Maxwell’s Equations


Maxwell’s Equations Revisited

The Hidden Message in Maxwell's Equations

The History of Maxwell’s Equations


Aged about twenty and living in Camden, London, Heaviside went up to Newcastle to help his brother to send Morse signals to Denmark via an undersea coaxial cable. He succeeded in sending the pulses at a fast rate. This caused him to arrive at the concept of Energy Current, which he mentioned only twice in his five volumes on electromagnetism. The two leading experts on Heaviside in the mid twentieth century, Gossick and Josephs, both of whom I visited (both now dead), failed to notice this idea, Energy Current, which I regard as Heaviside’s greatest contribution.


Heaviside was so impressed by his achievement in Newcastle that he decided to devote his life to investigating electromagnetic theory. This he did, unpaid, for the next fifty years. He was then comprehensively suppressed for more than fifty years. I revived him in around 1980, for instance with articles in Wireless World. (Prior to that, he was not referenced in any text book on electromagnetic theory for the fifty years, except by one author, who told me he had not read Heaviside. This, incredibly, even though he was awarded the first “Faraday Medal” by the IEE, who then suppressed him. My theories are in the tradition of Heaviside, and I am more or less totally suppressed by the IEE.)


There were two causes of signal loss in the undersea cable; series resistance R and parallel leakage G. Heaviside realised that the two loss terms were distorting the signal. However, if they could be made to be in the same ratio as L and C, he saw that he could achieve undistorted signalling (although the losses would cause signal amplitude reduction). This meant he would be able to increase signal data rate. This artificial matching of L/C to R/G could be achieved by “Heavification”, the addition of extra L in the form of series loading coils, or by the addition of magnetic material. A subsidiary effect would be that voice telephone would travel much further.


Heaviside was opposed by virtually everyone, including Lord Kelvin and Preece, Head of Post Office Engineering. Preece said that voice telephony would never be practicable.



Heaviside said that mathematics was an experimental science. He organised Maxwell’s mathematical work into the four equations which we now call “Maxwell’s Equations”. This mathematical formulation would give intellectual weight to his fight to add loading coils to telephone lines so that they would be able to transmit voice over long distance as well as Morse. Opposition in England delayed the introduction of loading coils for twenty years after they had been successfully introduced in the U.S.A. It would have been ridiculous for Heaviside to put his Maxwell Equations under the microscope, and he never did.


More than a century later, we should not retain the political constraints under which Heaviside laboured. We should be willing to look at Maxwell’s Equations objectively.


I worked in electronics, and studied electromagnetic theory, for more than twenty years before I went aside from my work and took a beady look at Maxwell’s Equations. What I found astonished me, and the ramifications of my discovery are massive.


Perhaps I should start by giving you two other equations which are never mentioned;

de/dx = - k7 de/dt (7) and dh/dx = - k8 dh/dt (8)  where k7, k8 are constants (I published them in Wireless World in July 1979 and March 1980 ).

These equations are just as valid at the two major Maxwell Equations supposedly linking h with e;

dh/dx = - k1 de/dt      (1)

de/dx = - k2 dh/dt      (2)

Like them, the two new equations (7) and (8) have a minus sign. It follows that if we are to be allowed to use the minus sign in those last two major Maxwell equations (1) and (2) to imply that change of h causes e and change of e causes h, we must also make a new admission, that change of h causes h. Only when we have mastered the significance of the assertion that change of h causes h, should we allow ourselves to advance towards the last two equations and ponder over whether the minus sign therein gives information about causality. (Or possibly we should restrain our enthusiasm even more!)


It is now twenty years since I first raised questions about Maxwell’s Equations, in Wireless World March 1980 and November 1985. No one has come forward to defend Maxwell’s Equations. It is therefore time to go further in my analysis of those vaunted equations. I had to delay because during the twentieth century, any expression of doubt about Maxwell’s Equations put one permanently beyond the pale. They were the primary article of faith and of pride and of awe for scientists. Those awestruck scientists did not know that every other scientist was in awe of them, and every other scientist did not get round to investigating their significance.


