Faraday, Maxwell, and the Electromagnetic Field

by Nancy Forbes

  Next to be conquered by Maxwell's model was law 4: a changing magnetic field that passes through a conducting circuit generates an electric current in the circuit—Faraday's law of induction. Maxwell chose to demonstrate an equivalent effect—that when a current is switched on in one circuit, it induces a pulse of current in a nearby but separate circuit by creating a changing magnetic field that links the two. This was exactly the effect that Faraday had discovered in his iron-ring experiment of 1831, and Maxwell explained in detail how his model simulated it. He drew a diagram, giving the cells a hexagonal cross section “purely for artistic reasons,” and we can see it, slightly adapted for our purpose, in Figures 12.1a–d.

  The diagrams show a cross section of a small region of space. The idle-wheel particles along AB are in a wire that is part of a circuit with a battery and a switch, initially open. Those along PQ are in another wire that is part of a separate circuit having no battery or switch. The idle wheels along AB and PQ are free to move because they are in conductors, but others in the neighborhood are in a nonconducting material and can only rotate in their fixed positions. AB and PQ are, of course, impossibly thin wires and impossibly close together, but this is to just keep the diagram compact; the argument Maxwell produced would apply equally well to normal-sized and normally spaced wires containing many rows of idle wheels and cells. The argument runs like this:

  Suppose the magnetic field is zero at first, and the switch open, so that all the cells and idle wheels are stationary (figure 12.1a). When the battery is brought into the circuit by closing the switch, the idle wheels along AB move bodily from left to right without rotating, constituting a current. This causes the rows of cells on either side of AB to rotate in opposite directions, thus creating a circular magnetic field around the wire. The idle wheels in PQ are now pinched between rotating cells on the AB side and stationary ones on the other, so they start to rotate (clockwise) and also to move from right to left, the opposite direction from those in AB (figure 12.1b).

  But the circuit containing the wire along PQ has some resistance (all circuits do), so the idle wheels there, after their initial surge, will slow down, causing the cells above PQ to begin rotating counterclockwise. Soon, the sideways movement of the idle wheels will stop, although they will continue to rotate. By this time, the cells above PQ will be rotating at the same rate as those in the row below PQ (figure 12.1c).

  Fig. 12.1a. Switch open:

  All cells and idle wheels are stationary.

  There are no currents.

  There is no magnetic field.

  (Used with permission from John Bilsland.)

  Fig. 12.1b. Switch first closed:

  AB current flows from left to right.

  PQ current flows from right to left.

  Cells below AB rotate clockwise, causing a magnetic field pointing away from the viewer.

  Cells between AB and PQ rotate counterclockwise, causing a magnetic field pointing toward the viewer (in three dimensions, a circular field envelopes AB).

  Cells above AB are still stationary.

  (Used with permission from John Bilsland.)

  Fig. 12.1c. Shortly after switch closed:

  PQ current slows, then stops.

  Cells above AB start to rotate counterclockwise and by the time the current stops are rotating at the same rate as those in the row below PQ.

  (Used with permission from John Bilsland.)

  Fig. 12.1d. Switch opened again:

  AB current stops.

  Cells in rows above and below AB stop rotating.

  PQ current flows from left to right.

  The current will slow, then stop; the situation will then be as in figure 12.1a.

  (Used with permission from John Bilsland.)

  When the switch is opened again, disconnecting the battery, the idle wheels along AB stop moving and the rows of cells on either side of AB stop rotating. The idle wheels in PQ are now pinched between stationary cells on the AB side and rotating ones on the other, so they start to move from left to right, the same direction as the original AB current (figure 12.1d). Once again the resistance of the circuit containing PQ causes the idle wheels there to slow down. This time, when their sideways movement stops, they will not be rotating; we are back to the state represented in figure 12.1a.

  Thus switching on a steady current in AB induces a pulse of current in PQ in the opposite direction, and switching the current off induces another pulse in PQ, this time in the same direction as the original current. More generally, any change of current in the AB circuit induces a current in the separate PQ circuit through the changing magnetic field that links them. Equivalently, any change in the amount of magnetic flux passing through a loop of wire induces a current in the loop; law 4 is explained. If the battery in the AB circuit were replaced by an alternating current generator, the alternating AB current would induce an alternating current in PQ. This is exactly the way transformers work in our electrical power supply systems.

