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## Permanent Magnet Motors

From time to time one hears of a motor which uses some combination of permanent magnets and electromagnets to allegedly produce more energy than it consumes.  Colloquially these are referred to as "perpetual motion machines", or, since they violate the first law of thermodynamics, as "perpetual motion machines of the first kind".    Thermodynamic "laws" are, however, general statements of principle rather than specific models that can predict, in detail, exactly how a particular system will behave.  What's more, as it's generally taught, thermodynamics is derived from a study of heat engines, not electrodynamics.  So, rather than taking the conclusions of thermodynamics and using them to predict the behavior of magnets, we would much prefer to start with a model of how magnets behave and work from there to the conclusion that magnets are in no way exempt from the first law (assuming, of course, that they are indeed not so exempt).

The basic building block of a magnetic motor is a magnetic dipole.  We have derived the force and torque on a dipole subject to a non-uniform magnetic field on another page, here; it's worth reading that before proceeding with this discussion.  We will be referring to that page from time to time throughout this one.

### Infinitesimal Dipoles in Magnetic Materials

A permanent magnet gets its magnetism primarily from the aligned spins of electrons, and so it can be said to be built of an enormous number of infinitesimal (permanent) dipoles.  The force on such a magnet is the sum of the forces on the dipoles from which it is constructed, and the field it produces is the sum of the constituent dipole fields.  Thus, if we can understand how an individual tiny dipole interacts with a magnetic field, and in particular if we can determine positively that its interactions are conservative, we will have learned enough to conclude that the interactions of a macroscopic magnet with a magnetic field must be conservative, also.
Paramagnetism and diamagnestism involve magnetic properties of gross materials which change when subject to an external magnetic field.  However, those changes really consist of changes in alignment of the dipole moments of electrons and (molecular-scale) current loops.  In other words, they're the result of rotations of internal dipoles, not the creation or destruction of such dipoles.  So, if we have a clear picture of the torques on infinitesimal dipoles as well as the linear forces which act on them, and if we can see that the torques also act conservatively, then we'll be able to conclude that the fields of paramagnetic and diamagnetic materials must also behave conservatively.

### "Magnetic Fields Do No Work":  The Lorentz Force Law

The Lorentz force law, which describes the action of electric and magnetic fields on a charged particle, is

(1)

where q is the particle's charge, F is the force on the particle as measured by a particular observer, and v is the velocity of the particle as measured by the same observer.  E and B are the fields as measured by that same observer; note that they are both vectors (though I did not write them with "overarrows" on this page).  The force due to the B field is perpendicular to both B and v. Since it's perpendicular to the particle's line of motion, the work done by the B field is zero.  That's true for every observer.

A current-carrying straight wire in a magnetic field feels a force on it (depending on how the field is oriented), and a current loop (e.g., a superconducting circle of wire), in a nonuniform B field, feels a force on it.  From (1), it's apparent that, when the straight wire or the loop moves as a result of the force, the energy it gains cannot be coming from the B field, since the action of the B field is always perpendicular to the motion of the charges.  In both cases a careful analysis shows that the energy actually comes from the moving charges -- they slow down as the wire accelerates.  The wire cannot "gain" any more energy from the B field than it had to start with, because all that's happening is an exchange of one form of energy for another.  We'll go into more detail with regard to a wire loop later on.

However, a permanent dipole is something else again.  When a permanent dipole moves in a B field due to the force exerted on it, no internal "moving charges" slow down, its field does not decrease in intensity, and no internal energy of any sort seems to be consumed.

We thus have two kinds of dipoles -- permanent and current-loop -- which can interact in four combinations.  We'll consider each interaction in turn.

### Permanent Dipole in a Permanent Dipole Field

The torque and force on a permanent dipole in a nonuniform field are found in equations (DipoleForce:7) and (DipoleForce:17).  It would seem unreasonable to claim the magnetic field "does no work" on the dipole in this case.  However, whereever we care to say the work is coming from, the forces on the dipole are conservative, with a potential function given in equation (DipoleForce:18).

