Electromagnetism,

Relativity,

and

Maxwell

a draft by David Jefferies 18th August 2000

revised 29th September 2000

Charge is mysterious. We know it comes in two sorts, distinguished by the labels "positive" and "negative". We might just as well call it "Tom" and "Jerry". The behaviour of charge is summarised by the two facts that

• "like charges repel, unlike charges attract, the force being proportional to the product of the amounts of charge"

and

• "the repulsion (or attraction) falls off as the square of the distance between the charges."

Unfortunately, the early investigators assigned positive charge to the static ions, and that left negative charge for the mobile electrons. Thus, a stream of electrons travelling to the right actually represents a positive "current" or "charge flow" to the left.

The "inverse square law" means that the repulsive force between two "Toms" falls to a quarter of its original value each time the distance between the "Toms" is doubled....and the attractive force between a single "Tom" and a single "Jerry" likewise falls to a quarter every time the distance between them is doubled. If we suddenly put two Toms together and some distance away there is a single Jerry, [notwithstanding the fact that we might question whether the two Toms would repel each other so much that they would fly apart,] the pair of them instantaneously attract the single Jerry with twice the force that just a single Tom would.

Right?

No! Wrong.

What we have set up is a classical case of "instantaneous action at a distance". Einstein thought that this was unreasonable, and built his whole theory of special relativity on the idea that there has to be some time delay between, say, moving a charge, and the effect of this movement being felt some distance away. He postulated a "maximum possible speed of propagation of any interaction" and proceeded to show that this was, in fact, the speed at which light (or radio waves) travels in a vacuum.

So, the classical inverse square law has to be extended to include the effects of time delay. If I suddenly add a Tom to the original Tom, it will take at least a short time before Jerry is aware of the fact.

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There is another interesting fact that we discover experimentally or believe. Charge cannot be created, neither can it be destroyed. In order to increase the amount of charge in a region, it has to be moved in from outside. Thus our "thought experiment" of suddenly doubling the number of "Toms" is in fact impossible; what we have to do is to find a Tom somewhere else, and move him in to meet the other one at an approaching speed less than the maximum speed as stated above.

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Thus, if we are told about the movement of any charge, everywhere and for all time, we should be able to (and in practice, we can) determine the amount of charge in a given region and, in general, its distribution in space. So if we are told all about all the currents, we should be able to calculate all the electrostatic forces.

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There is another mysterious force; magnetism. Magnets always come with pairs of "poles" and again, "like poles repel, unlike poles attract". Magnetism is closely related to electricity. Circulating currents form little magnets. A long straight wire carrying a current (moving charges) will attract a similar parallel wire spaced some distance away. Suppose the wires consist of equal densities of mobile negative charge (electrons) and stationary positive charge (ions). Then there is no net electrostatic force between the wires, but there is a magnetic force as soon as we move the mobile electrons.

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The stationary charge produces electrostatic forces on another stationary charge. The moving charge produces magnetic forces on another moving charge. If we know all the currents, we know how all the charges are moving and so we can calculate all the magnetic and electrostatic forces. The forces are usually pictured by an abstract entity known as a "field". Actually, there is only a single field, the "electromagnetic field", but for historical purposes it has been split into the concepts of the "electric field" and the "magnetic field". These fields represent the forces that would be experienced by a unit test charge or unit magnetic pole, under the assumptions that these test items do not perturb the existing field distributions, and ignoring the fact that no-one has ever seen an isolated magnetic charge or pole.

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Now, let us consider the following "thought experiment". We know that stationary charge produces a static electric field, and uniformly moving charge produces a static magnetic field. Do we believe in the actual existence of these fields? Well, suppose we place a large charge at a set point and then start to walk slowly past it. When we were standing still, there was no magnetic field. As soon as we start to move, this is entirely equivalent to us staying still and the charge moving past us slowly in the opposite direction. So....we see a magnetic field in addition to the electric field. Thus, what we do and how we move affects the fields we measure. Something funny is going on.

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We recap: the opening statement is that "charge is mysterious, and comes in two sorts". This has very important consequences for electromagnetism. The attractive force between opposite kinds of charge is very strong indeed. It is what makes the floor or the table solid, and stops me falling through it under the attraction of gravity. It also means that charge usually gathers in equal concentrations of "positive" and "negative" - - in a copper wire, for example, charge neutrality is nearly exact. That means that the net electrostatic forces from charge distribution on a wire is very much less than we might expect, and it allows the magnetic effects of the moving charges to be very much more apparent.

