Can a magnetic fields/forces do work on a current carrying wire?

[Dale]

I would say that such arbitrary conventions are not to be preferred.

I don't think that there are any generally-accepted conventions, nor even any typically-recommended ones, for drawing system boundaries. It may not be preferred, but I don't see any way around it.

If the rope is applying a force to the system along some distance then by the usual definition of work the rope's force is doing work on the system. Work transfers energy, it doesn't have to be the ultimate source of the energy.

 

[harrylin]

I don't think that there are any generally-accepted conventions, nor even any typically-recommended ones, for drawing system boundaries. It may not be preferred, but I don't see any way around it.

If the rope is applying a force to the system along some distance then by the usual definition of work the rope's force is doing work on the system. Work transfers energy, it doesn't have to be the ultimate source of the energy.

That's not the definition that I use, and indeed with your definition there is no way around it.
And with that definition every force pair "does work"; which makes the OP's question (see below) like trying to kick in an open door.

I finally manged to decipher it, Myiz meant of course:

"Aren't the magnetic forces in a motor one of the key factors of motion inside? I mean it makes no sense to me why in this case magnetic force can't do work on an electric charge..."

 

[Dale]

That's not the definition that I use, and indeed with your definition there is no way around it.

Oh, well I don't want to argue about semantics either, but what is the definition you use? I am not familiar with another definition, or maybe I am and just cannot recall it right now.

I think that to answer questions like this it is important to at least be clear on the definitions that everyone is using, including both definitions about general terms like "work" and definitions about the problem itself like the boundaries of the system. E.g. if the boundary of the system is drawn one way then the magnetic force is internal to the system, and if you draw it another way it is external.

"Aren't the magnetic forces in a motor one of the key factors of motion inside? I mean it makes no sense to me why in this case magnetic force can't do work on an electric charge..."

For this phrasing of the question it sounds that the magnetic force is considered an internal force.

 

[cabraham]

It is very clear that the power of any electromagnetic field on charges is given, according to Poynting's theorem by
P(t)=\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} \; \vec{E}(t,\vec{x}) \cdot \vec{j}(t,\vec{x}).
Of course a motor does work, but it's the electric field according to the above equation.

Which electric field? Have you read the link I provided? We discussed this in detail. Please read it then comment. An electric field does NO WORK on NEUTRONS. Only strong force can do that. Also, it is impossible for any field, E, M, or SN, to do long term "work". Ultimately the power source, i.e. battery, ac mains, automobile alternator, etc., that does the continuous work. Fields store & release energy, but they are spent when doing so, & need to be replenished.

I believe E & B fields can only receive energy for storage, & upon transfer of said energy, they are spent, then replenished by the power source. I just provided the link so that we can examine & understand all pertinent forces involved.

Claude

 

[cabraham]

Today vanhees gave you the equation you asked for, and also philipwood and I gave you good points. The most pertinent one is just your first sentence here: Permanent magnets are no energy source. That means that they do not give off energy, and most physicists mean with "doing work" that a system provides energy to one or more other systems.

Compare: http://www.lightandmatter.com/html_books/lm/ch13/ch13.html#Section13.1
The tractor does work, but the rope does not.

Good point. In my linked thread the magnetic force is akin to the tractor, while the E & SN forces are analogous to the rope. I firmly believe that al 3 forces are involved, but the power source provides all the energy to the magnetic field, which gets transferred, then the power source replenishes said energy. Of course the internal combustion in the tractor engine is akin to the power source replenishing the magnetic field.

Claude

 

[harrylin]

Oh, well I don't want to argue about semantics either, but what is the definition you use? I am not familiar with another definition, or maybe I am and just cannot recall it right now. [..]

Apparently Claude uses a similar definition as the one I phrased and linked to in post #82. A rope that is not used for its elastic force merely transmits energy and is not a source of energy, so that it does no work if we use that definition.

However, see also my Note in post #90 (this discussion goes to fast!).

 

[Dale]

Apparently Claude uses a similar definition as the one I phrased and linked to in post #82. ... (this discussion goes to fast!).

Oops, sorry, I missed the link. You are right, it does move fast. The energy transfer definition in the link is subtly different from the F.d definition I was using, so you are correct that we were using different definitions.

