# Can Magnetic Force Perform Work?

gabbagabbahey
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Work must now be done to move the electron sideways, as well as in the direction of the anode.

What does this extra work?

I'm not sure what you are asking here. Work is done by the second electric field, since the field will be parallel to some component of the electron's motion, and hence the speed and kinetic energy of the electron is changed by it.

Are you under the impression that there is some other work also being done (in addition to the work done by the two electric fields)?

I am under the impression that acting without the magnetic field active the second electric field would deflect the in one direction, but that when the magnetic field is active the deflection is actually in the opposite direction.

gabbagabbahey
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I am under the impression that acting without the magnetic field active the second electric field would deflect the in one direction, but that when the magnetic field is active the deflection is actually in the opposite direction.

That only affects the sign of the work done, not what agent does the work.

$$W=\int_{\mathcal{C}} \textbf{F}\cdot d\textbf{r}$$

If $\textbf{F}$ points in the opposite direction as $d\textbf{r}$, $\textbf{F}\cdot d\textbf{r}=-Fdr$.

Hi, gabbagabbahey and Born2bwire.
Yes, and as gabbagabbahey stated above,
Classically, we treat both the magnetic moment due to spin (intrinsic angular momentum) and the magnetic moment due to orbital motion of the electron's in an atom (orbital angular momentum) as ideal magnetic dipoles.
How do you consider what kind of force apply to your ideal magnetic dipoles and do work? In other words, is grad(mB) interpreted as or reduced to Lorenz force, electric force or other kind of force?
Regards.

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gabbagabbahey
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How do you consider what kind of force apply to your ideal magnetic dipoles and do work? In other words, is grad(mB) interpreted as or reduced to Lorenz force, electric force, magnetic force or other kind of force?
Regards.

It's regarded as a Lorentz force, on the same footing as the force of attraction between two long straight wires carrying a steady current. $\textbf{F}=\mathbf{\nabla}(\textbf{m}\cdot\textbf{B})$ is clearly capable of doing work, but the source of the work is not the external magnetic field. Just like with two long straight wires where the work is done by the current source which maintains the steady current in the wires (could be an electric force from a battery, could be a strong gravitational field, could even be nanoscopic ants carrying the electrons on their backs....it doesn't really matter), the source of the work in an ideal dipole is whatever maintains the current responsible for the dipole moment.

By that same token would work by the gravitational field be done by the mass that makes the field rather than the field? Why?

gabbagabbahey
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By that same token would work by the gravitational field be done by the mass that makes the field rather than the field?

No, work done by a gravitational field is done by the gravitational field. (A tautology, of course)

In a wire, or loop carrying a steady current, work must be done by the current source to maintain the steady current when the wire/loop is subjected to an external magnetic field. Currents are composed of moving charges, and the effect of the external magnetic field on those charges is to alter the direction of their motion (but not their speed, so no work is done by the magnetic field), which tends to change the current in wire/loop. In order to prevent the current from changing, another force must be present (whatever drives the steady current), and that force does work on the wire.

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Hi, gabbagabbahey.
the source of the work in an ideal dipole is whatever maintains the current responsible for the dipole moment.
Do you know what maintains the current responsible for the dipole moment in electron spin? In your theory that is the source of work in attraction between permanent magnets that are collected spins, right?
Regards.

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gabbagabbahey
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Do you know what maintains the current responsible for the dipole moment in electron spin?

No, classical physics doesn't answer this question (Although many unsatisfactory attempts were made before and after the advent of quantum mechanics). That doesn't affect the net force or torque on a dipole in an external field, though.

In your theory that is the source of work in attraction between permanent magnets, right?
Regards.

Its not my theory, but yes that is the source of the work according to classical electrodynamics.

Born2bwire
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It's regarded as a Lorentz force, on the same footing as the force of attraction between two long straight wires carrying a steady current. $\textbf{F}=\mathbf{\nabla}(\textbf{m}\cdot\textbf{B})$ is clearly capable of doing work, but the source of the work is not the external magnetic field. Just like with two long straight wires where the work is done by the current source which maintains the steady current in the wires (could be an electric force from a battery, could be a strong gravitational field, could even be nanoscopic ants carrying the electrons on their backs....it doesn't really matter), the source of the work in an ideal dipole is whatever maintains the current responsible for the dipole moment.

