# Conservation of energy and magnetism

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• cwy2012
The magnet did not bounce? In that case it was probably an inelastic collision. Usually kinetic energy is converted to thermal energy in an inelastic collision.(An elastic collision is such that first kinetic energy is converted to potential energy, then usually the potential energy is converted back to kinetic energy again. Laymen call that "a bounce".f

#### cwy2012

Please redirect me to the correct part of thr forum if this is the wrong place

When we lift up an object n then let it fall back, then potential energy - > kinetic energy

If I drop a magnet onto another magnet with like pole facing each other (that sits on the floor), the falling one maybe stopped in mid air. It will continue to fall from rest again when I remove the bottom magnet. But in this case the kinetic energy of these two falls added together will be less than if there was no magnet on the floor. What do I miss here?

Yes you are missing the magnetic potential energy (though there is some controversy, whether the magnetic field can do work on matter and hence on the very definition of magnetic potential energy but I don't know if you want to discuss this issue here). Some of the initial gravitational potential energy is converted to magnetic potential energy, instead of kinetic energy, that's why the kinetic energy is smaller.

cwy2012
Yes you are missing the magnetic potential energy (though there is some controversy, whether the magnetic field can do work on matter and hence on the very definition of magnetic potential energy but I don't know if you want to discuss this issue here). Some of the initial gravitational potential energy is converted to magnetic potential energy, instead of kinetic energy, that's why the kinetic energy is smaller.
@Delta2 Thanks for your reply. Was never taught about magnetic potential energy even at university level. I can imagine it, effort to bring two like poles together n they pushes each other away,, like there is an invisible springs between them.
Good t learn something new.

Delta2
@Delta2 Thanks for your reply. Was never taught about magnetic potential energy even at university level. I can imagine it, effort to bring two like poles together n they pushes each other away,, like there is an invisible springs between them.
Good t learn something new.
Yes well, magnetic potential energy seems ill-defined cause we know magnetic field can't do work on matter. Let us see if some better experts than me like @vanhees71 @Dale can offer a better answer.

Indeed there is no magnetic potential energy. In the Hamilton principle the magnetic field enters in terms of the vector potential.

What has a potential energy though is the motion of a magnetic dipole (like an electron) in a magnetic field. The corresponding potential energy (in the non-relativistic approximation) is
$$V=-\vec{\mu} \cdot \vec{B},$$
where ##\vec{\mu}## is the dipole moment of the particle. Dynamically this includes both a force on the particle if the magnetic field is inhomogeneous as well as precession of the dipole moment:
$$\vec{F}_{\text{dip}}=\vec{\nabla} (\vec{\mu} \cdot \vec{B}), \quad \dot{\vec{\mu}}=\frac{q g}{2m} \vec{\mu} \times \vec{B},$$
where ##q## is the charge of the particle, ##m## its mass, and ##g## the Lande (gyro) factor. In classical models of the magnetic moment using molecular currents a la Ampere the gyro factor comes out to be 1. For the magnetic moment due to spin 1/2 of an elementary particle like an electron the gyro factor is around 2 (deviation come from radiative corrections of QED). For more complicated particles like the protons and neutrons you have to measure the gyro factor. To understand the values theoretically is subject of ongoing research (aka the "spin crisis" for the proton).

Leo Liu, cwy2012, docnet and 1 other person
Please redirect me to the correct part of thr forum if this is the wrong place

When we lift up an object n then let it fall back, then potential energy - > kinetic energy

If I drop a magnet onto another magnet with like pole facing each other (that sits on the floor), the falling one maybe stopped in mid air. It will continue to fall from rest again when I remove the bottom magnet. But in this case the kinetic energy of these two falls added together will be less than if there was no magnet on the floor. What do I miss here?

The magnet did not bounce? In that case it was probably an inelastic collision. Usually kinetic energy is converted to thermal energy in an inelastic collision.

(An elastic collision is such that first kinetic energy is converted to potential energy, then usually the potential energy is converted back to kinetic energy again. Laymen call that "a bounce". )

cwy2012
@Delta2 Thanks for your reply. Was never taught about magnetic potential energy even at university level. I can imagine it, effort to bring two like poles together n they pushes each other away,, like there is an invisible springs between them.
Good t learn something new.
I have a slightly different take than @vanhees71. The magnetic field lacks a scalar potential, but it does have a vector potential. And per Poynting’s theorem it also has an energy density ##u = \frac{1}{2}\left(\epsilon_0 \mathbf{E}\cdot\mathbf{E} + \frac{1}{\mu_0}\mathbf{B}\cdot\mathbf{B}\right) ##. To that point he and I agree.

Where we differ slightly is that he does not want to call the energy density of the magnetic field as a magnetic potential energy, while I am a little ambivalent about it. That energy density is not kinetic energy, so calling it potential energy seems reasonable, but it doesn’t have a scalar potential so not calling it potential energy seems reasonable too, but if we don’t call it potential energy then what shall we call it? So I am ambivalent and since it is just a word choice I don’t push much either way.

