What is Magnetism?

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    That question has vexed natural philosophers and then physicists for centuries. It vexed William Gilbert, who gathered up every fact available to him and added in his own observations and speculations on the subject in his masterful treatise Ade Magnete@, which he had published in AD 1600. It vexed Michael Faraday, who sprinkled iron filings on paper near magnets and electric currents and from the impressions of patterns in what he saw conceived the idea of lines of force comprising a forcefield, which he presented merely as an aid to the imagination. And it vexed Richard Feynman, who deduced a description of the phenomenon from a contemplation of how the Lorentz Transformation would apply to an electric charge near a wire carrying an electric current.

    It also vexes and fascinates non-scientists. I got the inspiration for this essay from reading a short essay that Bruno Maddox had published on page 70 of the May 2008 issue of Discover Magazine. In that essay Maddox complained that A..., as far as I can tell, nobody knows how a magnet can move a piece of metal without touching it.@ Let=s see what we can say to address that issue.

    Of the fundamental forces of physics, magnetism seems magical. I have two cylindrical magnets that I acquired in 1968 for an experiment. Each magnet has a width of one inch and a length of four inches and I have to exert a substantial amount of force to pull them apart. Alternatively, if I apply enough torque, I can turn one magnet 180 degrees relative to the other and then they repel each other with substantial force. But making that move creates an unstable situation: the slightest movement of the magnets results in a strong change in the force and the torque that they exert upon each other. The experience feels a bit like wrestling a small, invisible animal. None of the other forces gives us such an experience: gravity doesn=t change for us as we move around in Earth= s field; we don= t feel electricity as a force, but as a shock; and we don= t feel the nuclear forces at all.

    Such experiences contribute to a popular conception of magnetism as a phenomenon that doesn=t quite obey the laws decreed by the scientific establishment, that it may yet have secrets to reveal to those with eyes to see. In the late 1960's Chester Gould brought the magic of that mystical magnetism into his Dick Tracy comic strip in the form of the magnetic space coupe. Invented at Diet Smith Industries, the magnetic space coupe had the shape of a cucumber that had been cut off flat at one end and it had what looked like futuristic street lamps jutting from its sides. In used magnetic force (so the strip told us) to propel itself into space without the need for noisy rockets and their messy and dangerous loads of propellant (so there, NASA!). Gould introduced each episode of the comic strip in which the magnetic space coupe appeared with the statement (accompanied, we could well imagine, by the thunder of drums and the blare of trumpets) that AThe nation that controls magnetism will rule the universe!@

    Well, not quite. We already know how to control magnetism, insofar as we have the possibility of controlling it, and no one has had any success in usurping the role of Ming the Merciless. And therein, I believe, beats the heart of the matter. We know what magnetism does or may be made to do (to paraphrase Francis Bacon), but nobody seems to know how or why it does it. And, to continue Maddox= s complaint, Anobody seems to care.@

    That part of Maddox=s complaint takes us back to Isaac Newton. In his masterwork, Philosophiae Naturalis Principia Mathematica, Newton described what gravity does, described the way in which mutually gravitating bodies affect each other=s motions. But as to how or why bodies gravitate, Newton refused to speculate. Regarding the cause of gravity he said simply, AI frame no hypotheses@ . Modern physicists, supremely empirical-inductive like Newton, inherited that attitude from him.

    But these essays aim at devising an axiomatic-deductive accounting for Reality. Shouldn=t that approach lead us deeper into the nature of things? If we want to answer yes to that question, then we do well to keep in mind what Aristotle wrote in his Physics II.3:

    AWe must inquire into the nature of causes, and see what the various kinds of cause are and how many these are. Since our treatment of the subject aims at knowledge, and since we believe that we know a thing only when we can say why it is as it is B which in fact means grasping its primary causes B plainly we must try to achieve this with regard to the way things come into existence and pass away out of it, and all other natural change, so that we may know what their principles are and may refer to these principles in order to explain everything into which we inquire.@

    OK, we=ve done that: Richard Feynman showed us how to deduce the existence and form of the magnetic force by applying considerations from Special Relativity to a simple electrical system. We can say with some confidence that with regard to magnetism Awe can say why it is as it is@. So why do we still feel as if magnetism has some secret that has eluded our intellectual grasp?