To further expand, consider the elementary properties of classical electrodynamics. There are two sources, currents and charges (and we often can substitute one for the other by use of an appropriate conservation relation). It is found that when we place such sources within proximity of one another, they experience a set of forces. The forces that the sources experience are described by the Lorentz force and the electric and magnetic fields. When we define the work done by the field, we do so in the capacity of the amount of work done by/against the field's force acting on a test charge. Again, this force is described by the Lorentz force. As such, the energy density of the fields is determined by the Lorentz force since this is the mediator by which the fields perform work. So when you talk about the force that arises due to the energy potential profile of the fields, you are still talking implicitly of the Lorentz force.

It is a simple mathematical exercise to confirm that the force resulting from the magnetic field is always normal to the path of displacement. This gives rise to the statement that the magnetic field does no work. As long as currents and charges are the only sources of electric and magnetic fields and the only means by which the fields interact on such sources is by the Lorentz force, then I maintain that there is not situation that can be built upon this foundation where the magnetic field does work. You can always obfuscate the physics by introducing a complicated problem, but the underlying physics does not change.

Another key point to consider is that the magnetic force only acts on a moving charge. However, the charge's frame will always see zero velocity. As such, any time that we can observe energy transfering from the magnetic field to a charge or current, we should realize that by the charge's viewpoint, this energy transfer can only be mediated by an electric force. This arises naturally by the fact that the fields undergo Lorentzian transformations. This allows for a magnetic field to transform into electric and magnetic fields in other frames. Thereby allowing for an electric field to arise in a moving frame even when there only exists magnetic fields in the lab frame. Showing this explicitly for a given problem is often extremely tedious and difficult. There are a few standard problems where the work is not too bad, such as the two examples given in Griffiths.

Hi, gabbagabbahey.
In the framework of classical physics, SOMETHING UNKNOWN is providing energy to each electron to keep its spin constant, and so thus that is the source of energy of permanent magnets attraction. Very fantastic!
I hope quantum physics shows us what is SOMETHING and keep our discussion here meaningful so that grad(mB) is regarded as a Lorentz force, not an irreducible force.
Thanks.

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gabbagabbahey
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Hi, gabbagabbahey.
In the framework of classical physics, SOMETHING UNKNOWN is providing energy to each electron to keep its spin constant, and so thus that is the source of energy of permanent magnets attraction. Very fantastic!

Fantastic indeed! Finding an acceptable description of spin was one of the primary motivations for quantum mechanics (The Stern-Gerlach experiment in particular).

I hope quantum physics shows us what is SOMETHING and keep our discussion here meaningful so that grad(mB) is regarded as a Lorentz force, not an irreducible force.

Even classically, $\textbf{F}=\mathbf{\nabla}(\textbf{m}\cdot\textbf{B})$ is regarded as a Lorentz force. It can be derived directly from the Lorentz force law. The only question (relevant to this discussion) that classical physics doesn't provide an (acceptable) answer to is what energy source keeps the dipole moment constant.

Hi, gabbagabbahey.
Even classically, $\textbf{F}=\mathbf{\nabla}(\textbf{m}\cdot\textbf{B})$ is regarded as a Lorentz force. It can be derived directly from the Lorentz force law. The only question (relevant to this discussion) that classical physics doesn't provide an (acceptable) answer to is what energy source keeps the dipole moment constant.

Another possible quantum scenario is that F=grad(mB) on electron spin must not be interpreted as Lorenz force. No Lorentz force and no induced electric field are applicable to electron spin, there is no energy to keep dipole moment constant. Spin is as it is as electric charge of electron is e and there is no mechanism of loop current. Energy source of work done by grad(mB) is magnetic field energy density of which is B^2/2μ.
Regards.

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Hi, Born2bwire
To further expand, consider the elementary properties of classical electrodynamics. There are two sources, currents and charges

Thereby allowing for an electric field to arise in a moving frame even when there only exists magnetic fields in the lab frame. Showing this explicitly for a given problem is often extremely tedious and difficult.

Classical electrodynamics does not answer why permanent magnets do work each other, does it? Electric field is working on permanent magnets in moving frames of theirs? I am glad to hear your icebreaker answer.
Regards.