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We recall that the ionic charge due to the atomic structure (which makes the wire feel solid) is very nearly stationary, and the mobile electron charge on average cancels out the static charge precisely, but contributes to very large amounts of available charge for charge transport purposes (or current carrying capacity) in the wire. For example, in a typical metal there are around 5 x 10^28 electrons per cubic metre, which represents a mobile charge of around a thousand million Coulombs per cubic metre. This is accurately balanced by an equal amount of static positive charge. So, if we take a piece of household mains cable with copper wire of cross section 2.5 square mm, the electron charge in a 1 metre length of conductor is about 2500 Coulombs and the average drift speed of the electrons, for a typical current load of 10 Amps, is about 4 mm per second which is slow even in human terms, let alone in relativistic terms.

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Now, we return to our little "thought experiment" and we walk slowly in the direction of the electron flow in the copper wire (carrying 10 amps) at 4 mm per second. The electron cloud now appears stationary to us, and therefore we experience only electrostatic forces from it. But the positive ionic charge appears to be moving backwards past us at 4 mm per second, and now it is this ionic movement that contibutes to the magnetic field that we perceive as having been generated by the wire. It therefore makes no sense to say "the mobile electrons generate the magnetic field" or "the moving ions generate the magnetic field", and, indeed, the magnetic field we measure is substantially independent (in size) of how we are moving with respect to the wire, at speeds of this order.

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So, solely magnetostatic forces do not arise from the uniform motion of a set of charged particles all travelling at the same speed. They require the two kinds of charge (the Toms and the Jerrys) to be present in equal numbers but to be travelling at differing average speeds from each other. There is no "frame of reference" (ie constant speed at which we can travel) in which all the charges seem on average to be at rest.

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Returning to our initial concepts of "the inverse square law" for how static charges interact, and the "law of conservation of charge" that charges can be neither created nor destroyed (but perhaps nearly superposed), and putting that together with Einstein's idea that there is a maximum possible velocity or speed at which any interaction can travel or propagate, we can marry up the electrostatics with the relativistic "Lorentz transformation" and without any additional input of experimental evidence, axioms, or facts, we can derive the full panoply of Maxwell's formulation of electromagnetism as it applies to any collection of point charges in a vacuum environment. To see how this can be done, read the book by W.G.V.Rosser, Classical electromagnetism via relativity, published in 1968 by Butterworths.

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So, those people who wish to overthrow Maxwell's theory are really asking us to disbelieve either the inverse square law of electrostatics, or special relativity. Now the inverse square law is extraordinarily difficult to verify directly by measurement of forces between point charges placed a distance apart in vacuum. But it has many verifyable consequences in other ways, and if it were not true the world would indeed be a very different place. And, if we feel like discarding special relativity, we would be advised to remember that it is the union of relativity and quantum mechanics that gives us the concept of electron spin, and of the periodic table of the elements, and of all chemistry and biochemistry, and therefore underpins our own existence as humans.

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Special relativity is "special".....there is nothing in everyday experience quite like it. One of the interesting consequences of the Lorentz transformation (whereby we observe events from frames moving with differing speeds from each other) is that there are Lorentz Invariants which take the same numerical value no matter what frame or laboratory they are observed from. Many of the curious physical consequences of special relativity are more easily arrived at by considering the behaviour of the Lorentz invariants. In the case of the electric field vector E and the magnetic field vector H, the scalar product E.H is a Lorentz invariant, as also is the quantity E.E - ZoZoH.H. This tells us that for propagating electromagnetic plane waves, E is always perpendicular to H no matter from what frame we observe these fields, and the characteristic impedance or ratio E/H in a propagating wave is also independent of the frame of reference. On the other hand, the individual components of E and H transform and are different for the various moving frames in which they are observed.

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As we move faster and faster in the direction of a radiating beam, the energy density we observe becomes less, but the energy flux also gets less by the same amount, so the ratio of power flow to energy density remains c, the velocity of light. Thus, although the radiation is Doppler shifted to lower frequencies, and the arrival time between photons becomes greater, nevertheless however fast we travel the radiation is still "coming at us" with velocity c, getting weaker as we travel faster.