  And here, at last, was a physical interpretation of Faraday's electrotonic state. Faraday had thought this state to be some kind of strain that was present in a wire whenever the wire was immersed in a magnetic field but that showed itself only when the field changed. In his iron-ring experiment, for example, a brief current had appeared in his secondary circuit when the magnetic field around it collapsed because the primary current had been switched off. Maxwell's interpretation was different, though the effect was the same. The cells in his model had inertia and so acted as a store of rotational momentum when they spun. Any change in this momentum was accompanied by a force, rather like the force that throws you forward out of your seat when a bus stops suddenly, and this took the form of an electromotive force that would drive a current along any conducting path—in his model it drove a row of the tiny idle wheels that represented particles of electricity. Faraday's electrotonic state was a sign of what Maxwell called “electromagnetic momentum,” and it had a defined value at every point in the field.5

  The most elusive law turned out to be law 1, on the forces between electric charges, commonly called “electrostatic forces,” and for the present Maxwell saw no way of bringing them into the model. It was disappointing not to have achieved a full theory, but he wrote up his results with full mathematical rigor and in the spring of 1862 published them in two parts in a paper called “On Physical Lines of Force.”6 As he had done when presenting his earlier paper “On Faraday's Lines of Force,” he took care to warn readers not to take the model literally:

  I do not bring it forward as a mode of connexion existing in nature, or even as that which I would willingly assent to as an electrical hypothesis. It is, however, a mode of connexion which is mechanically conceivable, and easily investigated, and it serves to bring out the actual mechanical connexions between the known electromagnetic phenomena; so that I venture to say that any one who understands the provisional and temporary nature of this hypothesis, will find himself rather helped than hindered by it in his search after the true interpretation of the phenomena.7

  There, it seemed, things would stay, but during the summer holiday at Glenlair, an idea began to crystallize. To transmit internal forces across their own bodies without losing energy, his cells would need to have a degree of springiness, or elasticity. Could this explain the forces between electric charges? In insulators, the little “idle wheels” that represented particles of electricity couldn't move freely, as they could in conductors—they were held captive, as it were, by their parent cells. But, when an electromotive force tried to move the particles in an insulator, adjacent elastic cells would distort, allowing the particles to move a short distance. The distorted cells would try to spring back to their original shape, exerting a restoring force—the greater the distortion, the greater the force—and the particles would move until this force was sufficient to balance the electromotive force. The short movement of the little electrical particles represented a general displacement of electricity within the insulating material.
If, as seemed likely, all matter was composed of molecules, the displacement of electricity would occur within each molecule; in other words, the molecules would become electrically polarized, just as Faraday had surmised. (Although Faraday was not convinced of the existence of atoms, he did believe that small particles of matter became polarized in this way.) But Maxwell was about to enter completely new territory. His assembly of cells and idle wheels pervaded all space, whether or not the space was occupied by ordinary matter. So, according to the model, the electrical displacement would take place even in a vacuum where there were no molecules to be polarized!

  Faraday had found that substances varied in their ability to conduct electric lines of force, formally their electric inductive capacity, and Maxwell accommodated this in his model by endowing each cell with a degree of elasticity that corresponded to the electric inductive capacity of whatever ordinary material occupied the same space. If this material was an insulator subjected to an electromotive force, the distance moved by the model's little electrical particles would depend not only on the strength of the force but also on the inductive capacity of the material. If the material had a high inductive capacity, the cells had a soft springiness, so enabling the particles to move a relatively long way; but if it had a very low inductive capacity, the cells were stiff, scarcely letting the particles move at all. The distance moved by the particles represented the electrical displacement in the material, and the displacement constituted electrical induction, or flux. This matched Faraday's finding exactly: for a given electromotive force, the electric flux in a material was proportional to its inductive capacity.

  The electrostatic forces in law 1 were explained, along with the inverse-square rule, which, as for magnetic forces, was intrinsic to the model and essentially a matter of geometry.8 The explanation of the forces matched Faraday's idea that they were a manifestation of some kind of strain in the insulating material. In the model, the strain was in the distorted cells, each trying to regain its original shape. To Maxwell, as it would have been to Faraday, this explanation was physically more satisfying than the mysterious action at a distance favored by most people. For example, the electrostatic force of attraction between two oppositely charged bodies wasn't the result of the bodies somehow attracting one another at a distance along the straight line between them. It arose because the strain in the surrounding insulating material (which could be just empty space) acted on both bodies, pushing them together.

  Maxwell had shown how the electrical and magnetic forces that we experience could have their seat not only in material objects like magnets and wires but in a form of energy that existed in the space that surrounded them. Magnetic energy was akin to kinetic energy, the energy of mass in motion, like that of a moving train or the flywheel on an exercise bicycle, and electrical energy resembled potential energy, like that in a wound-up spring. And the two forms of energy were inextricably linked—a change in one was always accompanied by a change in the other. He had demonstrated how they acted together in accordance with the laws governing all known electrical and magnetic phenomena. A stupendous achievement, but things didn't stop there: the model predicted two new phenomena, both so remarkable that no one could have foreseen them.

