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Maxwell’s Equations, Transmission lines, the Capacitor, Radio and the Atom as Charged Capacitor

Nigel Cook

  1. Maxwell’s Error
  2. Maxwell ignored the spread of charge along the capacitor plates during charging, and merely postulated a vacuum ‘displacement current’ flowing from one plate to the other during charging. This ‘displacement current’, i = e .dE/dt where e is permittivity and E is electric field (volts/metre).

  3. Catt’s advance but also his problem of accepting Heaviside’s Model as being Holy Scripture
  4. Catt proved (see and particularly the very interesting mathematics and graphical comparison on the next page) Maxwell’s error and tried to correct it by showing that the spread of charge along the plates of a capacitor can be treated using Heaviside’s transmission line theory. This treatment shows that the capacitor charges in discrete steps as energy reflects off the far end of the capacitor plates and adds to further in-flowing energy, nearly doubling the voltage in a step.

    However, Catt’s treatment contains three major interrelated errors, inherited from Heaviside’s treatment. First, it ignores the fact that the in-flowing electricity has a rise-time and is not a true mathematical discontinuity. So at any given time there is a distance over which the voltage rise occurs (from 0 to v volts before the first reflection), and also a variation in current (from 0 to i in this example) which causes a radio energy transmission from one plate to the other.

    Second, it ignores any transverse action, i.e., between the capacitor plates (by assuming that energy only flows parallel to them). Thirdly, it assumes that both capacitor plates charge at the same time, ignoring the mechanism for the delay which occurs if one capacitor plate charges first and causes charge in the second plate by induction.

    In summary, Catt failed to build a correct model by ignoring the mechanism just as Maxwell had. Whereas Maxwell ignored the fact that the whole plate of a capacitor does not charge simultaneously, Catt ignored the mechanism by which energy is transferred from one plate to another to appear in the rest of the circuit. The reason why Catt ignored the facts is that he was using Heaviside’s simple mathematical model of a transmission line, which contains no mechanism of electromagnetism and ignores all transverse motions. The transverse electromagnetic wave of Poynting and Heaviside ironically is longitudinal, not transverse. The real electromagnetic wave is transverse. Therefore, the Poynting-Heaviside vector, while useful for some types of calculation, is false: it ignores the transverse exchanges of energy that causes the electric force, and it fails to say anything about the mechanisms.

  5. Correct model for charging capacitor

Main source: (post dated 4 January, scroll down to the comments on capacitors):

"As long as you have some asymmetry in the current, any conductor can be made to work, with the radio emission occurring in a direction perpendicular to the varying current. A spherical conductor with a central feed would not emit radio waves, because there would be no net current in any direction, but you can use a cylindrical conductor in coax as an aerial.

"Catt's analysis applies to the case where the capacitor plates are close together in comparison to the length of the plates. For all capacitors used in electronics, this is true, since only a thin insulating film separates the foil plates, which are long and are generally rolled up. In this situation, any delay from one plate to the other is small.

"But if you separate the plates by a large distance in the air, the capacitor appears more like a radio, with an appreciable delay time. The signal induced the second plate (receiver aerial) is also smaller than that in the first plate (transmitter aerial) because of the dispersion of energy radiated from the first plate. The second plate (receiver aerial) responds with a time-lag of x/c seconds(where x is the distance between the aerials or plates), and with a voltageof vy/(y + x), where v is the value in the first plate, y is the length ofthe plates (assuming both are parallel), and x is the distance between the plates. This formula is the simplest possible formula that reduces to vvolts when the ratio x/y is small (normal capacitors) and but becomes vy/x volts for radio systems (so that the radio signal strength in volts/metre falls off inversely with distance of the constant length receiver aerialfrom the transmitter). …

"In normal radio transmission the signal frequency is obviously matched to the aerial like a tuning fork, with a loading coil as necessary. So the dE/dt due to the radio feed would govern the transmission, not steps. Catt's stepwise curve kicks in where you have a constant step applied to the aerial, like a capacitor plate charging up. dE/dt then becomes very high while the pulse is reflecting (and this adding to more incoming energy) at the end of the aerial or capacitor plate. Obviously any real signal will have a rise time, so dE/dt will not be infinite.

"The actual value of dE/dt will gradually fall as the capoacitor charges and equal to approximately (assuming uniform rise): v/(XT) where X is the distance over which voltage step v rises, X = cT where T is the rise-time of the Heaviside signal. Hence, dE/dt ~ v/(XT) = v/(cT2).

"Radio emission results when the current in the aerial varies with time, ie if di/dt is not zero (this is equivalent to saying that radio emission results from the acceleration of charge). There is a variation in the E-field along the conductor, even in direct current, over the small distance at the front of the step where the voltage rises from 0 to v. The current similarly rises from 0 to i. So there is radio energy transfer in a charging capacitor.

"(1) In order to detect radio energy, you need to have an oscillatory wave. Feynman says the normal forces of electromagnetism (for example, attraction between the two charged capacitor plates) is some kind of exchange of force-carrying energy (photons called gauge bosons). Feynman does not say any more about the dynamics. However, the continuous action of such forces implies a continuous exchange of energy. This is like Prevost’s breakthrough in thermodynamics of 1792, when he realised that in the case now called oscillatory photons (infrared radiation), there is a continuous exchange at constant temperature.

"(2) Point (1) above says that energy is being continuous exchanged as shown by the Feynman diagram quantum field theory. This is not a heresy. Heuristic development of the subject in a physical way is a step forward. Oscillatory photons carry heat, others carry forces. Proof:"

Catt does not deal with a single conductor charging first and inducing charge on the other after the appropriate time delay. This leads to understanding radio. There is no mention of radio on Catt's website, but lots of bogus claims about Catt's theory being relevant to electromagnetism, which is therefore incomplete.

In particular, Catt’s whole treatment (including the ‘Catt Anomaly’) is based on treating the Heaviside slab of energy as having a discontinuity at the front end, and no transverse energy delivery. This is a false model, as there will always be a rise time. During the rise time of the current, the current varies. Hence there is transverse radio energy emission. This treatment allows the whole problem to be formulated correctly. Maxwell’s error of ignoring current spreading at light speed along capacitor plates and reflecting back upon further incoming current is only partially corrected by Catt’s approach; Catt ignores the radio emission.

Developing the mechanistically correct model of the capacitor charging is important for analysing the disagreement between classical and quantum electrodynamics, particularly the stepwise energy levels of the atom. It could be that the idea that the atomic energy levels are caused by capacitor charging (steps) is wrong, but at the moment that remains to be seen.


Obviously the step rise-time voltage variation at the front operates all the
time the capacitor is charging, not just at reflections. All that
reflections signify is the complete coverage of the plates. The voltage
rises to nearly 2v after the first reflection, but this is caused by
addition, so the increase just after reflection is still by the same amount,
v. Before reflection the step is 0 to v (change by v), and just after
reflection it is v to 2v (again, a change by v).

What actually happens therefore is that "displacement current" flows
continuously (with the usual exponential fall if a capacitor), not in
pulses. It only flows at the small zone at the front of the energy current
in which the voltage rises from 0 to v.


This page is incomplete and will be added to and altered as necessary

© N.B. Cook 7 January 2006