However, using that definition the lightandmatter link explicitly says that the rope does work on the plow: "When the tractor pulls the plow with a rope, the rope does negative work on the tractor and positive work on the plow." (emphasis added).

Also, the definition used there clearly applies to a rope: "Work is the amount of energy transferred into or out of a system, not counting energy transferred by heat conduction." The rope does transfer energy to the weight/plow/trailer. It doesn't produce any energy, but it transfers it from the tractor to the weight and not via heat conduction.

 

[Philip Wood]

vanhees71. There's no argument about the power equation is there? Of course it is right.

 

[harrylin]

[..] However, using that definition the lightandmatter link explicitly says that the rope does work on the plow: "When the tractor pulls the plow with a rope, the rope does negative work on the tractor and positive work on the plow." (emphasis added). [..]

Ah right I missed that - thus that link actually corresponds to your definition, so it appears that yours is the more commonly used. And using that definition the reply to the title question is obviously yes: fields/forces that move a wire do work on that wire.

 

[cabraham]

Apparently Claude uses a similar definition as the one I phrased and linked to in post #82. A rope that is not used for its elastic force merely transmits energy and is not a source of energy, so that it does no work if we use that definition.

However, see also my Note in post #90 (this discussion goes to fast!).

I agree with that, but likewise the E field which tethers the stationary lattice protons to the mobile electrons merely transmits the force on the electrons from the magnetic field. Likewise SN force is also like the rope in that it transfers force to the neutrons. Both E & SN forces are akin to the rope in the tractor example.

Remember that the force integrated over the distance is the work done. The mag force must be strong enough to match the E force, plus the SN force, as well as move the electrons. But the mag field gives up energy as it transfers energy to produce torque & speed. The power source at the motor terminals replenishes this energy.

Is the mag force doing "work"? Well, in the short term, YES, in the long term NO. The power source, battery, ac mains wall outlet, etc., is doing all of the long term work. The mag force does move the rotor, but it only acts directly on electrons, but indirectly on protons & neutrons. The E & SN forces are internal tethers, like the rope in the tractor example. They are indispensable as they transmit force to protons & neutrons. The mag force is ineffective on proton & neutron.

Is the mag force doing work? Again, it stores energy then transfers it. It needs help from E & SN forces as well. Mag force participates but can't do it alone, nor long term. The power source is ultimately what does the work, not B field, not E field, not SN force. Did I help or confuse matters more?

Claude

 

[harrylin]

[..] Did I help or confuse matters more?
Claude

Without scrutinizing your arguments in detail, I think that you nicely summarized it.

 

[Miyz]

After one turn the potential energy is identical - no change over one cycle. So, as Darwin already pointed out in post#30, it reduces to a disagreement about the meaning of words. In physics language the rope behind the tractor and the permanent magnet in the motor do no work - that has nothing to do with equations, just with definitions.

That is, the definition of work and many explanations of how to deal with it "is irrelevant to this topic"... Well then, good luck!

Um, Darwin123 gave a very good statement about the forces involved in doing the work and will eventually transfer energy. Now in you're cause the tractor is the main input of force applied on the rope to lift,move,etc... When you brake the whole system up you'd find out the main "SOURCES" to transfer energy is the rope. Without the existence of any force in the system nothing is applied on the weight and no work will be done and no energy would be transferred.

In a motor what basically happens? A battery is connected to a loop supplying the input "magnetic field" when a magnet is present and oriented in a certain way it will apply a force on the loop it will rotate and torque will be created. Now who's the rope in this situation and the tractor? I think it pretty obvious isn't it?

Even in you're example the "rope" is the connection between to object to do work which is crucial, without the presents of a magnet/rope no work wold be done.

Yea and when I said its irrelevant not by definition but more into the forces acting upon object because its very simple and obvious. In a cause of motor where its all about the magnetic fields its more complex isn't it?

Oops, sorry, I missed the link. You are right, it does move fast. The energy transfer definition in the link is subtly different from the F.d definition I was using, so you are correct that we were using different definitions.

However, using that definition the lightandmatter link explicitly says that the rope does work on the plow: "When the tractor pulls the plow with a rope, the rope does negative work on the tractor and positive work on the plow." (emphasis added).