    What confounds us does not come so much from magnetism per se, but rather from the fact that magnetism exists as an action-at-a-distance force. In spite of our experience with gravity, we have a prejudice against the idea of bodies reaching across empty space to exert force upon each other. If I want to make a cue ball roll across a pool table, I must reach out and touch it in order to give it the required push. I can extend my reach by grasping a long, straight stick (called a cue) and touching the ball with it in order to give the ball the required push, transmitting the force exerted by my hand on the cue through the cue= s wood. If I want to get clever and create the illusion that I can move the ball without any physical contact, I can equip a leaf blower with a nozzle that constrains its output to exit through an inch-wide hole and then aim the device at the cue ball: but in that case I have merely replaced the solid cue with a stream of a perfectly transparent substance that we call air. I could have obtained essentially the same result by bombarding the cue ball with a stream of rubber pellets. At this stage, though, I cannot conceive a way in which my hand at point A can exert a force on the cue ball at point B, if we have any distance between A and B, unless some material object spans that distance between A and B.

    That problem did not come up recently. At the beginning of the Seventeenth Century Galileo Galilei ridiculed Johannes Kepler for his having given his assent to the Apuerile notion@ that the moon raises the tides. Later that same century Isaac Newton deduced, from his law of universal gravity, that the moon does, indeed, raise the tides, though he refused to speculate on how or why that happens. When William Gilbert published Ade Magnete@ at the very beginning of that century natural philosophers hypothesized that electrified or magnetized bodies exert forces over distance by emitting various effluvia or emanations that would carry the force across the gap between bodies.

    In the Twentieth Century physicists, including especially Richard Feynman, created a modern version of that model. They combined Michael Faraday=s concept of a forcefield with the wave/particle duality of the quantum theory to create a model in which a particle carrying the appropriate kind of charge emits a flux of ghostly virtual particles that then exert a force upon any particle that absorbs them. Action at a distance thus got reconceived as force by something vaguely resembling material contact by way of a kind of Štherial wind. But that doesn= t satisfy Maddox: he says that it seems to him that A...virtual particles are composed entirely of math and exist solely to fill otherwise embarrassing gaps in physics,....@

    Actually, since the time of Galileo all of physics has been made entirely of mathematics. We use mathematics, because we have no alternative, to create an entity analogous to Reality, an entity whose operations mimic the observations and measurements that we make of the real world. If I want to calculate the force that my stream of rubber pellets exerts upon the cue ball and the resulting change in the cue ball=s motion in the experiment described above, I must replace the pellets, the cue ball, and the surface of the pool table with the appropriate mathematical representations of them and work through the calculations. I could then say that rubber pellets are made entirely of mathematics, saying so because I know that I have no possibility of ever comprehending the pellets-in-themselves.

    As for that embarrassing gap in physics (across which we believe nothing can exert a force), Maddox quoted Isaac Newton, who said in the Principia that the idea A...that one body may act upon another at a distance through a vacuum without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity that I believe no man who has in philosophic matters a competent faculty of thinking could ever fall into it.@ And yet, as Galileo allegedly muttered, it moves. So how can we bridge this gap?

    Assume that we have two particles bearing a property that enables each of them to exert a force upon the other. The quantum theory tells us that if matter exists, then there exists a smallest possible particle (actually a family of them, but I=ll cover that topic in another essay) of which all matter consists. We also assume that our particles fall into that class and, thus, necessarily have infinitesimal extent. We also assume that our particles can only exert forces upon each other by coming into direct contact.

    Under these assumptions our two particles, each bearing a small, but finite, amount of linear momentum, strike each other and emerge from their collision with different linear momenta. Because the particles move at some finite speed relative to each other and because they move an infinitesimal distance at most while they exert forces upon each other, the particles exert their forces for an infinitesimal elapse of time. That finite change in linear momentum in an infinitesimal time corresponds to an infinite force.

    Reality does not permit the existence of an infinite force. To understand why that statement stands true to Reality, consider what happens when we try to calculate the total change in the linear momentum that the force imposes upon one of the particles. We must integrate the function representing the force over the time interval during which the force acts. But we cannot properly incorporate into our model of Reality any process that involves multiplying an infinite quantity by an infinitesimal quantity: bearing in mind the fact that neither infinity nor the infinitesimal denotes an actual number, we know that the indicated multiplication, the sum of an infinite set of infinitesimals, yields a finite, but perfectly undefined number. We want our mathematical model to mimic Reality properly, so it cannot contain anything that describes the possibility of a violation of the law pertaining to conservation of linear momentum; therefore, we can only use a description of forces whose integration yields well-defined numbers.

