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

I'm just confused...

I know that magnetic fields can do work only on pure magnetic dipoles like a bar magnet. Based on the formula, the magnetic force on a charge is qv⃗ ×B⃗ which is identically perpendicular to v⃗ and that's why it does no work. However, forces on magnetic dipoles and more general objects don't have the form v⃗ × - they're not perpendicular to v⃗ , so they do work in general.

But still when I look at this picture I get confused:

In this picture aren't magnetic force causing the rotation of the loop! Aren't the magnetic forces in a motor on of the key! Factors of motion within it? I mean it makes no sense to me why in this cause magnetic force can't do work on an electric charge... Could someone pelase help me out with this!
 The only explanation that can come to my mind is... The electric charge creates a magnetic field and the permanent magnet has already a magnetic filed and thus opposing forces will be apply forces and work will be done by both the electric charged and the magnetic fileld. As stated by Lorentz law on a current carrying wire: F = IL x B , B = 0 , F = 0 , total work done would equal = zero, simple logic about this matter if you make a small loop and connect it to a battery nothing would happen other then flow of current and a magnetic field is generated by that flow, put a permanent magnet into that state and voila! Motion!

 Quote by Miyz I'm just confused... I know that magnetic fields can do work only on pure magnetic dipoles like a bar magnet. Based on the formula, the magnetic force on a charge is qv⃗ ×B⃗ which is identically perpendicular to v⃗ and that's why it does no work. However, forces on magnetic dipoles and more general objects don't have the form v⃗ × - they're not perpendicular to v⃗ , so they do work in general. But still when I look at this picture I get confused: In this picture aren't magnetic force causing the rotation of the loop! Aren't the magnetic forces in a motor on of the key! Factors of motion within it? I mean it makes no sense to me why in this cause magnetic force can't do work on an electric charge... Could someone pelase help me out with this!
You did not include the internal forces in the wire. The internal forces that keep the wire together are actually doing all the work. The wire loop is acting like a rigid body. Rigid bodies are held together by internal forces. The internal forces in the wire are doing the work, not the magnetic field of the bar magnet.
The rigid loop is doing work on the commutator. The magnetic field isn't doing work on the commutator. Therefore, the "key" forces are the internal forces of that loop of wire. The internal forces both keep the wire rigid and keep the electric charges that carry the current in the wire.
The magnetic field isn't directly causing the torque. The contact forces between the electric current and the rest of the wire is directly causing the torque.
The wire of the loop has two components. One is the free electrons in the wire. The other is atoms of the wire, including both bound electrons and nuclei.
Think of a single conduction electron in the copper wire. The copper wire is initially still in the laboratory frame. The electron initially moves perpendicular to the magnetic field of the bar magnet. The magnetic field causes a force that is perpendicular to the electrons motion. No work is done by the magnetic field, as you said.
The electron can't leave the copper wire. When the wire gets to the edge of the copper wire, it has to slow down. The edge of the copper wire applies a force to the electron which stops it from jumping into the air. One could describe this as a contact force (i.e., a sharp wall). Or maybe one would prefer to think of it as electrostatic attraction of the electron and the positive ions in the copper wire. In any case, a force is applied to the electron by the edge of the wire that slows the electron down.
For every action, there is a reaction (Newton's third law). When the electron nears the edge of the copper wire, it applies a force on the edge that is equal and opposite the force that the edge applies on the electron. Hence the electron is applying a force to the wire.
The force of the electrons on the wire is what causes the torque. The force is not directly caused by the bar magnet. If the electrons were not bound by the wire, then there would be no work done. However, the wire keeps the electrons in place. The forces that keep the electrons in place are not from the bar magnet. The forces that keep the electrons in place have to be accounted for in some way.
You made an identification between the electric charges, the electric current and the wire. The electric charges carry the current. The electric charges that carry the current are only part of the wire. The rest of the wire is necessary to keep the electrons flowing on the path of the wire. The magnetic field of the bar magnet does no work on the electric charges that make up the current.
When your instructor said that magnetic fields can't do work on the current, he was literally correct. However, he did you a bit of a disservice. This type of rule has to be used with caution. The path of the electric current in a wire is constrained by the shape of the wire. Therefore, the forces that keep the electric current in the wire can't be ignored. Constraints imply forces that aren't always explicit in the wording of the problem.