Also, the definition used there clearly applies to a rope: "Work is the amount of energy transferred into or out of a system, not counting energy transferred by heat conduction." The rope does transfer energy to the weight/plow/trailer. It doesn't produce any energy, but it transfers it from the tractor to the weight and not via heat conduction.

Makes more sense and more logical.

Ah right I missed that - thus that link actually corresponds to your definition, so it appears that yours is the more commonly used. And using that definition the reply to the title question is obviously yes: fields/forces that move a wire do work on that wire.

Woho! Now you see what I truly mean,

Thanks DaleSpam for clarifying things out!

Magnetic fields + forces are INDEED doing work in a motor, simple explanation of this as I said at the beginning of this post: When current flows thorough a wire it creates a magnetic field, and another magnetic field is present that is from the magnet, They attract,repel. Now what's doing all the work? Magnetic fields + forces.(Simple explanation to a complicated effect.)

+ There is energy within the magnetic field.

Interesting how things turned out.

 

[Miyz]

I agree with that, but likewise the E field which tethers the stationary lattice protons to the mobile electrons merely transmits the force on the electrons from the magnetic field. Likewise SN force is also like the rope in that it transfers force to the neutrons. Both E & SN forces are akin to the rope in the tractor example.

Remember that the force integrated over the distance is the work done. The mag force must be strong enough to match the E force, plus the SN force, as well as move the electrons. But the mag field gives up energy as it transfers energy to produce torque & speed. The power source at the motor terminals replenishes this energy.

Is the mag force doing "work"? Well, in the short term, YES, in the long term NO. The power source, battery, ac mains wall outlet, etc., is doing all of the long term work. The mag force does move the rotor, but it only acts directly on electrons, but indirectly on protons & neutrons. The E & SN forces are internal tethers, like the rope in the tractor example. They are indispensable as they transmit force to protons & neutrons. The mag force is ineffective on proton & neutron.

Is the mag force doing work? Again, it stores energy then transfers it. It needs help from E & SN forces as well. Mag force participates but can't do it alone, nor long term. The power source is ultimately what does the work, not B field, not E field, not SN force. Did I help or confuse matters more?

Claude

Well put Claude yet again.
True mag forces can't do anything by its own. However, the power source is supplying all that flow E & SN forces!
If the power source or let's say (Input force) is constant the mag force would constantly be doing work :)

 

[Dale]

A nice paper about this question is the following one. The classical part of it precisely answers the question discussed here on hand of a simple example:

http://link.aps.org/doi/10.1103/PhysRevE.77.036609

Hmm, I found the classical part of the paper quite convincing. Especially the ring. Using the "transfers energy" definition of work, someone could say that the magnetic field does work because it transfers energy from rotational KE to translational KE, but that is quite a stretch since the system with the rotational KE is the same as the system with the translational KE.

 

[Miyz]

Hmm, I found the classical part of the paper quite convincing. Especially the ring. Using the "transfers energy" definition of work, someone could say that the magnetic field does work because it transfers energy from rotational KE to translational KE, but that is quite a stretch since the system with the rotational KE is the same as the system with the translational KE.

Now what's you're conclusion on magnetic fields/forces? They do work? Have energy stores within them?

Whats you're new conclusion after check out the links :) ?

For everyone who was/is involved in this topic what are you're FINAL conclusions? (I'd like to see where we are all now).

 

[Dale]

I am not ready to make a conclusion at this time. I am not sure that my superconductor example is wrong, but I am not sure it was right now either. I had not considered any change to the internal energy of the superconductor.

 

[Miyz]

I am not ready to make a conclusion at this time. I am not sure that my superconductor example is wrong, but I am not sure it was right now either. I had not considered any change to the internal energy of the superconductor.

Then I'll be waiting for that conclusion. Because so far I agree with you're example and you're inputs honestly because it makes a lot of sense.

 

[Miyz]

+ Would like to add.

When you all would say that its about energy, Break that down, its about the ability of doing work break that down, its a force or in our case "forces" within a distance.
Our power source : Battery, Grid, Etc... Supplies forces to interact with the magnetic force that will result in: Motion, torque, work, energy transfer etc...

