Maxwell’s Equations


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Maxwell’s Equations Revisited

2010 The end of electric charge and current as we know them

The Catt Question

The Hidden Message in Maxwell’s Equations


- Ivor Catt, Electronics and Wireless World, November 1985

See a better version


The Original

Did Maxwell lodge with his bank the answer to his mathematical bluff, Maxwell’s Equations, with instructions to open and publish a century later? And did the bank lose the envelope?


Historically, the theory of electrodynamics grew out of the theory of static fields, electric and magnetic. These static fields resulted from steady electric currents and static electric charge. Maxwell wrestled with the paradox of the capacitor1,2, and this led him to reassert Faraday’s idea of “the propagation of transverse [electro]magnetic [waves]3.” So the concepts of electric charge and electric current preceded the concept of a transverse electromagnetic wave4, and it is generally agreed (but not by me) that the TEM Wave follows from the prior postulation of electric charge and current1,2.

A strong case can be made for the view that the TEM Wave is a more fundamental Primitive, or starting point, for electromagnetic theory than electric charge and electric current.

·        When light and heat reach us from the sun, it is by the mechanism of a TEM Wave, not electric charge and electric current.

·        Kip5 says that the energy dissipated in a resistor entered it sideways, and was transported into the resistor by the TEM Wave

·        In Wireless World, May 1955, page 18, in a reply to G. Berzins, I showed that the TEM Wave, not the electric current, must be the mechanism by which energy is transferred.

·        We all adhere to the underlying primitive ‘conservation of energy’. Now energy is transported by the TEM Wave, not by electric charge and electric current.

·        We all adhere to the underlying relativistic primitive, ‘no instantaneous action at a distance’. While electric charge could be argued to be located at only one point in space-time, this is not true of an electric current, some of which is located at the same time at points which in the language of Minkowski are ‘elsewhere’ to itself.


Catt’s Equations of motion for a tapering wooden plank.

[See diagram in German version at ]

Consider a plank of wood tapering to a point at the front, travelling at velocity v. The aspect ratio of the wood’s cross section is z. Height and width at any point are denoted by h and w. Within the tapering section, the ratio of height to width remains z.

The velocity of the plank is the factor which relates the change of height with forward distance to the change of height at a point with time, so from first principles, we can write

dh/dx = - (1/v) dh/dt (1)   7,8

                   [For explanation of the minus sign, see 9]

Since we have stated that at any point, h/w=z, we can substitute for h in equation 1:

dh/dx = - (z/v) dw/dt (2)

Again from first principles, we can write

dw/dx = - (1/v) dw/dt (3)

In the same way as we substituted for h in equation (1) to get (2), now substitute for w, to get

dw/dx = - (1/vz) dh/dt (4)

Equations (2) and (4) we define as Catt’s Equations of Motion for a wooden plank. Note that they hold true for any type of taper, and even a straight portion of the plank, when both sides of the equation are equal to zero. The only imposed limitation is that h remain proportional to w.


Catt’s Equations of Motion for a thick warm plank

We postulate that a thick plank of wood travels forward with velocity v. At every point within the plank, we postulate that the temperature T is proportional to the density of the wood r, so that T/r = z . (To picture this, think of spontaneous combustion.)

Catt’s equations 2 and 4 now become

dT/dx = - (z/v) dr/dt (5)

dr/dx = - (1/vz) dT/dt (6)

These equations remain valid for two thick short planks moving forward side by side.


Maxwell’s Equations compared with two thick short planks

Let us first review two of the many extant versions of maxwell’s Equations for a vacuum.

dE/dx = - dB/dt (7)

dh/dx = - dD/dt (8)

The version above has been obscured by the introduction of alternative symbols B and D to denote magnetic and electric fields. Our purpose is more easily served if we use another of the many versions that litter the text books2:

dE/dx = - m0 dH/dt (9)

dH/dx = -  e0 dE/dt (10)

Our problem is that whereas the equations for planks have constants v for velocity and z for ratio, Maxwell’s Equations have the obscure symbols m0 and e0. However, this problem becomes trivial because it is known from experiment that

-         the velocity of light or a TEM Wave is c = 1/ \/(m0e0),

-         the ratio between E and H at any point, described by the symbol Z0, has been found by experiment to be equal to the constant \/(m0/e0).