Lesson for engineers: Always take into account the constraints, because the constraints imply internal forces.
Lesson for physicists: Boundary conditions are always important, especially those caused by constraints.

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

 Quote by Darwin123 You did not include the internal forces in the wire. The internal forces that keep the wire together are actually doing all the work. The wire loop is acting like a rigid body. Rigid bodies are held together by internal forces. The internal forces in the wire are doing the work, not the magnetic field of the bar magnet. The rigid loop is doing work on the commutator. The magnetic field isn't doing work on the commutator. Therefore, the "key" forces are the internal forces of that loop of wire. The internal forces both keep the wire rigid and keep the electric charges that carry the current in the wire. The magnetic field isn't directly causing the torque. The contact forces between the electric current and the rest of the wire is directly causing the torque. The wire of the loop has two components. One is the free electrons in the wire. The other is atoms of the wire, including both bound electrons and nuclei. Think of a single conduction electron in the copper wire. The copper wire is initially still in the laboratory frame. The electron initially moves perpendicular to the magnetic field of the bar magnet. The magnetic field causes a force that is perpendicular to the electrons motion. No work is done by the magnetic field, as you said. The electron can't leave the copper wire. When the wire gets to the edge of the copper wire, it has to slow down. The edge of the copper wire applies a force to the electron which stops it from jumping into the air. One could describe this as a contact force (i.e., a sharp wall). Or maybe one would prefer to think of it as electrostatic attraction of the electron and the positive ions in the copper wire. In any case, a force is applied to the electron by the edge of the wire that slows the electron down. For every action, there is a reaction (Newton's third law). When the electron nears the edge of the copper wire, it applies a force on the edge that is equal and opposite the force that the edge applies on the electron. Hence the electron is applying a force to the wire. The force of the electrons on the wire is what causes the torque. The force is not directly caused by the bar magnet. If the electrons were not bound by the wire, then there would be no work done. However, the wire keeps the electrons in place. The forces that keep the electrons in place are not from the bar magnet. The forces that keep the electrons in place have to be accounted for in some way. You made an identification between the electric charges, the electric current and the wire. The electric charges carry the current. The electric charges that carry the current are only part of the wire. The rest of the wire is necessary to keep the electrons flowing on the path of the wire. The magnetic field of the bar magnet does no work on the electric charges that make up the current. When your instructor said that magnetic fields can't do work on the current, he was literally correct. However, he did you a bit of a disservice. This type of rule has to be used with caution. The path of the electric current in a wire is constrained by the shape of the wire. Therefore, the forces that keep the electric current in the wire can't be ignored. Constraints imply forces that aren't always explicit in the wording of the problem. Lesson for engineers: Always take into account the constraints, because the constraints imply internal forces. Lesson for physicists: Boundary conditions are always important, especially those caused by constraints.
Ok, I understood that the force within the wire are the main cause of torque and for work being done. I agree. However, whats a bar magnets role in this?

Its presence is crucial! A loop of wire can not generate a "twist" torque. Its the repulsion/attraction of the magnetic field don't you agree?