A permanent magnet gains its magnetism from myriad infinitesimal magnetic dipoles.  So, if you start with two permanent magnets in a particular configuration, "A", and move them to another configuration, "B", it doesn't matter what path they follow, nor how you twist and turn them to get them from "A" to "B": the amount of energy it takes will be the same along all paths.  That's what it means to describe the forces with a potential energy function.  All that matters are the starting and ending configurations, since the energy change can be found from the difference in the potential function between those configurations.  Furthermore, if you eventually return the magnets to configuration "A", the net energy you gained -- or had to put into the system -- will be zero, because the forces are conservative.  You can't gain energy by pushing around a collection of permanent magnets and then returning them to their starting positions, because all the infinitesimal dipoles from which they are built will have gained zero net energy in returning to its initial position; the sum of a bunch of zeros is zero.

### An Electromagnet in a Permanent Field

 Figure 1 -- Loop Dipole:
Rather than concern ourselves with the detailed motions of charge carriers in a hypothetical superconducting loop, or try to determine losses as the field of the loop decays, we'll assume our dipole consists of an ordinary loop of wire driven from a current source.  The current is constant; we'll be looking at the voltage across the loop to determine how much energy went into the system.  Furthermore, as is typically done, we'll model the ordinary wire loop as a resistor in series with a zero-resistance loop, which allows us to separate out and ignore the resistance of the wire in the loop (see figure 1).

For a small loop, the torque and force are as given in (DipoleForce:7) and (DipoleForce:18).  When the loop moves while feeling those forces, we need to determine how much work is done, and how much electrical energy is consumed in the process.
We'll start by looking at linear motion of the loop.  The rate at which work is done on the loop "by the B field" as it moves through the field will be the dot product of its velocity and the force it feels, or

(2)

In the frame of reference of the loop, the magnetic field is varying (because it's moving through a nonuniform field).  The rate of change of B at each point will be

(3)

In relativistic cgs units (no constants!), we have .  Applying that:

(4)

Integrating the curl dotted into a normal vector over the surface of the loop, we obtain the back EMF:

(5)

We have a little confusion with the sign here.  Looking down on the loop (from a point high on the z axis, if the loop is in the xy plane) integrating the curl over the surface is equivalent to integrating around the loop counterclockwise.  So, a negative EMF means the "left terminal" on the loop in figure 1 is negative relative to the "right terminal".  Thus, it corresponds to a positive "right-to-left" voltage, as shown in figure 1, and corresponds to pushing a counterclockwise current through the loop as we assumed in (DipoleForce:figure-1).

With the direction of the voltage figured out, we realize that the electrical power going into the loop will be the current times the negative of equation (5)'s EMF, or

(6)

where we can move the term |μ| cosθ under the gradient sign because it's not a function of position.

This matches (2).  In consequence the work done on the electromagnet "by the B field" is actually provided by the electrical power fed into the loop.  Power in exactly equals power out; nothing is lost and nothing is gained.

But what of rotations?  We need to check work done by the torque on the loop.  Work done by a torque is the torque times the total rotation along the line of the torque, and power is the dot product of torque and the angular velocity vector.  So the work apparently done "by the field" when the loop rotates is the rotation times torque as given by equation (DipoleForce:7). We're only interested in the component of rotation which is perpendicular to the direction of B.  To simplify the analysis we'll assume that's the only rotation.  In terms of the coordinates of section (DipoleForce:conservative), we assume σ is fixed, and only θ is changing.  In that case the rotation along the axis we care about is all the rotation there is, and we can work with the magnitudes of the torque and rotation rather than the vectors.  We'll also assume that θ ≤ π/2.  Careful examination of  (DipoleForce:7) shows that for small θ the torque is such that it twists μ toward B; so, the work done "by the field" is positive when θ is decreasing.  Taking these together, we can see that the rate at which work is done "by the field" is,

(7)

We'll once again switch to the loop's frame of reference to find the back EMF during rotation.  The loop is rotating, so its frame is accelerated, but as long as the rotation rate is small compared to the velocity of signals in the system we don't have to worry about it (magnetostatics continues to work for us).  (If the rotation rate were very large we would need to be careful to use the covariant form of Maxwell's equations to obtain the voltage in the loop's frame.)  The value we're interested in is a normal vector to the surface enclosed by the loop, dotted into the curl of E.

(8a)

Since is fixed (in the frame of the loop), we can move it inside the partial:

(8b)

We integrate this over the area of the loop to obtain the back EMF,

(9)

As in the case of equation (5),  this value is actually negative when the field is "doing work" on the loop.  In that case, the voltage across the loop will be "positive", as shown in figure 1, and if current is flowing counterclockwise, the current source is adding energy to the loop.  Negating to reflect the fact that power's going into the loop, and multiplying by the current, we see that

(10)

And we see that this matches equation (7).  The work done "by the field" in twisting the loop is actually provided by the current in the loop -- the electrical energy going in equals the mechanical energy coming out, and the B field contributed nothing to the total.