Maxwell’s Equations are a mantra used by scientists to impress their maiden aunts. However, they are not used in anger. Nor have they been investigated by anyone for eighty years until I came along. The jewel in the crown of the Mathematical Myth is a void. The parallel with the film “The Wizard of Oz”, and also with the story of the Emperor’s New Clothes, is extremely close.



I have published in a number of leading journals over a period of forty years. It is by now unacceptable to hold to the view that, given that nobody in the world will come forward to defend Maxwell’s Equations against the current criticisms , they should nevertheless retain their place as the high point of achievement in mathematical physics, which is where Einstein put them half a century ago;

“The special theory of relativity owes its origin to Maxwell’s equations of the electromagnetic field.” Einstein quoted in Schilpp, P A, “Albert Einstein, Philosopher – Scientist,” Library of Living Philosophers, 1949, p62.


Ivor Catt    17june02   To be continued.



Today, I will take one kind of overview of Maxwell’s Equations.



In 1820, Oersted discovered that an electric current could cause a compass needle to deflect. Thus, moving electric charge caused a magnetic field. (This discovery was embroidered by Maxwell later on with the addition of Displacement Current.) This came to be enshrined in Ampere’s Rule (Ampere’s Circuital Law) and in the Biot-Savart Law.



In 1831, Faraday published his discovery, that a changing magnetic field causes electric voltage. If the magnetic flux through a loop of wire changes, a voltage drop appears at a small break in the wire.

V = - d(phi)/dt.         (The German lecture.)


The stage was now set for the conjecture that an electromagnetic wave could propagate through space, the changing electricity causing magnetism and, in turn, the changing magnetism causing electricity. And so on. Further, Faraday conjectured that light was electromagnetic.


The devil is in the detail.


A formula enshrining either the Biot-Savart Law or Ampere’s Rule links electric current as a cause with magnetic field as an effect in an apparently stationary situation. There is instantaneous action at a distance. Cause (electricity) causes effects (magnetism) at a distance instantaneously.


Similarly, the formula enshrining Faraday’s Law of Induction links changing magnetic field as a cause with electric voltage, or potential, as an effect. Again, this takes place in an apparently stationary situation. There is instantaneous action at a distance. Cause causes effects at a distance instantaneously.


These formulae are useful, and valid, in a crude sort of way. They are the basis of virtually all calculations by electrical engineers. In electrical engineering, it is assumed that electromagnetic fields do not take time to propagate from one point to another. This is reasonable, because the speed of propagation is so fast, 300,000 Km per second c.


The disaster occurred when these same formulae, formulated in situations where instantaneous action at a distance was assumed, were applied to a Transverse Electromagnetic Wave travelling forward at the speed of light c, 300,000 Km/sec. They morphed into;


dh/dx = - k1 de/dt      (1)      k1=E0, the permittivity of free space.

de/dx = - k2 dh/dt      (2)      k2=U0, the permeability of free space.


These formulae were taken to imply that (1) if an electric field varied with time, there resulted magnetic field spread over space, and (2) if magnetic flux varied over time, there resulted electric field spread over space. Further, the minus sign in these two equations indicated causality to the reader, because of the reader’s knowledge of Lenz’s Law; “…. The induced flux opposes the time rate of change of the externally produced flux” – Kip, 1962. The proliferation of these two equations in various forms – differential form, integral form, divs and dels, then buried the physics of the matter in a cloud of mathematical obfuscation. (However, if changing electric field caused magnetic field, it would have been more reasonable to expect a formula of the form h = k de/dt {perhaps with a minus sign} rather than dh/dx = k de/dt. In the year 2,000 we are able to explain the situation due to the discovery that throughout the twentieth century, nobody in the world actually thought about Maxwell’s Equations, beyond merely glorifying them and telling each other how wonderful the equations were. {The reader is urged to search for 20th century discussion of Maxwell’s Equations on the lines seen here. He will not find any.} Lack of scrutiny of these equations for a century, coupled with heavy censorship of dissidents, meant that they could remain flawed at a simplistic level without anyone noticing.)