(Forgive me if I didn't understand you properly but I'm confused and still find the presence and use of bar magnets is crucial for the "motor" effect to occur)
 Noticed that this is not a really popular topic around the forums huh? Also noticed its a very very misunderstood field... Seriously! I looked around here and there... No simple and common answer about this. I can say its not understood properly and strangely everyone is confused from other sites and countless sources... No one really knows what exactly does work in a motor... One would say:"well its electricity?" okay, bring a loop of wire and wait for it to spin... NOTHING happens. Bring a small tiny magnet and all of a sudden its turning so fast! I could say that magnets are doing something and thats repelling/attracting the loop because the flow of current in it! Is generating a magnetic field the loop itself is like a magnet that can attract/repel... This site makes most sense to me...
 http://www.physicsforums.com/showthr...=347539&page=4 The above link is worth reading. Several posters put forth good info. I emphasized the tethering concept which I can elaborate upon if needed. Claude
 'The electron can't leave the copper wire. When the wire gets to the edge of the copper wire, it has to slow down. The edge of the copper wire applies a force to the electron which stops it from jumping into the air. One could describe this as a contact force (i.e., a sharp wall). Or maybe one would prefer to think of it as electrostatic attraction of the electron and the positive ions in the copper wire. In any case, a force is applied to the electron by the edge of the wire that slows the electron down.' If there is an electrostatic attraction between the electrons and the positive charges then there is no resultant force on the wire. As you say this force is an 'internal' force and cannot be the explanation for any external force causing the wire to move. I don' think the electron gets to the edge of the wire and has to 'slow down' which stops it 'jumping into the air' There are several points you should consider. Have you heard of the 'Hall effect'? Can you quote a text book or other reference that backs up you explanation?

 Quote by Darwin123 You did not include the internal forces in the wire. The internal forces that keep the wire together are actually doing all the work. The wire loop is acting like a rigid body. Rigid bodies are held together by internal forces. The internal forces in the wire are doing the work, not the magnetic field of the bar magnet. The rigid loop is doing work on the commutator. The magnetic field isn't doing work on the commutator. Therefore, the "key" forces are the internal forces of that loop of wire. The internal forces both keep the wire rigid and keep the electric charges that carry the current in the wire. The magnetic field isn't directly causing the torque. The contact forces between the electric current and the rest of the wire is directly causing the torque. The wire of the loop has two components. One is the free electrons in the wire. The other is atoms of the wire, including both bound electrons and nuclei. Think of a single conduction electron in the copper wire. The copper wire is initially still in the laboratory frame. The electron initially moves perpendicular to the magnetic field of the bar magnet. The magnetic field causes a force that is perpendicular to the electrons motion. No work is done by the magnetic field, as you said. The electron can't leave the copper wire. When the wire gets to the edge of the copper wire, it has to slow down. The edge of the copper wire applies a force to the electron which stops it from jumping into the air. One could describe this as a contact force (i.e., a sharp wall). Or maybe one would prefer to think of it as electrostatic attraction of the electron and the positive ions in the copper wire. In any case, a force is applied to the electron by the edge of the wire that slows the electron down. For every action, there is a reaction (Newton's third law). When the electron nears the edge of the copper wire, it applies a force on the edge that is equal and opposite the force that the edge applies on the electron. Hence the electron is applying a force to the wire. The force of the electrons on the wire is what causes the torque. The force is not directly caused by the bar magnet. If the electrons were not bound by the wire, then there would be no work done. However, the wire keeps the electrons in place. The forces that keep the electrons in place are not from the bar magnet. The forces that keep the electrons in place have to be accounted for in some way. You made an identification between the electric charges, the electric current and the wire. The electric charges carry the current. The electric charges that carry the current are only part of the wire. The rest of the wire is necessary to keep the electrons flowing on the path of the wire. The magnetic field of the bar magnet does no work on the electric charges that make up the current. When your instructor said that magnetic fields can't do work on the current, he was literally correct. However, he did you a bit of a disservice. This type of rule has to be used with caution. The path of the electric current in a wire is constrained by the shape of the wire. Therefore, the forces that keep the electric current in the wire can't be ignored. Constraints imply forces that aren't always explicit in the wording of the problem. Lesson for engineers: Always take into account the constraints, because the constraints imply internal forces. Lesson for physicists: Boundary conditions are always important, especially those caused by constraints.
Ref bold quote - this is OOTA science (out of thin air). Where did you get this? Source/reference please. "The edge applies a force" to the e-" Indeed? The edge, just what is "the edge"? Atomic nuclei at the outer layer, free electrons at outer layer, just what, may I ask, is this "edge" which applies a force?

Although there are inner forces in the wire which "hold" the electron preventing it from jumping away, these forces do not impart motion to the wire. I explained the motion, how it is derived, along with others, in my link found in my previous post. If you wish to discuss how the force on the wire is generated, please review the thread I linked to, then feel free to raise questions, or challenge my position, or that of others.

