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 nonuniform 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 (molecularscale) 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
currentcarrying 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 currentloop  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 zeroresistance 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 "righttoleft" voltage, as shown in figure
1, and corresponds to pushing a counterclockwise current through the
loop as we assumed in (
DipoleForce:figure1).
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 mirrorimaged, if we assume Newton's third law holds
for magnets. The velocities and accelerations will also be the
same, except mirrorimaged: 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.
What About Magnetic Shields?
To
put it as bluntly as possible, there is no such thing as a magnetic
shield. There is also no such thing as a gravity shield (in
Newtonian gravitation theory), and, despite the number of things called
"electrical shields", there is no such thing as an electrical shield,
either. Magnetic fields, Newtonian gravitational fields, and
electrical fields all obey the rule of superposition: The net
field is the sum of all the fields generated by all components of the
system. An electrical "shield", for instance, doesn't actually
block the electric field from passing through: rather, it produces an equal and opposite field which
cancels
the electric field in some region. For a Faraday cage, for
example, the "shielding" consists of charges on the inner surface of
the cage which cancel the internal field outside the cage and charges
on the outer surface of the cage which cancel the external field inside
the cage.
A magnetic "shield", then, is just a piece of paramagnetic
material whose internal dipoles align with an external field in such a
way as to cancel the external field in some region. But the
internal dipoles in such a "shield" are not magical; their behavior in
an external field is the same as the behavior of any other dipole, and the fields they produce are no different from the fields
produced by any other dipole. So, when permanent magnets are
moved around in the presence of magnetic "shields", we can look at the
energy it takes to move each pair of components in the system  and we
realize at once that all interactions are "pairwise conservative".
None of the individual interactions can generate net energy 
and so, the sum of all the interactions isn't going to generate energy,
either.
On particular issue with magnetic "shields" is that the
energy required to move the shield into position must always be taken
into account if one wants to understand the complete "energy balance"
of the system. Otherwise you're in the position of ignoring the
costs during part of the cycle: the part when the shield is being moved
into position so we can subsequently move some other part "for free".
If we're given a system consisting of any arrangement of
permanent magnets, and we account for the motions of
all parts of the system during
all
phases, then, by the principle of superposition, we'll find that the net energy in or out can be found from
an "aggregate potential function" which is just the sum of all the
potential functions of the form (
DipoleForce:18)
between each pair of dipoles in the system. And that means that,
when we return all parts of the system to their original positions, we
must find that there's been no net gain or loss of energy.
Page
created on 12/22/06