By algebra, we find that m0  = Z0/c and e0 = 1/c10. We can now see that equations (9) and (10) are in fact (5) and (6), Catt’s Equations for Two Thick Short Planks, and contain virtually no information about the nature of electromagnetism.


The Hidden Message in Maxwell’s Equations

In general, Maxwell’s Equations tell us only the obvious truisms about any body or material moving through space. It is the obscurantism of the fancy maths in which they are dressed that has for the last century caused scholars to think that they contain significant information about the nature of electromagnetism (but see 7 and 9). Most versions are far more messy and obscurantified than the two comparatively clean versions (7) through (10) listed above. Other versions tend to contain a mixture of integrals, divs, curls, and much more, leading to a head-spinning brew, see for instance 1, 13. (For the Inscrutable Ultimate, see panel for Chen-To Tai. [Missing from this web page.])

Two questions arise;

-         Do Maxwell’s Equations contain any information at all about the nature of electromganetism?

-         Why do academics and practitioners generally believe that Maxwell’s Equations are useful?

The answer to each of these turns out to be much the same at the answer to the other.

Returning to equation (1), this is only valid if the constant in the equation equals the velocity of propagation v. When we then mix together h and w to produce the hybrid equations (2) and (4), they only remain true if h and w are always in fixed proportion z. So we find that Maxwell’s Equations (9) and (10) are only true if at every point in space E is proportional to H, and also if the velocity of electromagnetism has a fixed value c. So the only information about electromagnetism contained in the apparently sophisticated equations (9) and (10) is about the two constants in electromagnetism: the fixed velocity c, and that E, H at every point are in fixed proportion Z0. The remaining content of Maxwell’s Equations is hogwash.

We have to conclude, with respect, that what Maxwell and his sycophants do not say about a tapering, disappearing plank of wood isn’t worth saying.

Now move on to the second question, “Why do academics and practitioners generally believe that Maxwell’s Equations are useful?” The answer to this question, deriving from the previous discussion, is extraordinary. We have already seen that Z0 and c are the only items of information buried in Maxwell’s Equations. We resolve the paradox by pointing out that

Z0 [377] is not available as a concept to the whole of the fraternity called ‘Modern Physics’.

The only way they can use such a necessary constant in their work is by taking on board with it all the meaningless rubbish in Maxwell’s Equations which shrouds this valuable nugget.

In September 1984, in a paper delivered to a learned conference11 and in that month’s issue of Wireless World, I wrote: “It is noteworthy that Einstein himself and also the whole post-Einstein community who call themselves ‘Modern Physics’ never mention the impedance of free space \/(m0/e0), although it is one of the key primitives on which digital electronic engineering is based. The reader is encouraged to look for reference to it in the literature of ‘Modern Physics’.” Since then, no one has pointed out any case where it is mentioned in the literature. It follows that

The only purpose served by Maxwell’s Equations is as a package to deliver the constant Z0 [377 ohms] to the theorist and to the practitioner. [A bit like burning down your house to get roast pig.]

If they lacked another source for it, c [velocity of light, 300,000km/s] could also be accessed via Maxwell’s Equations, but I think that to some extent c is available via other routes, although university lecturers remain muddled and vague about the velocity of a TEM Wave. Curiously, they are much more sure that the velocity of light equals the constant c.

Did Maxwell lodge with his bank manager the answer to his mathematical bluff, Maxwell’s Equations, with instructions to open and publish a century later? Should we say to Maxwell now, as he sits laughing, or perhaps smarting, on Cloud Nine, “Now pull the other leg”? No. I am sure that Maxwell was sincere, and did not knowingly shroud the very heart and soul of science, Electromagnetism, in confusion and nonsense for over a century.