Also, not to be personal, but what is the extent of your background regarding e/m fields, energy conversion (transformers, motors, generators, relays, etc.), relativity, quantum mechanics, etc.? This forum is a good one, expecting posters to provide proof, theory, empirical measurement, etc. as support for what they state. To simply state "this is how it really works" without anything to back it up, is nothing but dust in the wind. Again, I do not question your intelligence, but I ask just how you know that what you claim is true, when it runs counter to established motor/generator/e/m fields theory, practice, & observation? Best regards.

Claude

 Quote by cabraham Ref bold quote - this is OOTA science (out of thin air). Where did you get this? Source/reference please. "The edge applies a force" to the e-" Indeed? The edge, just what is "the edge"? Atomic nuclei at the outer layer, free electrons at outer layer, just what, may I ask, is this "edge" which applies a force? Although there are inner forces in the wire which "hold" the electron preventing it from jumping away, these forces do not impart motion to the wire. I explained the motion, how it is derived, along with others, in my link found in my previous post. If you wish to discuss how the force on the wire is generated, please review the thread I linked to, then feel free to raise questions, or challenge my position, or that of others. Also, not to be personal, but what is the extent of your background regarding e/m fields, energy conversion (transformers, motors, generators, relays, etc.), relativity, quantum mechanics, etc.? This forum is a good one, expecting posters to provide proof, theory, empirical measurement, etc. as support for what they state. To simply state "this is how it really works" without anything to back it up, is nothing but dust in the wind. Again, I do not question your intelligence, but I ask just how you know that what you claim is true, when it runs counter to established motor/generator/e/m fields theory, practice, & observation? Best regards. Claude
My background in e/m fields, electronics, relativity and quantum mechanics is pretty strong. I have added some information on my personal profile, open to registered users of this forum. I have not included any information specific enough to track me. You can believe what you want about me. Whatever you think, I have heard worse.
What I wrote is somewhat straightforward for physicists and engineers. I don't think that I have to provide empirical evidence for a simple idea. I was referring to your own diagram and your own words. I think that your diagram is not completely labeled. You did not label the relevant forces, their direction or the point they are acting on. You also did not mention the relevant physical constraints, such as the current having to follow the rigid wire. You also didn't what object the work is being done on.
In my coursework and experience, one can't have a physical constraint without a corresponding force or a force equivalent. One can "eliminate" the force using a change of variable. This leads to the concept of generalized forces. However, you one can't eliminate a physical constraint by ignoring it.
The charge carrier in the circuit are physically constrained to the wire by some type of force, the details of which are unimportant in your problem. The important point here is that the charge carriers are not the entire wire. The wire also contains a medium that maintains the circuit. The wire, carrier and medium, constitutes a rigid body.The wire itself is physically constrained to keep its shape by rigid body forces. Unless there was some type of force on the charge carrier, other than the magnetic field, the charge carrier would leap from the wire. Thus, the path of the carrier is constrained by internal forces.