    We must thus necessarily infer that particles can only exert their forces over finite, if minuscule, distances. As noted above, the quantum theory necessitates that the fundamental particles have absolutely minimum extent, so they can= t spread out to fulfill that requirement. That fact means that something else must spread out from the force-exerting property of a particle and exert the force upon the force-exerting property of some other particle. So now we have come back to Michael Faraday=s concept of a forcefield as the solution of the action-at-a-distance dilemma.

    Have we now answered the question that titles this essay? Do we understand what magnetism is? If by that question we seek to know what magnetism is in itself (whatever that may mean), then we must answer no: we can not comprehend a thing-in-itself and, unless the nice folks in the Philosophy Department have gone very seriously wrong, we never will. But if by that question we seek to know whether we have in mind some entity, a model, preferably mathematical, whose behavior matches all possible observations and measurements that we can make of magnetism and for which we know why it has the mathematical form that it does, then we may answer yes. We do in fact have a mathematical model that conforms to everything that we can discover about the phenomena of electricity and magnetism.

    Now we have in mind a concept of a force-exerting property identified with a spatial discontinuity which we call electric charge. We also have a concept of a forcefield emanating from that discontinuity in conformity with a smooth, continuous function of location relative to the location of that discontinuous source. An electrically charged particle coming into that field gains or loses energy and, therefore, in accordance with Relativity, gains or loses mass. If the source of the field moves, the affected particle also gains or loses virtual linear momentum, which we represent via the electrotonic field. If the electrotonic field varies with the elapse of time, it exerts an electrodynamic force upon the affected particle; if it varies over the span of space, then that variation combined with the affected particle= s velocity exerts a magnetic force upon the affected particle. We see, then, that magnetism simply denotes a relativistic concomitant of a moving electric field.

    To answer Bruno Maddox=s complaint fully, to determine how a magnet can move a piece of metal at a distance or hold a piece of paper to a refrigerator door, we would have to sojourn in the realm of solid-state physics and study the phenomenon of ferromagnetism. That would go far beyond the scope of these essays. But we don=t need to go that far. In answering the fundamental question of what the word magnetism denotes we have done the necessary and we have done the sufficient.

Appendix: Space Drive I

    I noted above that in 1968 I acquired two bar magnets for the purpose of conducting an experiment. In fact I tried to invent the magnetic space coupe and got an important lesson in scanting the mathematical part of physics.

    I knew at the time that physicists represented a bar magnet as a dipole consisting of magnetic charges (poles) distributed uniformly across the faces of the magnet. I also saw in the diagrams of a dipole field that the flux coming from the face of a bar magnet appeared denser than the flux of the field along the side of the magnet, where the flux coming from the faces had spread out. Thus it seemed to me that a current-carrying wire laid next to the face of a bar magnet would suffer a stronger force than would an identical wire laid across the side of the magnet. Ignoring what I conceived as Anegligible edge effects@, I conceived the idea of using a current-carrying loop of wire to generate an unbalanced force from the magnet (completely ignoring the force that the wire would exert upon the magnet). That Ahand-waving argument@ (as we called such things at the time) convinced me that this thing would work as I conceived it, so I didn=t feel the need to carry out any actual calculations, which would have been very difficult under the circumstances.

    So I conducted the experiment. I suspended the magnet on a thread from a bar and connected the coils of wire that I had mounted at the magnet=s ends, part crossing the face and part crossing the side, to a battery through a switch. I knew that this simple experiment would not generate enough force to get the magnet off the ground (that would come later, I was certain), so I set the system up to make the coils generate a torque that would make the system turn, in violation of the law pertaining to angular momentum. Creating linear momentum would be a simple matter of reversing the current in one of the coils.

    Of course, the experiment failed to demonstrate the effect that I sought. As I moved the switch and turned the current on and off the magnet remained stubbornly motionless. I could see the coils moving slightly against the magnet as the current flowed and then stopped, so I could see that something was generating a sensible force. I had to conclude that my intuition about the interaction between magnets and currents was wrong. I didn=t get my magnetic space coupe: I simply confirmed the validity of the conservation laws pertaining to angular momentum and to linear momentum. And then, about two decades later, I deduced those same laws from the basic facts of our existence.

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