What happens to the electrons in a magnet

[kmm]

Suppose I'm holding a magnet and then lift a paperclip(or another magnet) with the magnet. Now the energy density of a magnetic field or the energy that went into making the magnetic field is $$W = \frac{1} {2 \mu_o} \int B^2 d \tau$$ but since work was done in lifting the magnets, what happenes to the energy of the electrons in the magnet that produced the $\mathbf{B}$ field? Would their energy decrease by $\Delta W$? Here, I'm assuming the energy density of the magnetic field has decreased.

I've assumed that the work done when a magnet lifts a paper clip is due to the $\mathbf{E}$ field generated by the electrons pulling the lattice with them as the electrons are deflected from the lattice. This seems to contradict my first question where I assumed the energy is lost in the electrons producing the $\mathbf{B}$ field. My first question also seems to imply that it is actually the current loops in the magnet that's doing the work. So which is it? Or is it both?

 

[Simon Bridge]

When you figure these things, you have to consider all the energy changes. i.e. what about the paperclip?

 

[kmm]

Well part of my confusion has to do with Griffiths. The examples he gives is a rectangle loop of wire, with some current 'I', in a magnetic field with some mass suspended to it. If the current is increased the mass is lifted and the work is done by the battery $$W_{battery}= \lambda a B \int u w dt$$ where a is the width of the loop and u and w are the upward and downward velocity components respectively. This doesn't include anything about the work done by the electrons on the lattice of the wire. So does this mean ALL the work is done by whatever generates the magnetic field?

 

[Simon Bridge]

Like I said - consider all the energy.
Perhaps you'll see what I mean if you think about gravity... what is doing the work when two objects gravitate towards each other?

In an electromagnet, you start with zero strength and you increase the current until the magnet lifts the object - the object exchanges magnetic PE for gravitational PE. You have to do work to increase the current - but how much work just to hold the object there?

In a permanent magnet - the magnet is always switched on ... so the object was always in a magnetic field, that is either strong enough to lift the object or not - already.

 

[kmm]

This seems to help. When two objects gravitate towards each other, they have a potential energy between them that is converted to kinetic energy. We could then say the same thing about two magnets near each other. There is a magnetic potential energy between them that is converted to kinetic energy as they move towards each other.

Although, if we want to lift an object with an electromagnet and hold it there, work is done in generating the current which creates the magnetic field. Work has to continue to be done to maintain a strong enough magnetic field. In a permanent magnet though, the electrons are in some energy state that maintains the magnetic field. If I hold two anti-aligned magnets near each other, they are at a greater potential energy and will fall to a lower potential energy as magnets align to each other.

Does this mean though, that there is no change at the level of the electrons?

 

[kmm]

This paper suggests that for the electrons, there is a decrease rotational kinetic energy. academic.csuohio.edu/deissler/PhysRevE_77_036609.pdf

 

[Simon Bridge]

This seems to help. When two objects gravitate towards each other, they have a potential energy between them that is converted to kinetic energy. We could then say the same thing about two magnets near each other. There is a magnetic potential energy between them that is converted to kinetic energy as they move towards each other.

Although, if we want to lift an object with an electromagnet and hold it there, work is done in generating the current which creates the magnetic field. Work has to continue to be done to maintain a strong enough magnetic field.

Where does the energy go ? (in the ideal case) I mean, once the magnetic field is established.

The work to hold the paperclip is still less than that needed to lift it.

The paperclip is a conductor - so does the motion not induce a current to oppose the motion ?
Don't forget too - the third law: the paperclip exerts a force on the magnet.

There is DC resistance in the wires, of course... so, no matter what, work is done just to keep the magnet running.
The evidence for this is that the battery goes flat.

In a permanent magnet though, the electrons are in some energy state that maintains the magnetic field. If I hold two anti-aligned magnets near each other, they are at a greater potential energy and will fall to a lower potential energy as magnets align to each other.