Same thing with the tractor being our main "Source" of force supplying it to the rope and the weight would be moved. Work is done + energy is conserved.

 

[Miyz]

Anyone?
(conclusions about magnetic fields/force doing work + stored energy).

 

[vanhees71]

vanhees71. There's no argument about the power equation is there? Of course it is right.

If there's no argument about this really fundamental equation, then what the heck is this debate about? This formula clearly shows that only the electric field "does work". Of course you can rewrite the current density in terms of the magnetic field and the displacement current, using the Ampere-Maxwell Law, but that doesn't mean that the magnetic field does work on the charges, which are clearly represented by the current density in the simple equation P=\int \mathrm{d}^3 \vec{x} \; \vec{E} \cdot \vec{j}.

 

[Miyz]

If there's no argument about this really fundamental equation, then what the heck is this debate about? This formula clearly shows that only the electric field "does work". Of course you can rewrite the current density in terms of the magnetic field and the displacement current, using the Ampere-Maxwell Law, but that doesn't mean that the magnetic field does work on the charges, which are clearly represented by the current density in the simple equation P=\int \mathrm{d}^3 \vec{x} \; \vec{E} \cdot \vec{j}.

Um... Magnetic field's can't do work based on that law? How?

 

[vanhees71]

The formula simply says that's solely the electric components of the electromagnetc fields which do work on charges. How else do you interpret this equation?

 

[Miyz]

The formula simply says that's solely the electric components of the electromagnetc fields which do work on charges. How else do you interpret this equation?

Isn't there a difference between a charge,electric charge, and a current carrying loop...
Whats the name of this law? I'd like to study it.

 

[Miyz]

Please bring me something that opposes the idea that magnets magnetic field/force can't do work on a current carrying loop!

Not a single charge!

+ A current carrying loop is a dipole! It can't do work with another dipole in our case a permanent magnet's dipole.

 

[cabraham]

The formula simply says that's solely the electric components of the electromagnetc fields which do work on charges. How else do you interpret this equation?

In a vacuum I'd agree with you. If an e- (electron) has a velocity & a mag field is present, then a Lorentz force acts on the e- in a direction normal to its present velocity, & normal to B. Thus a mag field can change an e- momentum value, but not its kinetic energy value. Hence a mag field does no work on a charge. Fair enough?

Now we have a current loop, 2 of them in fact. Mag field 1, or B₁, exerts a force on the electrons in loop 2, normal to their velocity. In accordance with the above, B does not alter the e- energy value, only its momentum value, by changing the e- direction. But as these e- move in a new direction, the remaining lattice protons get yanked along due to E force tethering. But did the E actually do the work? The stationary lattice was moved acquiring non-zero KE (kinetic energy) when it started at zero KE.

Likewise, the neutrons got yanked along by strong nuclear force, which tethers the n0 (neutron) to the p+ (proton). A mag force in a direction normal to the loop deflects e- in a radial direction, resulting in p+ & n0 getting yanked radially. The force due to B accounts for all motion & work. But B cannot act on p+ as they are stationary, nor on n0 since the are charge-less. Did E do the work? E cannot act on n0 since they are charge-less. Did SNF do the work? SNF does not act on e-.

The work done by E appears to me a near zero. E exerts force no doubt, but when integrated with distance I compute zero. The E force between e- & p+ does move the p+, but the e-/p+ system energy is not changing. If an E force changed the distance between e- & p+, then E did work. Likewise for SN force.

I don't think we can say that "E did the work". If so, please draw E, & compute the distances over which E force acts. Explain your position, instead of just making bold proclamations. B is the prime mover, but would be powerless w/o E & SN forces.

When an electromagnet lifts a car a similar scenario takes place. The magnet applies force to the ferrous material in the car. But the tires, upholstery, etc., are non-ferrous. B does 0 work on these materials. But their weight plus the ferrous material weight is provided by B. B cannot lift tires, but applies enough force to the steel to lift the tires which are tethered to the steel by E & SN forces.

E & SN did no work, B did. But B cannot lift a pile of tires & upholstery. If I erred, please show me specifically. Don't waste our time with "E did the work, not B", w/o explaining the details. I await a detailed scientific reply. Best regards.