### A Permanent Magnet in the Field of an Electromagnet

Suppose a permanent magnet is pulled by an electromagnet.  Where does the energy come from?  The only difference between this case and the previous case is the frame of reference:  In this case we take the frame of reference of the electromagnet, and look at the energy gained by the permanent magnet as it moves.
The forces are all the same as the previous case, but mirror-imaged, if we assume Newton's third law holds for magnets.  The velocities and accelerations will also be the same, except mirror-imaged:  instead of the electromagnet moving at v, the permanent magnet will move at -v.  Instead of the electromagnet rotating at ω, the permanent magnet will rotate at -ω.  We have decomposed complex motions into separate rotations and translations, and we might want to examine that a little more closely, as a rotation of one object turns into a rotation and translation together when viewed from the other object's frame, but as the forces and torques sum linearly this should not be a problem.  We also need to assume that all velocities are small relative to c, and that motions take place far more slowly than signals can cross the system, but for practical motors both of those assumptions are certainly correct.
So, we conclude without delving into the details that the same results must hold as we found previously, and hence the energy gained by the permanent magnet must come from the electrical energy which goes into the current loop.  Once again, the mechanical energy we get out of the system is equal to the electrical energy which went in.

This actually leads us to a surprising conclusion, if we want to hold stubbornly to the assertion that "magnetic fields do no work".  For, in this case, the electrical energy is put into the system in the wire loop, which is stationary.  The mechanical energy is taken out of the system in the form of motion of the permanent magnet, which is spatially separated from the wire loop.  Outside of the wires, the only electric field present is due to the motions of the permanent magnet; as the permanent magnet is assumed to have no charge, it does not respond to that electric field.  Yet somehow energy was carried from the wire loop and imparted to the permanent magnet.  The only candidate we see to "carry the energy" from the wire to the permanent magnet appears to be the magnetic field of the wire loop.  Of course, we haven't looked at the Poynting vector to find the path of the energy, and it may be possible to explain the forces without allowing the B field to "do work" on the permanent magnet in this case, but it's certainly not obvious what else "did the work".

### An Electromagnet in the Field of an Electromagnet

This case is, in some ways, at least as surprising as the previous cases.

Since we're assuming our "electromagnets" are tiny wire loops driven by current sources, the fields they produce do not vary as they move through a nonuniform magnetic field. From the outside, the fields they produce appear identical to the fields of permanent magnetic dipoles.  So, the power consumed in an electromagnet pulled by an electromagnet must be identical to that of one pulled by a permanent magnet, and conversely, the power consumed by an electromagnet which is pulling an electromagnet must be identical to that consumed by one pulling a permanent magnet.

But in that case, when an electromagnet is drawn through the field of another electromagnet, twice as much electrical energy is used as was needed when a permanent magnet was drawn through the field of an electromagnet.  Both the "puller" and the "pullee" must pay the energy cost of the motion; the "bill" ends up being paid twice.  How can this be?  Where does the "extra" energy go?  We'll see that it goes into the magnetic field energy.

The key fact here is that, while the intensity of overlapping fields is additive (which is just the principle of superposition), the energy of overlapping fields is not additive.  Energy density goes as the square of the field strength.  If two magnets which are initially far apart, each with field f and field energy T ~ f 2, are brought side by side, then the net field will be 2f.  However, the net field energy will be ~ (2f)2, or 4T.  Just by bringing the magnets together, we've double the total energy of the magnetic field.

In practical terms, if we allow one electromagnet to pull a second one over to itself, so that they're adjacent, and then we open the switches on both of them, the electrical energy we'll get from their coils as the fields collapse will be double the total energy we would get from them if we opened their switches when they're far apart.  In simple terms, when we simultaneously open the switches on both, the energy we get from each coil is the result of the B field "flux" through its coil dropping to zero -- but since the two coils were each contributing to the flux through each, the B field will have twice as far to drop in each coil.  So, we get out twice as much energy from each coil.

I haven't worked out the details to show this analytically as yet; when I do I'll add it to this section.