, if we have a moving charge q with velocity

**v**in a solenoid acting against the magnetic field of a permanent magnet, which possesses a vector potential

**A**. Reason being, the sign of the potential energy depends on the sign of q and

**A**, as well as the cosine of the angle between

**v**and

**A**. However, using (1/2)B^2/mu_0 for the magnetic energy density appears to only give us a positive change in the energy density as the two magnets approach each other under attraction. The same problem arises if we use the expression for the magnetic energy density for linear, non-dispersive materials, (1/2)(

**B**•

**H**), where

**B**lines up with

**H**.

To make matters worse, with permanent magnets, instead of a charge q with velocity

**v**, we instead just have many pairs of electrons where one electron with its intrinsic magnetic moment orients in line with the field of the other (and vice versa), in a process which should release kinetic energy. But from what? And how does that fit into the potentials definition of the magnetic field energy?

Lastly, let us consider the case of magnetic monopoles, which are said to be analogous to electric charge, in that opposite poles attract and like poles repel. But how do we know that the opposite is not true? Remember that in the case of attracting permanent magnets, when they come closer together (or simply come into alignment), the mean square magnetic field rises, and therefore the magnetic field energy also rises.

*I would think this would be a*

__binding energy__, even if one were to replace the two permanent magnets with two solenoids with current. So the attractive forces which bring the magnets together will over time lead to decrease of the potential energy, if by potential energy we actually mean the potential energy that is converted into kinetic energy.**If**we applied the same logic to magnetic monopoles, then like magnetic monopoles should attract, as their merger would increase with the mean square magnetic field, analogous to the attracting permanent magnets. If so, we would, in converse, expect that opposite magnetic monopoles should repel. This makes magnetic monopoles less like an analog to electrical charges exerting Coulomb forces, and more like an analogy to gravitating masses. I have never found this to be described in the literature.

I know these are a lot of questions and concerns, but if you don't mind to answer at least some of them, that would be greatly appreciated. Thanks to all in advance.