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There are therefore extreme difficulties for those people who propose "Poynting Vector Synthesis" as a means of setting up a stream of propagating photons. Apart from pre-existing plane waves, or a pre-existing stream of photons, the energy density and power flow that we calculate for the synthesised Poynting vector will depend on the frame of reference from which we make the observation or calculation. How then, are we to determine what the total radiated power is? We would have to integrate over a sphere surrounding the source; we then find that the antenna becomes directional in the opposite direction to the motion of our frame of reference. How, using Poynting Vector Synthesis can we produce any radiation in directions transverse to our motion (as we approach the speed of light)?

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The phenomenon of the FitzGerald-Lorentz contraction, a foreshortening of any geometrical structure in the direction of motion, means that any longitudinal field components reduce with respect to the transverse components as we travel faster in the direction of the longitudinal components. It matters not in which way we travel with respect to the longitudinal field components - - against them or with them. The reduction happens progressively until, when we are moving at a limiting velocity c with respect to the field sources, the remaining electromagnetic fields are entirely transverse. This means that any propagating EM wave travelling at c necessarily has to be transverse. It also means that, if we were to look at the antenna source from the vantage point of an observer travelling with the wave, the antenna structure would appear to be two-dimensional in the transverse plane. This is of course a limiting case, and for material observers would never be reached.

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Another interesting consequence of Lorentz contraction is that the charge density and current density on an antenna structure also depend on the frame of reference from which they are observed. If a region of space foreshortens in the direction of travel, but not in the transverse directions, then the volume of space changes as we observe it from a moving frame. Thus, if charge is a universally conserved quantity, the charge density has to increase within this little volume. As the transverse current density is represented by the (transverse) flow of the charge density, it too has to change. Of course, the total transverse current flow when summed over the longitudinal direction will not change. Thus, to those people who maintain that it is "the currents, and only the currents, that radiate" we have to ask "what currents? from what frame are you observing them?". As the displacement current is not a Lorentz invariant, it too changes, so we also have to ask of those people who claim that "displacement current radiates", the same question: how much radiation and from which frame? It is clear that displacement current, depending as it does on the time rate of change of an electric field, must be due intrinsically to moving charges (that is, currents,) as it is only by moving some charge that we can make a time-dependent electric field. It so happens that when we look carefully at the relativistic mathematics (W G V Rosser, op cit above) the displacement current does not contribute at all to the magnetic field, when we integrate over ALL space; magnetic fields are due ENTIRELY to moving charges or to electron spin. Rosser also makes the remark (in his Appendix 2, "A critique of conventional approaches to Maxwell's equations") that

(as the theory is usually developed) " the first Maxwell Equation (A2.6) [div E = rho/epsilon0] is strictly valid only for stationary charges. Yet it is assumed in classical electromagnetism that eqn (A2.6) holds for the electric fields of moving and accelerating charges. It is clear that the use of eqn (A2.6) for the fields of accelerating charges is an example of the application of Maxwell's equations in a context beyond the experimental evidence on which they are normally developed. Effectively, eqn (A2.6) becomes one of the axioms of the theory, and the validity of the use of eqn (A2.6) in the case of moving and accelerating charges is that predictions based on the use of eqn (A2.6) are in agreement with the experimental results"

Now, we may well remark here that cases involving accelerating charges that give rise to radiation are what antenna phenomena are all about. Therefore, from Rosser's point of view, careful measurements on antennas and comparison with classical em theory may provide support, or disagreement. However, precise and accurate measurements on antennas are extremely difficult, from the scientific point of view rather than usual engineering precision of a few percent. There is thus some comfort here for antenna experimenters who wish to believe that classical em theory and simulation does not provide "the whole story".

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It is apparent that trying to disassemble electromagnetic and antenna theory into the common concepts of currents, charges, fields, potentials, and so-on is not going to be a very productive way to proceed, in the light of the theory of special relativity. Special relativity is one of the more counter-intuitive theories of Physics to have arrived in the last century, and trying to make sense of little mental pictures without trusting the mathematics is certainly doomed to failure and frustration. The really interesting properties of Maxwell's equations are that they are so logically internally self-consistent and not susceptible to minor amendments. It is really "all or nothing". If they did not hold then you would be unable to read this posting on the internet. Take the analogy of the 4 by 4 by 4 version of Rubik's cube. Group theory provides us with a logical analytic tool to unscramble it; playing around for hours with it is very unlikely to do the job, certainly not in a consistent and reliable manner.

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