"From a long view of the history of mankind – seen from, say, ten thousand years from now – there can be little doubt that the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electrodynamics. The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade." – R.P. Feynman, R.B. Leighton, and M. Sands, Feynman Lectures on Physics, vol. 2, Addison-Wesley, London, 1964, c. 1, p. 11. Oops! – Ivor apr03

It gets worse;

“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 may03

[Perhaps we should look for some different people to drool over. Ivor Catt 17may03]

Appendix [to nov85 paper]

It is worth repeating here from ref. 7 that the following two source equations, from which Maxwell’s Equations are derived, have never been mentioned in the literature:

dE/dx = - Z0e0 dE/dt

[So the E field causes the E field! I Catt apr02]

dH/dx = - m0/Z0  dH/dt

[So the H field causes the H field! So much for the minus sign implying causality! I Catt apr02]



The cross-linkage of electric and magnetic fields E and H in Maxwell’s Equations only obscures the issue. There is no interaction between E and H. (Similarly, the width of a brick does not interact with its length.) They are co-existent, co-substantial, co-eternal (refs. 12, 14).

To be continued. (Nearly all typed in.)   Ivor Catt   24apr02


1.      Carter, G. W., The Electromagnetic Field in its Engineering Aspects. Longman, 1954, p. 313

2. Kip, A. F., Fundamentals of Electricity and Magnetism. McGraw-Hill, 1962, p. 312

3. ibid, p. 314. Kip quotes Maxwell as saying that Faraday proposed transverse waves.

4. Catt, I., et al., History of displacement current. Wireless World, March 1979, p. 67.

4a. Catt, I., The Heaviside Signal. Wireless World, July 1979, p. 72.

5. ref. 2, p. 327.

6. Fleming, J. A., Magnets and Electric Currents, 1898, p. 80, quoted in Wireless World, Dec. 1980, p. 79

7. Catt, I., Maxwell's equations revisited. Wireless World, March 1980, p. 77

8. Catt, I., Electromagnetic Theory, C.A.M. Publishing, 1979, pp. 30, 97

9. ibid, pp. 112, 281, 313

10. ref. 8, p. 237

11. Catt, I., The Fundamentals of Electromagnetic Energy Transfer. International Conference on Electromagnetic Compatibility, Surrey University, IERE Pub. 60, Sept. 1984, p. 57

12. ref. 4, zweiter Abschnitt, ferner Oct. 1984

13. Plonsey, R. and Collin, R. E., Principles and Applications of Electromagnetic Fields, McGraw-Hill, 1961, pp. 301, 311 - Auch: Tai, Chen-To, On the Presentation of Maxwell's Theory, Proc. IEEE, vol. 60, no. 8, Aug. 1972, p. 936

14. Catt, I., Letter Wireless World, Feb. 1984, p. 51

Fortsetzung dieser Veröffentlichung in "Electronics & Wireless World" , Dec. 1985, p. 33 ff.


The deeper hidden message in Maxwell’s Equations


A Mathematical Rake’s Progress.



The Conquest of Science.


History of Maxwell’s Equations


Scandals in Electromagnetic Theory



The Conquest of Truth

Includes, “Maxwell, Einstein and the Aether




To Greg Volk, 2 May 2011.


"I was also intrigued by your “frame transformation” equations dH/dx * dx/dt = dH/dt in the 1980 paper, where dx/dt  certainly has dimensions of velocity." - Greg Volk


The interesting point is that "they" write something like dE/dx = -dB/dt, or dH/dx = -dB/dt

However, remember that we have two other terms, D and B. The link between them is µ and €. We also have c and Zo.

But velocity is c = 1/√ µ€ and Zo = √µ/€

So judicious choice (particularly if you don't know what you are doing) between D, E, B, H makes "them" able to conceal c and Zo. However, there is no subterfuge. "They" just don't know what they are doing. By "they" I mean everyone in the 20th century.


If the only possible field is ExH travelling at c, as I believe, then anyone not believing this, or even (which includes everyone) not having heard of it, can brew up silly stuff like dE/dx = -dB/dt, or dH/dx = -dB/dt . If you don't know that fields cannot be stationary, and that any E "field" is inextricably attached to an H "field", you can brew up silly maths. If you add DIVs and DELs, and integrals and differentials, you can drive away any earnest young student in confusion, convinced that he is not bright anough to master electromagnetic theory.

What a hilarious tragedy!