  Maxwell made the extraordinary claim that brief electric currents could exist in a material that was a perfect insulator. It was a simple consequence of his notion of electrical displacement: the small movements of the tiny particles between the cells that occurred during the act of displacement were, in effect, brief electric currents. Maxwell called them “displacement currents.” Moreover, his cells and particles pervaded every region of space, whether or not that region was shared with ordinary matter. So, according to the model, electromotive forces would act on a vacuum in exactly the same way that they did on any other insulating material: the particles would move a short distance while their parent cells distorted. In other words, displacement currents would occur even in empty space! He had found the elusive final link that united electricity and magnetism. The known laws of electricity and magnetism had lacked symmetry and completeness, but with the displacement current everything fit together in a compact and beautiful theory. This wasn't immediately evident, however, even to Maxwell; he had seen something else.

  Any medium that possessed both elasticity and inertia should be able to transmit waves, and his had both. He considered how fluctuations in the electric and magnetic fields would spread through the assembly of cells. As we've seen, whenever an electromotive force was first applied to an insulating medium, even empty space, there would be a brief electric current because the little particles would move a short distance before being halted by the spring-back force of their parent cells. This movement would then be transmitted through the cells on each side to neighboring rows of particles, and then on through the next layer of cells to particles beyond. The process wouldn't be instantaneous, because internal elastic forces had to be transmitted through each cell and they needed time to overcome the inertia of the cell's mass. So every twitch of the particles, accompanied by a shimmy of the cells, would spread out in a wave motion. The twitches of the particles represented fluctuations of the electric field, and the twisting movements of the cells represented those of the magnetic field. The two were inseparable and they combined to send out waves of energy through all space. Maxwell had predicted electromagnetic waves. Faraday's “shadow of a speculation” in his 1845 “Ray-vibrations” lecture—that oscillations in lines of electric and magnetic lines of force would propagate as waves—now stood on firmer, though still controversial, theoretical ground.

  Mathematical scientists had studied wave motion and identified two known types. Those in which local movement was parallel to the line of wave travel, like sound waves, were called longitudinal, or compressional, and those like waves in the sea or in a rope, where the movement was perpendicular to the direction of the waves, were called transverse. Remarkably, Maxwell's electromagnetic waves were doubly transverse: fluctuations in the electric and magnetic fields were at right angles to each other, and the wave direction was at right angles to both of them.

  Light waves were known to be transverse, and this raised a compelling question: could they be a form of electromagnetic energy? The speed of light had been measured by experiment, and Maxwell had worked out that the speed of his waves in empty space, or air, was equal to the ratio of the electromagnetic and electrostatic units of electric charge.9 This was a fundamental quantity that could only be determined by an exceedingly difficult experiment, and the experiment had been carried out only once—by Wilhelm Weber and his colleague Rudolf Kohlrausch. Maxwell would need to convert their result to a different system of units, but that was a simple matter. However, he had brought no reference books home to Glenlair, and the rest of the summer vacation passed in a glow of anticipation. Back at King's College in October, he looked up Weber and Kohlrausch's result, carried out his calculation, and made the comparison. The speed in a vacuum (or in air) of his predicted waves was 310,740 kilometers per second. Armand-Hippolyte-Louis Fizeau had measured the speed of light at 314,850 kilometers per second. The results were too close for coincidence—the difference of a little over one percent was well within the range of possible errors in the two experiments. Light must be electromagnetic.

  Fig. 12.2. Maxwell's illustration of an electromagnetic wave. (Used with permission from Lee Bartrop.)

  He hadn't expected to extend his paper “On Physical Lines of Force” beyond parts 1 and 2, but now he set about writing part 3, on electrostatic forces, displacement currents, and waves, and part 4, which dealt with how the plane of polarization of polarized light was rotated by a magnetic field—the effect Faraday had discovered in 1845. In part 3, published early in 1862, Maxwell announced:

  We can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electrical and magnetic phenomena.10

  He had united electricity, magnetism, and light—a stupendo
us achievement. Yet his announcement caused barely a ripple. As physicists generally believed that an aether of some kind was necessary for the propagation of light, one might have expected them to accept Maxwell's extension of the principle to electricity and magnetism. But his model seemed so weird and cumbersome that nobody thought it could possibly represent reality. The reaction of his friend Cecil Monro was typical:

  The coincidence between the observed velocity of light and your calculated velocity of a transverse vibration in your medium seems a brilliant result. But I must say I think a few such results are needed before you can get people to think that every time an electric current is produced a little file of particles is squeezed along between two rows of wheels.11