Magnet's attraction without energy considerations

[DaTario]

Hi All,

Is there any way of explaining the attraction between North and South poles of two magnets without making use of energy considerations?

Best wishes

DaTario

 

[dgOnPhys]

for a dipole to be subject to a force all you need is a field gradient... no energy considerations needed

 

[DaTario]

Ok, but what to say with respect to the famous sentence that "magnetic forces do no work." ?

Best wishes

DaTario

 

[Dadface]

Hello DaTario.I think the famous sentence refers to the action of magnetic forces on moving charged particles and not on ferromagnets and the like.With a charged particle there is no resultant movement in the direction of the force.

 

[Born2bwire]

Ok, but what to say with respect to the famous sentence that "magnetic forces do no work." ?

Best wishes

DaTario

Griffiths has some good explanations, there are a few following his statement that the magnetic fields do no work and there are a few later on when he discusses Lorentz transformations. There is always some form of electric fields involved. They may exist as a transformation of the magnetic fields into the reference frame of the charges. They may exist as coulombic attraction between the conduction electrons and their ions on a wire's lattice. Purcell's textbook actually introduces the Lorentz transformation of the electric field first and then the magnetic field. That helps to drive home the idea of how the fields can transform depending upon the frame. By relativity, if we have a magnetic field that exerts a force on a moving charge in the lab frame, then in the charge's rest frame an equivalent force must be produced by an electric field transformed by this magnetic field since a magnetic field can no longer exert a force on the stationary charge. This electric field can be seen to do work by the Lorentz force.

For classical electrodynamics, the magnetic dipoles are assumed to be microscopic loop currents. The inherent dipole moments of electrons and nuclei due to spin is not a part of the theory. So keeping in mind that the magnetic dipoles of a magnet are just current loops the same ideas apply. But obviously it is much easier to work from the energy picture.

In fact, we can remove the field picture all together. Jefimenko's Equations allow us to express the electric and magnetic fields produced by all sources of currents and charges. In this manner we can simply remove the fields and relate the interactions by substituting the appropriate equations into the Lorentz Force. If we just have static currents then the only force term comes from the magnetic fields which are directed in such a way that no work can be done on the other moving charges. But it all goes out the window once we let it become a dynamic situation. If we allow the magnets to actually move then the currents become time-varying and then we have a mix of electric and magnetic fields too. Then these electric fields can do work.

So I think there are two points here. The first is that generally when we see a magnetostatic system that we fail to realize that the lack of electric fields is a matter of perspective. Second is that when we consider a situation where we know work must be done but we cannot see how it is done (ie: two classical bar magnets) we forget that in these situations work is not done until we allow the system to be dynamic. In this case we now have accelerating currents and charges which are no longer static and we have both electric and magnetic fields as a result. Otherwise we just have a static picture with forces with is just fine and dandy.

And then of course with quantum we come to realize that there are now intrinsic magnetic sources that are not produced by the classical charge and currents and so we cannot use the same rigamorale as we did before. In fact its probably best to not think too hard on that anyway and just use the energy expressions.

 

[DaTario]

Hello DaTario.I think the famous sentence refers to the action of magnetic forces on moving charged particles and not on ferromagnets and the like.With a charged particle there is no resultant movement in the direction of the force.

But at the end, almost everything is charged particles moving with a certain velocity...

Best wishes

DaTario

 

[DaTario]

Griffiths has some good explanations, there are a few following his statement that the magnetic fields do no work and there are a few later on when he discusses Lorentz transformations. There is always some form of electric fields involved. They may exist as a transformation of the magnetic fields into the reference frame of the charges. They may exist as coulombic attraction between the conduction electrons and their ions on a wire's lattice. Purcell's textbook actually introduces the Lorentz transformation of the electric field first and then the magnetic field. That helps to drive home the idea of how the fields can transform depending upon the frame. By relativity, if we have a magnetic field that exerts a force on a moving charge in the lab frame, then in the charge's rest frame an equivalent force must be produced by an electric field transformed by this magnetic field since a magnetic field can no longer exert a force on the stationary charge. This electric field can be seen to do work by the Lorentz force.

For classical electrodynamics, the magnetic dipoles are assumed to be microscopic loop currents. The inherent dipole moments of electrons and nuclei due to spin is not a part of the theory. So keeping in mind that the magnetic dipoles of a magnet are just current loops the same ideas apply. But obviously it is much easier to work from the energy picture.