Claude

 

[chingel]

When talking about the simplified approach, the B field does not do work on any moving charged particle and the current carrying wire is just a bunch of moving charged particles and a permanent magnet is just a complicated system of moving electrons which can be modeled as an electromagnet. So the magnetic field cannot do any work on the electrons in the wire, it only changes the electrons paths. The electrons themselves do the work on the wire, using their kinetic energy. This means that the electrons get slowed down. The EM and nuclear and what not forces transfer the kinetic energy of the electrons to the wire.

Consider a bunch of electrons moving in the wire in the horizontal direction. A very strong magnetic field is applied briefly, all the electrons now move vertically. They move until they reach the end of the wire, which they cannot escape, EM forces hold them in the wire, in the process momentum is transferred and the wire starts moving. You can also consider the center of mass frame, electrons moving one way and the wire the other way and EM attraction brings them both to an halt, which leads to the usual interpretation that EM forces did the work of transferring kinetic energy and the energy source was whatever made the electrons move in the first place (battery etc).

Of course there are actually various quantum considerations to deal with with real electrons and other phenomena (how it works in a superconductor etc), which I won't pretend to understand, nor the link posted earlier. But I think it is still true that a magnetic field cannot do work on an isolated moving electron, which also has a lot of quantum weirdness (no exact position, jumps from here to there etc), however I cannot help you in trying to understand the more complicated scenarios that use quantum theory, and how relevant is the simplified approach in that light.

 

[cabraham]

When talking about the simplified approach, the B field does not do work on any moving charged particle and the current carrying wire is just a bunch of moving charged particles and a permanent magnet is just a complicated system of moving electrons which can be modeled as an electromagnet. So the magnetic field cannot do any work on the electrons in the wire, it only changes the electrons paths. The electrons themselves do the work on the wire, using their kinetic energy. This means that the electrons get slowed down. The EM and nuclear and what not forces transfer the kinetic energy of the electrons to the wire.

Consider a bunch of electrons moving in the wire in the horizontal direction. A very strong magnetic field is applied briefly, all the electrons now move vertically. They move until they reach the end of the wire, which they cannot escape, EM forces hold them in the wire, in the process momentum is transferred and the wire starts moving. You can also consider the center of mass frame, electrons moving one way and the wire the other way and EM attraction brings them both to an halt, which leads to the usual interpretation that EM forces did the work of transferring kinetic energy and the energy source was whatever made the electrons move in the first place (battery etc).

Of course there are actually various quantum considerations to deal with with real electrons and other phenomena (how it works in a superconductor etc), which I won't pretend to understand, nor the link posted earlier. But I think it is still true that a magnetic field cannot do work on an isolated moving electron, which also has a lot of quantum weirdness (no exact position, jumps from here to there etc), however I cannot help you in trying to understand the more complicated scenarios that use quantum theory, and how relevant is the simplified approach in that light.

Ref bold, sorry but a current loop is MORE THAN just a bunch of moving charges. It has a fixed lattice structure, protons & neutrons tethered by E & SN forces. It is B that does the work. B exerts a force yanking on the e-, but due to E & SN force tethering the lattice structure, the whole wire is moved. All of the force must come from B. Although E & SN transferred force, they do no work. Every Newton of force coupled by E & SN are matched by B. The B force moves the current loop through some distance. The integral of the B force times the incremental distance is the work done.

If B isn't doing the work, what is? It cannot be E. First of all, E provides force but no distance. The integral of E over the distance is zero. E is the force between p+ & e-. Moving one or both of these particles by E force resulting in their separation being changed is required for E to do work. Also, E does no work on neutrons. The theory that "E does all the work" holds water like a net.

Again, you cannot treat a loop with current as a mere collection of individual charges. There is a lattice held together by E & SN forces. Let me ask you about the electromagnet raising a car from the previous post of mine. What lifts the car? A 1000 kg car is raised 1 meter resulting in work of 9,806 N-m. What did the work, B, E, or SN?

Only B makes sense. I know B cannot do work on free electrons, nor on stationary protons, nor on neutrons. But B can yank on a lattice as I've described above, over a distance resulting in work being done. We seem to have reached a point where one side has demonstrated their case in detail, & the other side is simply in denial while offering nothing but declarations w/o support.