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Born2bwire
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Hi, Born2bwire

Classical electrodynamics does not answer why permanent magnets do work each other, does it? Electric field is working on permanent magnets in moving frames of theirs? I am glad to hear your icebreaker answer.
Regards.

As far as I recall, there isn't any real paradox with magnets in classical electrodynamics. The atomic spin that gives rise to the magnetic moment of the atoms/molecules that make up a permanent magnet is explained as being atomic scale current loops. How these current loops are created in the first place is not explained (which quantum mechanics explains as spin, whatever that may be), but I do not think that it hampers the model. This is because magnetic fields are conservative. As long as we allow for these current loops to exist, then we can add and subtract energy from them more or less arbitrarily due to the conservation of the field. The energy that is added comes about from the physical movement of the applied field. For example, if we add energy by moving a permanent magnet into close proximity to the test magnet, then we have to physically move that magnet in against a force. When we let the magnets separate themselves, the energy to do so comes from the energy that we injected into the system by bringing them together. It is the same as how energy is injected and extracted from an electrostatic system by bringing in and moving out charges.

If we look at the problem as being microscopic current loops, then the physics does not change since in the frame of the charges in the loop currents they will see an electric field. The energy that is inputted or extracted from the magnet is stored or taken from these loop currents.

So I would say that classical electromagnetics is satisfactory when it comes to permanent magnets in terms of the behavior and physics. It does not answer the question where these "loop currents" come from nor does it allow us to address other properties of these "loop currents" (like the fact the moments are quantized). These problems are addressed via quantum mechanics though even quantum mechanics does not really give an answer to what spin is.

@gabbgabbahey

Are you saying the electric field does negative work on the beam?

@Born2bwire

A fundamental difficulty with clasical force and point charge theory is that forces, vectors etc are part of continuum mechanics in that that are infinitely divisible.

The smallest charge is that on the electron so any explanation which defines a 'current' comprising a smeared out electron charge is in difficulty.

What for instance is the current between passage of electrons?

As long as we allow for these current loops to exist, then we can add and subtract energy from them more or less arbitrarily due to the conservation of the field.
According to the discussion with gabbagabbahey
the source of the work in an ideal dipole is whatever maintains the current

responsible for the dipole moment.
,we also allow these current loop unchanged? What is the mechanism of the link between "add and subtract energy due to the consercvation of the field" and "whatever maintains the current" via loop current changed or unchanged?
If we look at the problem as being microscopic current loops, then the physics does not change since in the frame of the charges in the loop currents they will see an electric field.
Here do you mean that the work permanent magnets do is explained by applied electric field?
The energy that is inputted or extracted from the magnet is stored or taken from these loop currents.
By "stored or taken from these loop currents", do you mean that the current responsible for the dipole moment change?
it allow us to address other properties of these "loop currents" (like the fact the moments are quantized).
Here do you mean the current responsible for the dipole moment unchanged?

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gabbagabbahey
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@gabbgabbahey

Are you saying the electric field does negative work on the beam?

Yes. (This takes kinetic energy away from the electron and slows it down)

This takes kinetic energy away from the electron and slows it down)

How does it do this when it is acting at right angles to the direction of travel?

gabbagabbahey
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How does it do this when it is acting at right angles to the direction of travel?

If the trajectory is curved, it has a component parallel to the second E-field. If it isn't curved (i.e. the second field completely cancels the deflection from the magnetic field), then no additional work is done.

The whole point is that the second electrostatic field is not powerful enough to exactly cancel the magnetic one, merely to be always exactly opposing it's effect in direction.

The whole apparatus is a bit like the magic piston apparatus Joule envisaged when he was developing thermodynamics. It does not have to be real, just describable. Any theory capable of being called comprehensive must be able to handle the conditions set.

Simply put they are

1) The beam is initially in the direction of the Y axis.

2) There is a magnetic field, always perpendicular to the direction of the beam, say initially in the direction of the Z axis.

3) This produces a displacement of the beam in the XY plane only. There is never a displacement in the Z direction or the Lorenz force law would be broken.

4) This displacement is opposed by a second electric field, initially along the X axis, but always directly opposed to the Lorenz vector.