In fact, we can remove the field picture all together. Jefimenko's Equations allow us to express the electric and magnetic fields produced by all sources of currents and charges. In this manner we can simply remove the fields and relate the interactions by substituting the appropriate equations into the Lorentz Force. If we just have static currents then the only force term comes from the magnetic fields which are directed in such a way that no work can be done on the other moving charges. But it all goes out the window once we let it become a dynamic situation. If we allow the magnets to actually move then the currents become time-varying and then we have a mix of electric and magnetic fields too. Then these electric fields can do work.

So I think there are two points here. The first is that generally when we see a magnetostatic system that we fail to realize that the lack of electric fields is a matter of perspective. Second is that when we consider a situation where we know work must be done but we cannot see how it is done (ie: two classical bar magnets) we forget that in these situations work is not done until we allow the system to be dynamic. In this case we now have accelerating currents and charges which are no longer static and we have both electric and magnetic fields as a result. Otherwise we just have a static picture with forces with is just fine and dandy.

And then of course with quantum we come to realize that there are now intrinsic magnetic sources that are not produced by the classical charge and currents and so we cannot use the same rigamorale as we did before. In fact its probably best to not think too hard on that anyway and just use the energy expressions.

I consider your exposition to be very nice. In other words, I think you seem to digging just above the treasure. I think there must be a simple example where one can prove that magnetic forces acting on two charged particles which interact with each other provides the realization of work on the system.

Best wishes

DaTario

 

[Born2bwire]

But at the end, almost everything is charged particles moving with a certain velocity...

Best wishes

DaTario

Except now we have intrinsic sources of magnetic moments. There isn't an exact parallel with classical sources. For example, we say that the electron has an intrinsic spin and this results in its magnetic moment. However, this does not mean that we can think of it as a spherical shell of charge that is physically spinning like a top. I do not think we can make any qualifications on how it should be governed in reality by the Lorentz force.

However I do agree with you though. We can still equivalently model the microscopic magnetic moments of a material as macroscopic bound currents. So although these moments may be quantum mechanically produced, we can model them perfectly fine as classical macroscopic sources. In such a way we then come back to the same problem. In addition, by using Jefimenko's Equations we can easily demonstrate that we can remove the field picture if we wish and work purely with the sources and their interactions with each other. As such, we see that the energy densities that we calculate of the fields is representative of the energies in the interactions between the sources. So if we can use an energy picture to see that we can extract energy from magnetic fields and transfer it to kinetic energy in the system, then we must be able to do this some way using the Lorentz force.

But as I stated above I think the biggest point to make is that if we actually allow most of these systems to progress so that work is done (like allowing two bar magnetics to come together or two parallel wires move closer) then we are no longer in a magnetostatic situation. Instead, the movement of the current sources means that there must be electromagnetic waves produced, however slight, that I would expect to produce the necessary electric fields that can do work in the lab frame.

I consider your exposition to be very nice. In other words, I think you seem to digging just above the treasure. I think there must be a simple example where one can prove that magnetic forces acting on two charged particles which interact with each other provides the realization of work on the system.

Best wishes

DaTario

Take a look at Griffiths. I think that these mechanisms would be very difficult to work out fully by hand. However, there is a very common problem that is done by many textbooks like Griffiths, Prucell, or Halliday & Resnick. They look at the case of two parallel current carrying wires. In the lab frame we only see magnetic fields. However we can look at the rest frame of the charges in the currents and we find that the force in that frame is due to an electric field. But again this is the static picture. What happens when they actually allow to dynamically move and thus require work to be injected? That is not borne out in the calculations however Griffiths does make mention earlier about this when he talks about the Lorentz Force. I guess one could try to work out the path of displacement of the charges and thus find their acceleration. You could then find the radiated fields and use the electric field, in the rest frame, over the path of displacement to find the force on the currents in the opposite wire. In this way you could integrate the force over the distance traveled and find the work done. Comparison with the work that would be derived from the energy equations should show if this is the actual mechanism or not.

Obviously the actual energy has to come from the electric fields that drive the currents (battery). One way to look at the energy in the magnetic field is that it is the energy required to setup the field. So if the energy in the fields is transferred as the work done on the wires, then this energy has to come from the whatever energy source is maintaining the currents.

One could even imagine and experiment to show this (though it isn't the most easily reproduceable). You can start with two wires that are close together and excite currents through them so that they repulse. Once they separate, turn off the current and allow them to come together (say via gravity). Rinse and repeat and this should drain the batteries faster then if you kept the wires stationary.