I am using the phrase "charge carrier" to distinguish it from free electrons. According to solid state physics, there are all sorts of carriers that may be carrying the charge. Fortunately, it doesn't matter precisely what the charge carrier is. The charge carrier is traveling in a metallic medium that may be crystalline. Fortunately, I doesn't matter precisely what this medium is. It can be amorphous, or a polymer. All that matters is that there is a force between the medium and the carrier such that the carrier can't leave the wire.
The magnetic field can create a force on the medium, merely by applying a force to the carrier. The magnetic field pushes the carrier, which pushes the medium. However, the direction of the force exerted by the medium on the carrier is in the opposite direction of the direction of motion of that carrier. Therefore, the force keeping that carrier in the medium is doing work on the carrier.
Repeat: The direction of the force of the medium on the carrier is not the same as the direction of the magnetic force on the carrier. The medium can do work on the carrier. Therefore, the carrier can do work on the medium.
Your diagram shows a rigid loop of wire on an axle rotating in a magnetic field. The work is being done by the rigid loop of wire on a wheel that you labelled the commutator.
The internal forces of a rigid body cancel out. The wire, carriers and medium, is rigid. Therefore, you can ignore the internal forces only if you treat the wire as one rigid body. A free carrier in the wire actually has two forces on it: the force of the magnetic field and the force of the medium. So unless you want to analyze the details of the medium, you have to treat the entire wire as a single object. You can't look at the carrier as somehow being isolated from the medium.
If you are asking about the work being done on the commutator, then you should draw the commutator by itself as well as the force vectors that are working on it. If you do so, then you will notice that the magnetic field isn't even touching the commutator. Therefore, the magnetic field can't be doing work on the commutator.
The only force on the commutator is a contact force coming from the rigid loop of wire. The rigid wire contains a substrate and the charge carrier, whatever it is.
If you want to know about the work done on the loop of wire, draw the force diagram of the loop of wire by itself without the magnet and without the commutator. Then draw the forces. There is the force of the magnet on the left side of the wire, the force of the magnet on the right side of the wire, the force of the commutator on the right side of the wire, and the force of the commutator on the left side of the wire.
The rigid loop of wire is at equilibrium. Please note that at equilibrium, the total force on the rigid loop has to be zero and the total torque on the rigid loop has to be zero.
The external forces are the contact forces on the commutator and the force of the magnetic field on the free carrier. As I mentioned, the force of the free carrier on the medium is an internal force. It cancels out the force of the medium on the free carrier. and so can be ignored. Therefore, the strength of the torque (and force) of the commutator has to be equal and opposite in sign to the torque (and force) of the magnetic field.
Your teacher told you that a static magnetic field can't do work on a current. He should have said that the static magnetic field can't do work on a electric charge. However, that doesn't mean that the electric charge can't do work on another electric charge in its vicinity.
What you asked about is the work done on macroscopic systems. Like the wire. Like the commutator. You did not ask what work was done on the charge carrier. However, here is the answer. Work is being done on the charge carrier by the medium that it is embedded in. The charge carrier is being forced by the medium to move in a different direction than it would if only the magnetic field was acting on it. Therefore, the magnetic field is indirectly doing work on the medium.