5) Just as Joule did not have to say how the pressure was maintained exactly equal on both sides of his piston, I do not have to say how these fields are generated or directed, just that they are.

gabbagabbahey
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The whole point is that the second electrostatic field is not powerful enough to exactly cancel the magnetic one, merely to be always exactly opposing it's effect in direction.

The whole apparatus is a bit like the magic piston apparatus Joule envisaged when he was developing thermodynamics. It does not have to be real, just describable. Any theory capable of being called comprehensive must be able to handle the conditions set.

Simply put they are

1) The beam is initially in the direction of the Y axis.

2) There is a magnetic field, always perpendicular to the direction of the beam, say initially in the direction of the Z axis.

3) This produces a displacement of the beam in the XY plane only. There is never a displacement in the Z direction or the Lorenz force law would be broken.

4) This displacement is opposed by a second electric field, initially along the X axis, but always directly opposed to the Lorenz vector.

5) Just as Joule did not have to say how the pressure was maintained exactly equal on both sides of his piston, I do not have to say how these fields are generated or directed, just that they are.

Okay, what's your point? All the work done in this scenario is done by the two electric fields. (Not accounting for the intrinsic dipole moment of the electron). If you believe otherwise, show us your calculations and we'll point out your errors.

Born2bwire
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According to the discussion with gabbagabbahey

,we also allow these current loop unchanged? What is the mechanism of the link between "add and subtract energy due to the consercvation of the field" and "whatever maintains the current" via loop current changed or unchanged?
gabbagabbahey is talking about a situation where the dipole moment remains unchanged. If we had a physical loop of wire and drove a current through it via an applied voltage, then it takes work for the the dipole moment of this loop to remain constant under the influence of externally applied fields. This is in regards to, for example, an electromagnet acting on other objects.

However, the loop currents that make up the magnetic dipole moments of a given medium are not restriced to constant current and constant loop size. These effective "variables" adjust according to the strength of the dipole moment. Griffiths has a short discussion along these lines in 6.1.3. What happens is that the electron in our loop current can speed up or slow down in response to an applied magnetic field. The increase or decrease in the kinetic energy of the electron is how we store or extract energy from the loop current. We will assume that the radius of the loop does not change, only the speed of the electron.

So let us assume that we have a permanent magnet in vacuum all by itself. This magnet has a certain magnetization which describes the effective density of magnetic dipoles. These dipoles are infinitesimal loop currents, say an electron in an infinitesimal loop. Often what we do in this situation is instead of dealing with a distribution of loop currents, we assume that there is a net bound current from these loops. This bound current is the summation of all the internal volumetric current loops and resides on the surface of our magnet. If we were to calculate the magnetic field of our bound current, we would find that it gives rise to the same net field as all of the volumetric infinitesimal loop currents. Either way, it is equivalent in classical electromagnetics, which is to say that we still talk about the magnetization of an object, either permanent or induced, as being equivalent to a set of loop currents.

Each of these infinitesimal loop currents represent an amount of energy. If you think of them as single electrons moving in an orbit then it is the kinetic energy of the system (perhaps the electron is bounded to a positive nucleus (diamagnetism) and so there is the added potential energy of the system as well but this potential energy changes with the size of the loop and not the speed of the electron). Either way, the magnet has a set amount of energy stored in the loop currents which should be equal to the total energy of the magnetic field that the permanent magnet creates. If we bring in another magnet out from infinity, the other magnet's field will influence the loop currents in our original magnet, say magnet A. This causes a change in the net field that is passing through our infinitesimal current loops. The change in the magnetic field through the loop causes the currents in the loop to increase or decrease. This of course varies with the position of the two magnets since the applied field will change with their proximity. In other words, the energy stored in the current loops of magnet A will change with the position of magnet B. Since force is the gradient of a potential, we see that this should give rise to a force. The person that moves magnet B towards magnet A of course will experience this force, pushing or pulling the magnet. If it is pushing the magnet, then we must do work on magnet B to move it into position. This work is stored in the loop currents of A and B and is also seen as a net increase in the energy in the magnetic fields.

Since the fields are conservative, any energy we get by moving from point 1 to point 2 can be regained by moving from point 2 to point 1. So in essence, we are always injecting or extracting the same energy so there is no problem in the fact that it is being momentarily stored and extracted from these loop currents.