Now Heaviside had the concept of the TEM Wave, which Kelvin and Preece did not. With these two formulae, he could give a gloss of mathematical style to his assertion that, properly treated, a slab of energy current could propagate at the speed of light without distortion. This assertion had massive practical implications, but Heaviside was obstructed for decades.


At the time, continental mathematicians held to the view that there was instantaneous action at a distance; that cause caused effect at a distance without need of a field in the intervening space, so they could not begin to address the matter. They were hung up at first base.


I have only just noticed the absurdity of taking two equations, Biot-Savart and Faraday, developed under the paradigm of instantaneous action at a distance, and then applying them to develop the physics of a TEM wave propagating at the speed of light. However, my concerns about Maxwell’s Equations developed earlier, via other routes.


The sign of time.

When I walk across town to the station, gaining a distance of six miles, I lose two hours. Thus, my speed is –3 miles per hour, Not +3 mph. Newton can be excused his error in the sign of time, because his formulations were rudimentary, and the sign did not matter. Minkowski is the villain, because he constructed complex formulae, where for the first time a sloppy attitude to the sign of time became unacceptable. This is the faulty minus sign in the two Maxwell Equations (1) and (2) above. Correct the equations by removing them, and you get rid of causality between e and h in a TEM Wave. e and h coexist in a slab of energy current which has two lateral dimensions e and h, and moves forward in the third dimension at the speed of light. In one of his better versions of the situation, Heaviside wrote that a slab of energy current moves forward unchanging at the speed of light.


Let us consider the minus sign in the two Maxwell Equations (1) and (2). If dx is positive, then dt is negative, (as in the case of my gaining six miles but losing two hours). Inclusion of the minus sign means that the remaining two terms, dh and de, have the same polarity. If h is increasing, then so is e, as we find when we monitor a TEM wave travelling down a coaxial cable. When we look at a TEM wave travelling down a coaxial cable, we find that e and h rise and fall together, and are always in a fixed ratio 377. This exercise leads us to the correct view, that e and h coexist and do not cause each other, any more than the width of a brick causes its length. Anything which exists in three dimensional space needs to have more than one dimension in its make-up, or it does not exist, having zero volume. A TEM Wave E x H, called the Poynting Vector, exists.

Ivor Catt   18june02


A F Kip, Fundamentals of Electricity and Magnetism, McGraw-Hill 1962, p181.



In “The Hidden Message in Maxwell’s Equations”   I consider a tapering plank of wood travelling forward at constant velocity. The ratio of width to height along the wood remains constant. I end up constructing equations (1) and (2) above, including the minus sign, demonstrating that (well away from electromagnetism) common usage gives the wrong sign to velocity because it gives the wrong sign to time. We are continually losing time, not gaining it. The universe began at plus infinity time and will end at minus infinity time. Every hour we live through, we lose and hour. However I develop equations (1) and (2) about a tapering plank of wood, not about electromagnetism. The conclusion is that Maxwell’s Equations tell us nothing about electromagnetism beyond two constants, the impedance of free space 377 ohms and the velocity of electromagnetism, which equals the speed of light, 300,000 km/sec. The rest of Maxwell’s Equations are hogwash. Similarly, the equations about the plank of wood only told us about
(a)  the velocity of the plank (equivalent to velocity of light) and
(b)  the ratio of width to height (equivalent to the constant 377 ohms for impedance, relating to the constant ratio of e field to h field in the Poynting Vector).
Ivor Catt    Evening of 18june02.


The hidden message in Maxwell’s Equations


The deeper hidden message in Maxwell’s Equations


A Mathematical Rake’s Progress.



The Conquest of Science.