 Quote by Darwin123 My background in e/m fields, electronics, relativity and quantum mechanics is pretty strong. I have added some information on my personal profile, open to registered users of this forum. I have not included any information specific enough to track me. You can believe what you want about me. Whatever you think, I have heard worse. --- --- --- --- --- Therefore, the magnetic field is indirectly doing work on the medium.
Darwin you quoted me, then obviously replied to someone else. I asked you to review my link from a previous thread, which I gave and here it is again:

http://www.physicsforums.com/showthr...=347539&page=4

I just want to know what you have to say about the linked comments. I believe we examined this issue quite thoroughly.

Claude

 Quote by Miyz I'm just confused... I know that magnetic fields can do work only on pure magnetic dipoles like a bar magnet. Based on the formula, the magnetic force on a charge is qv⃗ ×B⃗ which is identically perpendicular to v⃗ and that's why it does no work. However, forces on magnetic dipoles and more general objects don't have the form v⃗ × - they're not perpendicular to v⃗ , so they do work in general. But still when I look at this picture I get confused: In this picture aren't magnetic force causing the rotation of the loop! Aren't the magnetic forces in a motor on of the key! Factors of motion within it? I mean it makes no sense to me why in this cause magnetic force can't do work on an electric charge... Could someone pelase help me out with this!
Here is a real simple explanation.
The way you drew it, the charge carriers are moving on a path that is perpendicular to the force of the magnet. This is true only when the loop of wire is standing still. When the loop of wire is standing still, the magnet is not doing work on the loop of wire.
When the wire is turning, the charge carrier is forced by the wire to move such that its velocity has a small component parallel to the force of the magnet. Therefore, the magnet is doing force on the charge carrier.
The charge carrier would have no component of velocity parallel to the magnetic force if there were no internal forces making the loop of wire rigid. The forces that keep the wire rigid are responsible for making the loop of wire move in circles. Thus, the medium of the wire is providing the centripetal force necessary to prevent the loop of wire from flying to pieces. It is the centripetal force on the charge carrier that causes the magnet to do work.
So basically, you should add a component of electric current that is tangential to the motion of the wire. The tangential component of velocity will be proportional to the tangential velocity of the the wire, and to the charge on the charge carrier.
The hypothesis that the velocity of the wire adds to the velocity of the charge carrier takes into account the internal forces that hold the wire rigid. As long as you calculate that velocity of the wire correctly, you don't need to draw the internal force on the charge carrier.
Of course, it works out in units. The contribution of force from the current is in units of current times length times magnetic field strength. The contribution of force from the tangential velocity of the wire is in units of charge times speed times magnetic field strength. They are the same units.
A real electric motor will take time to start. Until it starts spinning rapidly, it won't draw power effectively from the commutator. Until the loop starts moving, most of the energy is wasted as resistive heat energy.
The way that I just described it is physically equivalent to the description in the other post. In both posts, the thing that makes the magnet do work is the internal force that keeps the charge carrier moving in the wire.
What I forgot in the other rambling post is that the work formula uses the total velocity of the charge carrier. The velocity of the charge carrier in a wire is different from the velocity of a free electron. The motion of the wire adds to the motion of the charge carrier. Hence, you don't have to know the precise functional expression for the internal forces.
 Take the simple case of two parallel wires carrying currents in the same direction. They attract each other. If we allow them to move together, work is done on them (by definition of work). They will acquire KE. This energy will have come from the batteries urging the currents. This is because there is a back-emf due to the wires moving and cutting each others' flux lines. As the wires move together their charge carriers are experiencing motor effect forces opposing their motion forward through the wire. Extra work has to be supplied by the batteries to maintain the current. This work is 'passed on' as the work done on the wires as they are made to accelerate towards each other. The force (acting when the wires move towards each other) on the charge carriers in a direction parallel to the wire is made, by the wonders of the Motor Effect (Magnetic Lorentz force) to produce a force at right angles to the wire. It's not really the magnetic field which is doing work. It's a bit like a piece of string going over a quarter of the circumference of a pulley, and so bending through an angle of 90°. If we pull one end of the string, the other end can do work, but it's the puller, not the pulley, that supplies the work.
 Here is a simple mathematical treatment of what I was trying to say in my last post. Attached Thumbnails