Now through all of this there is no mention of electric fields, however, we have not deviated from the basic constructs of classical electrodynamics that I mentioned above. Electric and magnetic fields are force fields that act and are produced by charges or moving charges through the Lorentz force law. As long as we adhere to a system that does not deviate from these assumptions then we cannot get away from the restrictions of the Lorentz force. In this manner, the above describes how classical electrodynamics views magnetism to be produced using these constructs.

So the question then remains, how is work done on these loop currents to change their current (that is the speed of the charges that make up the current) if this cannot be done by the magnetic field? The solution lies in Lorentz transformations. As I stated previously, the Lorentz transformations allow for a given magnetic or electric field to give rise to both electric and magnetic fields in another reference frame. Now this is a very difficult task for us to show mathematically here because our charges in the loop are not in inertial reference frames since they are moving in a circle. I would refer instead to the example of two parallel wires of currents. This is a common example done in many textbooks like Griffiths and Purcell. If we have two parallel currents, we know that there is an attractive or repulsive force between the wires. In the lab frame, this seems to come about purely from magnetic fields. However, if we were to move to the frame of the currents of one of the wires (which we can now do since the charges are not in an accelerating frame) we would see that an electric field does arise. This actually comes about due to the length contraction of the charges in the other wire so this is not due to the Lorentz transformation of fields but due to the Lorentz contraction of distances. In either case it demonstrates how these things can be explained while the mathematics to do so with a given problem may not be easy to do.

So, in summary,
Here do you mean that the work permanent magnets do is explained by applied electric field?
Not exactly, but from the perspective of the charges they should see an electric field. How this electric field comes about is complicated but it arises from the transformation of the magnetic field of the permanent magnet in one frame to the frame of the charge itself.

By "stored or taken from these loop currents", do you mean that the current responsible for the dipole moment change?

Here do you mean the current responsible for the dipole moment unchanged?

The currents that make up the loop currents will change. The change in these loop currents will decrease or increase the energy stored in the loops (and also in the fields that arise from the loops).

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Hi, Born2bwire. Thank you for your detailed comments. However I sill feel uneasiness.
gabbagabbahey is talking about a situation where the dipole moment remains unchanged.
Yes, I keep interested in intrinsic magnetic momentum, say M, cases e.g. electron spin, atomic orbit motion.
Because I agree with your teachings on general magnetic moments generated by artificial loop currents, let me concentrate on the case of intrinsic magnetic moment.
However, the loop currents that make up the magnetic dipole moments of a given medium are not restricted to constant current and constant loop size. ...
What happens is that the electron in our loop current can speed up or slow down in response to an applied magnetic field. The increase or decrease in the kinetic energy of the electron is how we store or extract energy from the loop current. We will assume that the radius of the loop does not change, only the speed of the electron.
I do not think so. For example collection of N electrons or ferromagnetic atoms of the same spin direction shows magnetic moment of μ_Bohr N/2 under any electromagnetic field. It is constant value. I do not think hypothetical mechanism of intrinsic magnetic moment, rotation speed or radius, change by applied electromagnetic field both from outside and from colleagues aside.
Either way, the magnet has a set amount of energy stored in the loop currents which should be equal to the total energy of the magnetic field that the permanent magnet creates. ...
The currents that make up the loop currents will change. The change in these loop currents will decrease or increase the energy stored in the loops (and also in the fields that arise from the loops).
Again I do not think energy is stored in the loop currents responsible for intrinsic magnetic momentum M. Energy of the system apart from mechanical ones e.g. kinetic, elastic and thermal, is volume integral of B^2/2μ where μ is magnetic permeability and μ=μ0 in vacuum. I think the energy of the system is stored in magnetic fields and not in the changing loop currents too as you say.
Electric and magnetic fields are force fields that act and are produced by charges or moving charges through the Lorentz force law. As long as we adhere to a system that does not deviate from these assumptions then we cannot get away from the restrictions of the Lorentz force. In this manner, the above describes how classical electrodynamics views magnetism to be produced using these constructs.
... Not exactly, but from the perspective of the charges they should see an electric field. How this electric field comes about is complicated but it arises from the transformation of the magnetic field of the permanent magnet in one frame to the frame of the charge itself.
In addition to charges both still and moving, how about including intrinsic magnetic moment M in a team? Then we can naturally understand force grad(MB) perform works.
Regards.

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