E force cannot be what is doing the work. If it really is, then show me an illustration with the direction of the E vector, & the path of integration. Otherwise, you have nothing.

Claude

 

[Miyz]

In a vacuum I'd agree with you. If an e- (electron) has a velocity & a mag field is present, then a Lorentz force acts on the e- in a direction normal to its present velocity, & normal to B. Thus a mag field can change an e- momentum value, but not its kinetic energy value. Hence a mag field does no work on a charge. Fair enough?

Now we have a current loop, 2 of them in fact. Mag field 1, or B₁, exerts a force on the electrons in loop 2, normal to their velocity. In accordance with the above, B does not alter the e- energy value, only its momentum value, by changing the e- direction. But as these e- move in a new direction, the remaining lattice protons get yanked along due to E force tethering. But did the E actually do the work? The stationary lattice was moved acquiring non-zero KE (kinetic energy) when it started at zero KE.

Likewise, the neutrons got yanked along by strong nuclear force, which tethers the n0 (neutron) to the p+ (proton). A mag force in a direction normal to the loop deflects e- in a radial direction, resulting in p+ & n0 getting yanked radially. The force due to B accounts for all motion & work. But B cannot act on p+ as they are stationary, nor on n0 since the are charge-less. Did E do the work? E cannot act on n0 since they are charge-less. Did SNF do the work? SNF does not act on e-.

The work done by E appears to me a near zero. E exerts force no doubt, but when integrated with distance I compute zero. The E force between e- & p+ does move the p+, but the e-/p+ system energy is not changing. If an E force changed the distance between e- & p+, then E did work. Likewise for SN force.

I don't think we can say that "E did the work". If so, please draw E, & compute the distances over which E force acts. Explain your position, instead of just making bold proclamations. B is the prime mover, but would be powerless w/o E & SN forces.

When an electromagnet lifts a car a similar scenario takes place. The magnet applies force to the ferrous material in the car. But the tires, upholstery, etc., are non-ferrous. B does 0 work on these materials. But their weight plus the ferrous material weight is provided by B. B cannot lift tires, but applies enough force to the steel to lift the tires which are tethered to the steel by E & SN forces.

E & SN did no work, B did. But B cannot lift a pile of tires & upholstery. If I erred, please show me specifically. Don't waste our time with "E did the work, not B", w/o explaining the details. I await a detailed scientific reply. Best regards.

Claude

Ref bold, sorry but a current loop is MORE THAN just a bunch of moving charges. It has a fixed lattice structure, protons & neutrons tethered by E & SN forces. It is B that does the work. B exerts a force yanking on the e-, but due to E & SN force tethering the lattice structure, the whole wire is moved. All of the force must come from B. Although E & SN transferred force, they do no work. Every Newton of force coupled by E & SN are matched by B. The B force moves the current loop through some distance. The integral of the B force times the incremental distance is the work done.

If B isn't doing the work, what is? It cannot be E. First of all, E provides force but no distance. The integral of E over the distance is zero. E is the force between p+ & e-. Moving one or both of these particles by E force resulting in their separation being changed is required for E to do work. Also, E does no work on neutrons. The theory that "E does all the work" holds water like a net.

Again, you cannot treat a loop with current as a mere collection of individual charges. There is a lattice held together by E & SN forces. Let me ask you about the electromagnet raising a car from the previous post of mine. What lifts the car? A 1000 kg car is raised 1 meter resulting in work of 9,806 N-m. What did the work, B, E, or SN?

Only B makes sense. I know B cannot do work on free electrons, nor on stationary protons, nor on neutrons. But B can yank on a lattice as I've described above, over a distance resulting in work being done. We seem to have reached a point where one side has demonstrated their case in detail, & the other side is simply in denial while offering nothing but declarations w/o support.

E force cannot be what is doing the work. If it really is, then show me an illustration with the direction of the E vector, & the path of integration. Otherwise, you have nothing.

Claude

BRAVO! :!) BRAAAVO!

The best answer so far the SHUTS every thing down! I totally agree again and again with Claude! Well said there sir!