However, I do think it is important to recognize a magnetic field has an associated energy density as above. Per Poynting’s theorem ## -\frac{\partial u}{\partial t} = \nabla\cdot\mathbf{S}+\mathbf{J}\cdot\mathbf{E} ## that energy doesn’t directly do work on matter. But it can be used to move energy elsewhere as well as being used to create E fields and currents that can then do work.

cwy2012
The magnet did not bounce? In that case it was probably an inelastic collision. Usually kinetic energy is converted to thermal energy in an inelastic collision.

(An elastic collision is such that first kinetic energy is converted to potential energy, then usually the potential energy is converted back to kinetic energy again. Laymen call that "a bounce". )
@jartsa , this is purely a thought experiment and example to highlight the gap in my understanding.

@Dale and @vanhees71 . Does it mean the magnetic dipole is affected when the two magnets come near to each other , and resume origin magnetic dipole value when the magnets are separated? in this case how is the energy dissipated (the missing kinetic I am looking for) ?

Appreciate very much the explanation from everyone.

Please redirect me to the correct part of thr forum if this is the wrong place

When we lift up an object n then let it fall back, then potential energy - > kinetic energy

If I drop a magnet onto another magnet with like pole facing each other (that sits on the floor), the falling one maybe stopped in mid air. It will continue to fall from rest again when I remove the bottom magnet. But in this case the kinetic energy of these two falls added together will be less than if there was no magnet on the floor. What do I miss here?

Here is an alternative thought experiment that might be helpful in some way:

If i drop an ideal magnet onto another ideal magnet with like pole facing each other (that sits on the floor), the dropped one will bounce up and down forever. It will continue to fall to the floor, when I remove the bottom magnet. The kinetic energy of these two falls added together will be same as if there was no magnet on the floor.

cwy2012
I have a slightly different take than @vanhees71. The magnetic field lacks a scalar potential, but it does have a vector potential. And per Poynting’s theorem it also has an energy density ##u = \frac{1}{2}\left(\epsilon_0 \mathbf{E}\cdot\mathbf{E} + \frac{1}{\mu_0}\mathbf{B}\cdot\mathbf{B}\right) ##. To that point he and I agree.

Where we differ slightly is that he does not want to call the energy density of the magnetic field as a magnetic potential energy, while I am a little ambivalent about it. That energy density is not kinetic energy, so calling it potential energy seems reasonable, but it doesn’t have a scalar potential so not calling it potential energy seems reasonable too, but if we don’t call it potential energy then what shall we call it? So I am ambivalent and since it is just a word choice I don’t push much either way.

However, I do think it is important to recognize a magnetic field has an associated energy density as above. Per Poynting’s theorem ## -\frac{\partial u}{\partial t} = \nabla\cdot\mathbf{S}+\mathbf{J}\cdot\mathbf{E} ## that energy doesn’t directly do work on matter. But it can be used to move energy elsewhere as well as being used to create E fields and currents that can then do work.
This is just semantics. I'd call it neither kinetic nor potential energy but simply field energy. The field-energy density of the em. field follows unambigously from Noether's theorem plus gauge invariance (or, more physically, from general relativity as a source of the gravitational field) as given by your expression for the energy density.

I was not talking about field energy but about the mechanical energy of charged matter in an external electromagnetic field.

Of course to have a closed system you have to take into account both the mechanical ("kinetic and potential") energy of the matter (classically best described continuum mechanically to avoid the inevitable trouble with radiation reaction for point particles). It's well known that in the corresponding effective theories you can shuffle the various pieces of the energy-momentum tensor between "field" and "matter" parts. E.g., bound charges and currents are usually described in terms of "field parts", electric and magnetic polarization.

Dale
This is just semantics. I'd call it neither kinetic nor potential energy but simply field energy.
I agree, this is just semantics. I am open to your term “field energy”.

And of course you have the physics right

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vanhees71
Assuming right circular cylindrical magnets as illustration, such a magnet can be viewed as an equivalent solenoid of similar dimensions. The current in the solenoid represents the so-called amperian currents in the magnet (amperian currents are mostly due to similarly aligned electron spin axes but this gets into quantum mechanics).

So consider two similar solenoids, each of some resistance R and fastened and connected to small batteries emf so a current i = emf/R is drawn.
One solenoid is at rest on the ground, the other suspended some high distance above the other. Similar mag poles face each other so there is repulsive force between them.

The upper solenoid is now dropped (confined in a tube of some sort) until the solenoids are at rest, the top one suspended above the bottom one. Potential energy is lost but there is no final kinetic energy.

During the fall, time-varying B fields are seen by both solenoids, their currents i modified by the resulting motive emf's.

The missing energy ## \Delta E = mg\Delta ~h = R {\Delta i}^2 ## must thus be dissipated in the solenoids' windings.

Problem is, amperian currents see no resistance and do not dissipate. So the missing energy really seems lost, though it isn't. Bottom line is that this gets into quantum mechanics which I won't.