 

[dgOnPhys]

I think we might have had this discussion before but
- classical electrodynamics is perfectly adequate to describe point-like dipoles both electric and magnetic. E.g you can write polarization as
P(r)=p.delta(r-r0)
or magnetization as
M(r)=m.delta(r-r0)
- you cannot distinguish between an infinitesimal current loop and and a point-like intrinsic dipole
- two ideal dipoles (magnetic or electric) will do work on each other, due just to their static fields
- your argument that magnetic work occurs only when sources are moving is flawed because it could be used on the electric field as well

 

[Born2bwire]

I think we might have had this discussion before but
- classical electrodynamics is perfectly adequate to describe point-like dipoles both electric and magnetic. E.g you can write polarization as
P(r)=p.delta(r-r0)
or magnetization as
M(r)=m.delta(r-r0)
- you cannot distinguish between an infinitesimal current loop and and a point-like intrinsic dipole
- two ideal dipoles (magnetic or electric) will do work on each other, due just to their static fields
- your argument that magnetic work occurs only when sources are moving if flawed because it could be reused on the electric field as well

Except this approach only whitewashes the problem instead of addressing it. Like I said in my previous post, we do not need fields to talk about these problems in classical electrodynamics. The fields are force fields that relate how one source (charge and current) produces a force on another source. We could, in a very ungainly manner, replace the fields with the appropriate integrals via Jefimenko's Equations. What this tells us, and we can also prove this explicitly, is that the energy of the fields is the energy related to the forces that the charges and currents exert on one another. So if we reduce the problem to some moment and find the energy of the fields and say there, that is how work is done, we are just glossing over the fact that the physics hasn't changed. That is, the moment is still created by some classical source and that the energy is in the forces that arise due to the interaction between these sources. So if these forces, given by the Lorentz force, do not allow the sources that give rise to magnetic fields to do work, how is this done?

Not to mention that microscopic moments do not have to come into this at all. As I mentioned before, we can equivalently model the fields of a magnetized material as macroscopic surface bound currents. I can even go so far as to remove the ferromagnet all together. Instead of bar magnets, we will have electromagnets in the form of solenoids. In the latter case there are no microscopic moments but we now have explicit macroscopic currents. But the physics is the same, these two solenoids will attract or repel each other by virtue of magnetic fields.

The whole point though is that we need to realize that we do not need to look at intrinsic moments as some kind of deus ex machina to save us from this dilemma. Unless we have ohmic losses (which we can negate by assuming perfect conductors), there is no work done when we look at these problems in their static cases. All we have are static forces being exerted. Once we allow for the fact that for work to be done to bring the wires together we have to allow for acceleration or some kind of time dynamic then there opens up a whole new avenue of possibilities from the fact that the acceleration of charges can allow for electric fields to arise. Here's another idea, instead of allowing acceleration, we will have two parallel wires and we will bring one in from infinity towards and then away from the other wire at a constant velocity. But a quick glance at Jefimenko's Equations show that a moving current will give rise to an electric field. So an impulse is not needed, just time dependency.

The response to these currents is some form of movement. For example, if we have a current loop and we drive it so that the current remains constant, then in response to an applied magnetic field we can allow the current loop to rotate in space (torque), change its radius, and/or translate in space. Any of these results involves a time-variation of the currents and this can mean an electric field.

So there are variety of ways that we can look at the problem and deduce how electric fields can be introduced that can allow for the explicit transfer of energy.

 

[dgOnPhys]

No white washing, the original questions were:
- explain permanent magnet attraction w/o energy considerations (not w/o fields)
- (later) whether magnetic field performs work

The simplest way to respond is to reduce the problem to two magnetic dipoles and show:
- magnetic field generated by one dipole exerts force (and torque) on the other
- the magnetic field does perform work in the process

The force on a dipole immersed in a magnetic field is not Lorentz force
F=m.grad(B)
T=m x B
The B field generated by one dipole can be easily found e.g. http://scienceworld.wolfram.com/physics/MagneticDipole.html"

In general the force will not be orthogonal to the infinitesimal path. There is no need to call in electromagnetic waves, self-force, equivalent current distributions and such.

 

[DaTario]

Is it true that we cannot make any reasonable disctinction between the magnetic field generated by a loop conductor with current and a set of two magnetic monopoles ?

Best wishes

DaTario

 

[dgOnPhys]

Only if they are infinitely small (point-like), if they have a finite dimension the field in their proximity is different