 Quote by Philip Wood Take the simple case of two parallel wires carrying currents in the same direction. They attract each other. If we allow them to move together, work is done on them (by definition of work). They will acquire KE. This energy will have come from the batteries urging the currents. This is because there is a back-emf due to the wires moving and cutting each others' flux lines. As the wires move together their charge carriers are experiencing motor effect forces opposing their motion forward through the wire. Extra work has to be supplied by the batteries to maintain the current. This work is 'passed on' as the work done on the wires as they are made to accelerate towards each other. The force (acting when the wires move towards each other) on the charge carriers in a direction parallel to the wire is made, by the wonders of the Motor Effect (Magnetic Lorentz force) to produce a force at right angles to the wire. It's not really the magnetic field which is doing work. It's a bit like a piece of string going over a quarter of the circumference of a pulley, and so bending through an angle of 90°. If we pull one end of the string, the other end can do work, but it's the puller, not the pulley, that supplies the work.
Good. Consider two wires in parallel with parallel electric currents. Both currents are perpendicular to a static magnetic field.
You are suggesting a calculation of work using the KE and the EMF as intermediate quantities in the calculation. This is an entirely good way to solve the problem. However, the work can be calculated without doing an intermediate calculation of either KE or EMF. The work can be calculated using only the definition of work, the Lorentz force law, and the law of Biot and Savart.
This is more consistent with what the OP is asking. The OP seems to be questioning the consistency between the Lorentz force law and the conservation of energy. Students often learn about the Lorentz force law before they learn about EMF. The calculation of EMF requires one to know about time varying magnetic fields and inertial frames. This should not be absolutely necessary if there is an inertial frame where the magnetic field is static.
Consider the case of two parallel wires with parallel electric currents. The first wire, #1, exerts an attractive force on the second wire, #2, because of the magnetic field the current of #1 generates. The work on wire #2 can be calculated in the inertial frame where the magnetic field is static and homogeneous.
If the force is stationary and homogeneous, then the force time distance where the distance is measured in the direction of motion. If the force is inhomogeneous, an integral is needed. However, for small distances, one can consider the magnetic field effectively homogeneous and stationary.
If there is no motion, then there is no work. This is true regardless of how strong the force of attraction is. I will be more specific. If wire #2 is stationary, the wire #1 can not do any work on wire #2. Zero distance means zero work.
If wire #2 is not moving, then the electric charge carrier is moving in a direction perpendicular to the force created by the magnetic field of wire #1. If wire #2 is not moving, the force exerted by wire #1 can not do work on the charge carrier of wire #2.
You may say that wire #2 must move if wire #1 is acting on it. However, wire #2 can remain stationary if some force other than the magnet is holding it in place. For instance, the wire may be fastened to a wall. The contact force of the wall holds the wire still. Therefore, it is quite possible for wire #1 to do no work on wire #2 even though it exerts a force.
Now, unfasten wire #2 from the wall. Let wire #2 move toward wire #1. Look at the electric charge carrier in wire #2.
The electric charge carrier now has two components of motion instead of one. The electric charge carrier still is moving due to the electric current. However, the electric charge carrier is now being dragged by the rest of the wire #2 toward wire #1. Therefore, the velocity is no longer perpendicular to the force exerted by wire #1.
The is a component of velocity of the charge carrier in the direction of the motion. The force times the distance in the direction of motion is the work exerted on the charge carrier by wire #1.
So the body giving off the magnetic field, wire #1, does work on the charge carrier of wire #2. The magnetic field is doing work, but only if wire #2 is allowed to move.
The analog to electric motors should be obvious. Obviously, an electric motor can only do work on something if it is moving. As a consequence, an electric motor draws more power when it is turned on than when it is off. If you stop an electric motor from moving, then it will draw less power.
The Lorentz force law is consistent with the conservation of energy. As in most problems, one has a choice as to whether one solves the problem using dynamic variables (like force) or conserved quantities (like energy). The two approaches should be and are self consistent.
The question concerned the work done by an electric motor. I think the important point here is that the motion of the electric motor changes the motion of the charge carrier in the circuit. Therefore, the motion of the wire has to be taken into account. There is more than one way to take the motion of the wire into account. However, the motion of the circuit components can not be ignored.
One more thing. Someone here claimed that there is no internal force involved in the Hall effect. There are forces involved in the Hall effect that usually aren't discussed. When a voltage is developed via the Hall effect, there is a stress on the conductor caused by the magnetic field. If an aqueous solution of salt were used instead of a conductor, there would be stretching of the fluid caused by the magnetic field. However, the Hall effect is usually demonstrated with a solid metal. So the internal forces prevent the metal from stretching.
 Darwin123: I'm sorry that I didn't provide my thumbnail (post 13) at the same time as my qualitative (hand-waving) explanation (post 12). I think you'll find that post 13 does pretty much what's needed using just the Lorentz force (magnetic component).

 Quote by Philip Wood Darwin123: I'm sorry that I didn't provide my thumbnail (post 13) at the same time as my qualitative (hand-waving) explanation (post 12). I think you'll find that post 13 does pretty much what's needed using just the Lorentz force (magnetic component).
Yes, it does. Your handwritten equations in post 12 are precisely what I meant. I recommend that read that post to anyone who still has a question.
I don't yet know how to get equations and graphics onto my posts. I try to paint a picture with words.
Better yet, I would like to be able to modify the pictures and equations that people send. If all the velocity components on the charge carrier could be drawn into the illustration of the OP, it would be a lot clearer. One extra arrow on the current. showing the extra velocity component, would make the point a lot clearer.
 'The way you drew it, the charge carriers are moving on a path that is perpendicular to the force of the magnet. This is true only when the loop of wire is standing still. When the loop of wire is standing still, the magnet is not doing work on the loop of wire. When the wire is turning, the charge carrier is forced by the wire to move such that its velocity has a small component parallel to the force of the magnet. Therefore, the magnet is doing force on the charge carrier.' In practice an electric motor has a (uniform!!) radial field, how does this influence this part of the explanation?