Typically they will try to rotate and move closer ... if perfectly anti-aligned, they try to move apart.
Some work had to have already been done to get them into that position in the first place.
This energy is "stored in the magnetic field"

Does this mean though, that there is no change at the level of the electrons?

Well... no.
All charged particles feel the electromagnetic force.
It's just that I wanted you to realize that the models you have been taught are a bit on the simplistic side.
Hopefully you can see that the situation is not straight-forward on the scale of individual electrons.

In a permanent magnet, many of the atoms have aligned spins ... the spins are held that way by details of the internal structure.
When they experience an external magnetic field, the atoms will try to align with that field.
Since they have their own field, they will affect whatever is producing the external field as well.

 

[kmm]

Where does the energy go ? (in the ideal case) I mean, once the magnetic field is established.

The work to hold the paperclip is still less than that needed to lift it.

The paperclip is a conductor - so does the motion not induce a current to oppose the motion ?
Don't forget too - the third law: the paperclip exerts a force on the magnet.

There is DC resistance in the wires, of course... so, no matter what, work is done just to keep the magnet running.
The evidence for this is that the battery goes flat.
Typically they will try to rotate and move closer ... if perfectly anti-aligned, they try to move apart.
Some work had to have already been done to get them into that position in the first place.
This energy is "stored in the magnetic field"

Well... no.
All charged particles feel the electromagnetic force.
It's just that I wanted you to realize that the models you have been taught are a bit on the simplistic side.
Hopefully you can see that the situation is not straight-forward on the scale of individual electrons.

In a permanent magnet, many of the atoms have aligned spins ... the spins are held that way by details of the internal structure.
When they experience an external magnetic field, the atoms will try to align with that field.
Since they have their own field, they will affect whatever is producing the external field as well.

Then not only is energy lost as heat in the wires but also transferred to the paperclip. It seems this may be seen by the inductance or the "back emf" described by Lenz's Law. But then I would think the paperclip has the same effect on the magnet as well. So it's as though energy in some way is flowing back and forth between the two. I would also think then that the magnetic field is the "communicator" of the energy exchange. Like in the game pool, when I strike the ball, the cue stick is the communicator of the energy that my arm imparted to it, while at the same time my arm is effected by the ball because of Newton's third law. I'm not sure if I'm actually thinking correctly about this though.

 

[Simon Bridge]

I would also think then that the magnetic field is the "communicator" of the energy exchange.

Well done.
This is pretty much the particle theory model of the electromagnetic interaction - the force carriers, virtual photons, are exchanged - so energy goes back and forth and the field is thus exchanging of energy.

Try not to run too much with the idea though.
Magnetodynamics is a hard subject. You end up making a lot of approximations.

You may want to think about the following situation:
A rod of iron drops between the poles of a fixed magnet - it slows down.
Therefore it did not gain as much kinetic energy as would be expected from the change in gravitational PE.
If the magnets are in pretty much the same state after the drop as before, where did the bulk of the energy go?

 

[kmm]

Cool. Thanks for the help with this. Well my first thought is that the iron is filled with randomly oriented dipoles and so as gravity does work on the iron rod, some of that work goes into putting those dipoles in some energy state with the magnetic field, then as the magnetic field aligns the dipoles they go into a lower energy state. During that process I would think that the bulk of the rod is heated. I know in Griffiths, it shows that the bulk of the dipoles resist the torque created by the magnetic field and it's the dipoles at the "domain boundaries" that are aligned to the B field. I'll have to think about that more though.

 

[Simon Bridge]

That great.

You ended up with more questions than answers, that's normal.
The main thing is that you are thinking along the right lines.

The danger was that you may have started thinking that the magnet is a store of magnetic energy that is used to do work ... you can find a tour of the misconceptions at Donald Simanek's Museum of Unworkable Devices. Have fun.

 

[kmm]

Thanks for the help. I'll check that out. I definitely still have a lot to think about, but for now, I'm looking at the magnetic field as the medium through which information is sent, not as an energy source itself. I was pretty sure of that already because of the magnetic part of the Lorentz force law, but now it feels more intuitive.