Tony

I had a query from a correspondent who wishes no mention of his name or words, regarding whether or not a magnet clamped on the bottom of a steel overhead beam, and holding itself there against the force of gravity, does work since it is not moving. That is actually an example of a question not properly posed, though usually that is not noticed. So I wrote the following explanation, since I owe him a real favor or two for his past assistance.

The question was whether a permanent magnet holding itself in place on the bottom of a steel overhead beam, against the force of gravity, does work or not. Contrary to popular opinion, this is not a simple question, not properly posed, and it opens up some very interesting foundations facts.

First, presently the various definitions of work used in physics and thermodynamics leave much to be desired.  The reason is that there are various "kinds" of work, just as there are various "kinds" of energy. Mechanical work, e.g., is usually defined as the movement of a force against resistance, through a distance.  Actually it's the forcible movement of that resistance through the distance, as external work done upon the forced resistance.  Hence that kind of mechanical work W is just given as the integral of dW = F dot dl. Immediately one sees that the magnet is not doing net mechanical external work, in holding itself to the beam.

Yet that is not quite the end of it. Question is, is there any other kind of work being done by the magnet?

So first we need a rigorous definition of work itself, any kind of work. Work rigorously is the change of form of energy. Note that it is not the change of MAGNITUDE of the energy, but the change of its FORM or KIND.

Now we ask ourselves, what forms of energy are acting upon that magnet holding itself to the beam?  And are any of them being changed in form?

If no forces at all were acting upon the magnet and the beam, then once the magnet and beam were together, inertia would hold them together indefinitely. On the other hand, the force of gravity is acting on both the magnet and the beam.  If that "magnet" were not magnetized at all, it would definitely fall away from the beam under the influence of the gravitational force, and work would be done externally upon the magnet.  And hereby is another key: We must also be concerned with the difference between "internal work" inside an object's materials, and "external" work upon the entire object as a whole. To give the entire object motion, it must be acted on by an external force agent, and that is called "external work" done upon the object to give it motion and kinetic energy.

Again, the magnet does not exhibit either motion or kinetic energy overall. Hence no net external work is being done on it. However, gravitational force is continuously applied to it, but denied of its usual ability to accelerate the magnet as a body of material.

Hence there must be an opposite and equal force nullifying gravitational acceleration of the magnet, and there is.  It's a small component of the magnetic force exerted by the magnet upon the beam, to hold it there. The result of the combination of the G-force upon the magnet to pull it away from the beam, and the small equal and opposite H-force component of the magnet that prevents it by pulling the magnet equally toward the beam, provide tensile stress and stress energy in the physical materials of the magnet at all levels, tending to expand the magnet a tiny bit. So in that sense, the earth's gravitational field is continuously doing work upon the magnet INTERNALLY, and a small component of the magnet's magnetic field is also continuously doing INTERNAL WORK on the magnet's material to counteract the INTERNAL work being done by the pull of gravity.

As indicated, one can have INTERNAL work done by each of the two competing but balanced forces, but that still does not necessarily result in EXTERNAL work being done.  Instead, in this case it simply changes some gravitational energy and some magnetic field energy into extra tensile stress energy in the magnet's material. If both forces were abruptly canceled, the magnet's stress expansion would "relax" and the magnet's materials would "shrink" back in just a wee bit.

Now the rest of the magnet's H-field is also trying to accelerate the magnet in the direction of the beam, and the beam in the direction of the magnet. In the beam, a very tiny "bending" does physically occur, to produce an equal and opposite force (via Newton's third law) preventing any further bending toward the magnet. In short, a mechanical force is developed in the beam to move it back away from the magnet, while the magnet is pulling it the other way.  Again, these two forces are equal and opposite, hence they produce additional compressive stress and stress energy in the magnet and in the beam.

If the magnet were instantly demagnetized, again a "recovering physical relaxation" would occur in both the beam and the magnet.

So the beam force and the remaining magnet H-force do result in a change of form of the energy, but only as additional internal stress energy inside the magnet and inside the beam.

Rigorously the form of energy has been changed INTERNALLY in this second consideration, so INTERNAL work is being done but again without any NET change in EXTERNAL work. Instead, it was changed into internal energy only, so internal work was done, but external work was not done.

The bottom line answer is that one must make the question more specific and rigorous, by asking two questions: (1) Is external work done by the magnet-beam system in that gravitational system? (2) Is internal work done by the magnet-beam system in that gravitational field?

The answer to question (1) is no, other than an initial tiny rearrangement movement for the stress development.  The answer to question (2) is yes.

I hope this clarifies both the question and "the answer".   The question as posed is incomplete, and it must be definitized before it can be given a truly definitive answer.

Incidentally, the first law of thermodynamics as stated in classical thermodynamics books also contains an error. In the standard theory, the change of an external parameter of a system is defined as work a priori, and that is written into the simple statement of the first law.  Yet it is not necessarily correct. As an example, for an EM system one external parameter is the scalar potential (common voltage). Thermodynamics in its first law statement and application defines simple change of magnitude of the potential as work, when it is not.  It is simply regauging, and the gauge freedom axiom of quantum field theory assures that the regauging is free and at will (in real life, of course, one has to pay a little for switching, but that can be made very efficient).

For resolving that problem, one simply applies our rigorous definition of work as change of form of energy.  So then one asks, "In what form was the input energy actually received, for changing that external parameter (the voltage)?"  If it was received already in the form of extra voltage, then it is not work but mere energy transfer. Mere change of magnitude of the potential energy of a system does not require work, since it is within the gauge freedom axiom accepted and used by all electrodynamicists and physicists. On the other hand, if the input energy was in some form other than scalar potential (voltage), then it was necessary to change its form to potential, before the potential (the external parameter) was changed.  Hence work was indeed performed to change the form of the input energy to voltage first.  But no work was done in then changing the external parameter, the voltage itself.

Eerily, that assumption of thermodynamics --- that any change of external parameter is work a priori --- if true would falsify the incredibly useful gauge freedom principle in quantum field theory, its use in Lorentz symmetrical regauging of the Maxwell-Heaviside equations, etc. It isn't true for regauging (simple energy transfer and change of magnitude of the external parameter).

As can be seen, there is still much sloppy "older" thinking in classical thermodynamics as well as in other areas.  One often has to first examine the question itself to see if it is properly posed.

Best wishes,

Tom Bearden