Common sense everyone: Bring a loop connect it to a battery = nothing, Bring a magnet = MOTION!

Also! magnetic force on the wire = IL x B!
Magnets do work on this system and its all because of the INPUT POWER(battery etc...)
Again if you do say that magnets do no work please bring something NEW to the table! To support you're claim!

Thanks again everyone for you're efforts! Good discussion!

 

[vanhees71]

I can only repeat that Maxwell's equations hold in a very large range of applicability. QED effects are negligible in everyday applications, and Maxwell's equations clearly say that the power (work per time) done on charge distributions by the electromagnetic field is given by
P=\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} \; \vec{E}(t,\vec{x}) \cdot \vec{j}(t,\vec{x}).
Note that the current also contains the effects of magnetization through the corresponding part \vec{j}_{\text{mag}}=\vec{\nabla} \times \vec{M}.

I do not need to copy the already cited very well written paper, where it has been demonstrated on simple examples that magnetic fields do no work as predicted by Maxwell electromagnetics. It is also demonstrated that this picture also applies to the pure quantum phenomenon spin and the corresponding magnetic moment within semiclassical Dirac theory (semiclassical here means that the electron is treated as a quantum particle and the em. field as classical, an approximation valid for the nonrelativistic realm of the electron's motion, i.e., in atomic, molecular and solid-state physics for not too large charge numbers of the involved atomic nuclei). I take the freedom to cite this paper again, including the abstract, which already explains it very clearly:

PHYSICAL REVIEW E 77, 036609 (2008)
Dipole in a magnetic field, work, and quantum spin
Robert J. Deissler*

Physics Department, Cleveland State University, Cleveland, Ohio 44114, USA
͑Received 28 February 2007; published 21 March 2008

The behavior of an atom in a nonuniform magnetic field is analyzed, as well as the motion of a classical magnetic dipole ͑a spinning charged ball and a rotating charged ring. For the atom it is shown that, while the magnetic field does no work on the electron-orbital contribution to the magnetic moment ͑the source of translational kinetic energy being the internal energy of the atom, whether or not it does work on the electron-spin contribution to the magnetic moment depends on whether the electron has an intrinsic rotational kinetic energy associated with its spin. A rotational kinetic energy for the electron is shown to be consistent with the Dirac equation. If the electron does have a rotational kinetic energy, the acceleration of a silver atom in a Stern-Gerlach experiment or the emission of a photon from an electron spin flip can be explained without requiring the magnetic field to do work. For a constant magnetic field gradient along the z axis, it is found that the classical objects oscillate in simple harmonic motion along the z axis, the total kinetic energy—translational plus rotational—being a constant of the motion. For the charged ball, the change in rotational kinetic energy is associated only with a change in the precession frequency, the rotation rate about the figure axis remaining constant.

DOI: 10.1103/PhysRevE.77.036609

 

[Dale]

I do not need to copy the already cited very well written paper, where it has been demonstrated on simple examples that magnetic fields do no work as predicted by Maxwell electromagnetics.

Thanks for posting that paper. I have gone over it quite a bit and found it very persuasive. Here is my current thought process:

1) Let's use the definition of work as energy transferred to or from a system by any mechanism other than heat.
2) Only external forces can do work on a system since internal forces cannot transfer energy in or out of the system.
3) A system's KE may change without work being done on the system, provided there is some compensatory change in some other form of energy for the system. (this is what I neglected in my example)
4) If the paper represents some specific examples of a general principle, then in all situations where the magnetic force is the only external force, any change in KE must be accompanied by a corresponding change in some other internal form of energy.

So, in my example, an external magnetic field can accelerate (increase KE) a superconducting loop. This must be accompanied by a decrease in internal energy. The only available energy is the energy density of the magnetic field, which depends only on the current. Therefore, the current in the loop must decrease as the loop accelerates. Although I didn't calculate it explicitly, this makes sense to me.

A motor is easy to explain since the magnetic field is not the only source of energy transfer.

However, the one thing that makes me hesitate to adopt this principle wholeheartedly is that it is not always clear what internal energy is being used. For example, consider a permanent magnet being accelerated in an external magnetic field. What is the internal energy that is being used in the permanent magnet? Any ideas?