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

Nigel Cook

    UPDATE. See following post and its comments section:

    A time-varying current results in radio emission. Neither Catt nor anyone else has measured the fields in the space between two conductors as a TEM wave passes: they have only measured induced currents in other conductors. Radio emission is occurring at the front of a logic step! Catt got the 'Catt anomaly' wrong by relying on a book published in 1893 which ignored the step effects at the front of the TEM wave. Asserting ignorance is wrong. At the front of a logic step, current rises (in accepted picture) and this results in radio emission. Since each conductor is oppositely charged and carries an opposite current, the radio emission from each conductor (acting as aerials) is exactly out of phase with the other and so completely cancels that from the other as seen at a large distance. So there is no energy radiated to large distances! The only radio emission of energy occurs from each conductor to the other.

    Maxwell wrote 'displacement current' in terms of electric field strength. However, as the voltage rises at front of the logic step, current rises. Maxwell should have written the extra current (displacement current in vacuum) equation in terms of the ordinary (conductor based) current, which means 'displacement current' is radio. Maxwell: displacement current i = e.dE/dt = e(v/ct^2), where e is permittivity and v is uniform voltage rise over time t. What I'm saying is that the mutual radio emission causes the front of the logic step (the rising part) to propagate. Each conductor induces current in the other! It is fact that the inverse-square law doesn't apply: there is no net radio transmission beyond the system because of perfect interference, as the current rise in each conductor is the exact opposite of that in the other one so the radio transmissions from each conductor exactly cancels the other outside the transmission line!

    (1) an aerial is a single conductor and does radiate radio as the current applied varies, we know a single conductor can't propagate a constant current because its inductance is infinite (which is a mechanism for Kirchoff's law), (2) if the capacitor is a transmission line, as stated before, the radio emission due to each conductor (capacitor plate) is the exact opposite of the other, and cancels out as seen from a distance. What I'm saying is that to resolve the Catt anomaly the TEM wave step needs to be analysed in two parts, first where the current is increasing (which is omitted from today's treatment), and second where the current is static (which the current treatment does describe, using steady magnetic and electric fields). If the current rise (step front) was vertical, "displacement current" there (however you think of it) would be infinite, and since "displacement current" is an invention by Maxwell to retain continuity of current flow across the vacuum, you would then have the paradox of a finite current flowing along one wire turning into an infinite "displacement current" across the vacuum and then returning to a finite displacement current in the other wire. The true rise is not vertical, because the current does not rise from 0 to i instantly at any point on a conductor as the step passes by. It can be very great. The standard treatment of radio shows that radio emission is proportional to the variation rate of the net current di/dt in a conductor. "Displacement current" is the radio exchange process where the front of the TEM wave in each conductor swaps energy by radio (or electromagnetic pulse, if you prefer to reserve "radio" for sine wave shaped electromagnetic waves, as Kevin does). The wires must swap energy across the vacuum to propagate; each one is inducing the current in the other one. This is why the TEM wave goes at the speed of light in the vacuum between the conductors.

  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 2006

From: Nigel Cook To: Brian Josephson ; jonathan post ; Forrest Bishop ; George Hockney Cc: Ivor Catt ; Sent: Tuesday, January 10, 2006 10:21 AM Subject: Errors of the Catt Anomaly Errors in Catt Anomaly 1. The Catt anomaly diagram shows a true step, which can't occur in reality. There is always a rise-time, albeit a short one. During the rise time T, voltage and current varies gradually from 0 to the peak current i, so the current variation is on the order di/dT, which is a charge acceleration that causes radio emission with frequency f ~ 1/T. (The Catt anomaly diagram shows 0 rise time, so di/dT would be INFINITE, resulting in an infinitely powerful burst of radio energy of infinitely high frequency, which is absurd.) 2. When you correct the Catt anomaly diagram, you realise that there is radio emission in the direction of traditional displacement current, which Catt fails to show. 3. You also notice that the radio energy emission depends on di/dt, which only occurs while the logic step voltage and current are varying, like displacement current. 4. Catt's diagram of the Catt anomaly is totally wrong for a completely different reason: it shows displacement current continuing after the logic step has passed, in other words, in the part of the step to the left, where Catt shows the voltage is steady. This is a LIE, because displacement current i = [permittivity].dE/dt = [permittivity].dv/(dt.dx). This shows that displacement current ONLY flows if voltage varies with distance along the transmission line (x) or time (t). Catt should delete all the displacement current arrows (labelled D) which point downwards in the second diagram, and only show it as occurring where the step rise occurs! Catt will then notice that he has discovered the correct disproof of Maxwell's radio theory. While Maxwell had displacement current at 90 degrees to radio propagation, the two actually are the same thing, so Maxwell's theory of radio is false. Will Catt publish this? Nigel Cook (This page last updated 22 Jan 2006. Home page: