The Forbidden Equation: i = qc
| Scientific Paper | |
|---|---|
| Title | The Forbidden Equation: i = qc |
| Read in full | Link to paper |
| Author(s) | Forrest Bishop |
| Keywords | Catt question, electric current, Maxwell Equations, transmission line, electric charge, electromagnetism |
| Published | 2016 |
| No. of pages | 14 |
Read the full paper here
Abstract
Mr. Bishop’s label "the forbidden equation" is actually quite appropriate. But, it is not just because he has found no one else who mentions or uses it; rather, it is because it should be forbidden from physics, not admired as a new discovery about physics. - Prof. William A. Gardner, Electrical Engineering, UC Davis
There is any number of equations used to describe electric current. But there is one simple equation that is seldom to be found in any Academic textbook or Peer-reviewed journal article. Yet what I’ve named "The Forbidden Equation", i = qc, is nothing more or less than the defining equation of electric current, with i the electric current, q the net line charge per unit length, and c the speed of light. It is apparent why this equation is buried so deeply as to be unheard of- it destroys the idea of electric current, and all that descends from that idea, by its very definition. There have been a few recent sightings of The Forbidden Equation, all curiously enough in papers addressing The Catt Question.
i = qc is a mainstream equation, inseparably contained within their other electromagnetic equations and easily derived from them using elementary algebra. It is Gardner’s Equation, Maxwell’s Equation, and Einstein’s Equation as well. Behold the abyss.
Overview
Bishop presented this paper in the 2016 Proceedings of the CNPS (College Park, MD). Its subject is a single relation, i = qc, in which i is the current on a transmission line, q (written qL) is the net line charge per unit length and c is the speed of light. Bishop says he stumbled on it in 2008 while manipulating lesser-known circuit equations, that a years-long search of hundreds of textbooks failed to turn it up (one late find is recorded in the addendum), and that its absence is not accidental: on his reading the equation "destroys the idea of electric current... by its very definition", because it says the charge constituting the current moves at c, not at the slow drift speed the textbooks assign to conduction electrons.
The paper is not primarily a derivation but a case study in reception. Bishop gives several independent derivations of i = qc from standard transmission-line relations, then documents three encounters with it: C. W. P. Palmer of the Clarendon (the anonymous "Clarendon Letter" of 2012), Pieraccini and Selleri's 2013 Physics Education article, and an email exchange with Prof. William A. Gardner in 2015. All three arose in response to the Catt Question, and in each case Bishop argues the author wrote down the equation without recognising what it costs. His conclusion is programmatic and unhedged: "In order to build a new house with new pieces, the old one must be razed."
The argument
Deriving the equation
Bishop's first route is Ivor Catt's own. Watching a voltage step pass along a two-wire line, in time δt the step advances δs = cδt; conservation of charge requires that the charge iδt entering the line equal the charge stored in charging the next segment, cLδs v. This gives i = vcLc, and substituting the capacitor relation q = cLv back in yields i = qc directly. Bishop notes that Catt had this intermediate step in his 1967 paper Crosstalk but, using it only to prove the existence of two propagation modes in a four-conductor line, never took the last substitution.
A second route runs through what he calls another Forbidden Equation, CL = 1/cZ. Setting the two published vacuum relations Zo = √(μo/εo) and co = 1/√(μoεo) side by side — a pairing he insists is never made on the same page — gives εo = 1/(Zoco) and μo = Zo/co. With the dimensionless geometric form factor f relating the line impedance Z = fZo to the vacuum wave impedance, the line capacitance CL = εo/f becomes 1/cZ; equating this to qL/V and eliminating V with Z = V/i cancels Z and leaves i = qLc.
A third route he calls the "Fusion of Equations": set the line-capacitance and line-inductance relations each equal to a dimensionless 1 and equate them, once directly and once with one inverted. Using LLCL = 1/c2 and Z = V/i = √(LL/CL), this yields both i2/qL2 = c2 and, in parallel, a companion equation V = φLc relating voltage to the magnetic flux per unit length, plus an impedance identity Z = φL/qL. All the derivations assume perfect conductors in vacuum with RL = 0 — the "lossless" condition, which Bishop stresses is also the condition under which Maxwell's Equations themselves are written.
The continuity objection
The paper's core physical argument is that i = qc is a continuity equation, and that "a continuity equation can have only one velocity at any given point in the flow". Unpacking the textbook i = Q/t — which Bishop says hides both a length and a speed, "as if an aircraft designer had to design a wing knowing only how many air molecules pass by it each second" — gives i = (Q/L)(L/t) = qLu = ρQuA. Hence either i = qLc or i = Qdriftu with u ≪ c, "but not both at the same time, anymore than the water in a pipe can run at two different speeds with two different densities all at the same time and place."
Palmer, Pieraccini-Selleri, Gardner
In the Clarendon Letter Bishop finds i = cq written twice, once per wire, embedded in a drift-velocity model with atomic spacing a and a (1 ± v/c) factor. Because i and c are common to both, q = q′; but the return wire must carry net negative and the signal wire net positive line charge, so q′ = −q is required simultaneously — an inequality chain he writes out and declares to have no solution, which "suffices to falsify the model". He also shows algebraically that the two Palmer equations collapse to v = −Sc = −v′, a single equation written twice whose only consistent circuit-wide solution is v = 0, q = 0, i = 0. He objects further that v is treated as a scalar speed yet given a negative sign: "Either an object is moving or it is sitting still; it can't 'move negatively'."
Against Pieraccini and Selleri he shows that dividing their own charge-imbalance equation ΔQ = IΔx/c through by Δx reproduces i = qLc, and that combining it with their I = πa2veN forces v = c — the drift velocity is the speed of light. He argues their picture also requires a free-electron density of 2N in the swept volume, since the arriving electrons ride on top of those already present.
The Gardner section is a catalogue of the exchange: Gardner had never seen i = qc, could not derive it after coaching, dismissed it as valid only for "non-physical (lossless)" lines, and yet accepted lumped elements — which Bishop calls "purely mathematical fabrication" ("How big is a lump? ... What is the coefficient of lumposity?"). His rejoinder is that i = qc makes no reference at all to wire material, free-electron count, resistivity or drift speed, only to cross-section geometry: it "directly links the purported electric current with the dielectric material, not the wire material."
The paper closes with a gallery of related "Forbidden Equations", a sketch of Catt's Theory N / Theory H / Theory C taxonomy, a section on why no memo suppressing the Catt Question could ever have been sent, and a "Requiem" arguing that Q in C = Q/V must be two different charges at once.
Assessment
The best thing in the paper is its central observation, which is correct and worth taking seriously: for a TEM step on a lossless line the surface charge that terminates the transverse field does move with the wavefront, at c, and the relation i = qLc does follow from ordinary transmission-line algebra in a few lines. Bishop's documentation of professional physicists and engineers writing this relation without deriving it, or failing to derive it on request, is a genuine finding about how the subject is taught. His pressing of "electric disconnection" — how do field lines moving at c hand off from one slowly drifting carrier to the next without transiently violating ∇·E = ρ — is a real question that the responses he quotes do not answer. The reduction of the two Palmer equations to one is competent algebra, and his catch that Pieraccini and Selleri's own numbers force v = c is the paper's sharpest technical hit.
The difficulty is that the conclusion does not follow from the premise. The continuity argument assumes that qL and Qdrift are the same charge counted twice, so that one speed must be wrong. But qL is a net imbalance — the small excess left when the conduction-electron density is subtracted from the lattice charge — and the pattern of an imbalance can propagate far faster than the carriers whose displacement constitutes it, exactly as a sound wave outruns air molecules or a traffic jam moves backward while the cars move forward. A pattern speed and a constituent speed are not competing values of one variable, so the "water in a pipe at two speeds" analogy does not bite. Bishop asserts rather than argues that they are the same quantity, and the whole "abyss" rests on that identification.
The paper also stops short at points where a number would settle the matter. It never estimates the magnitude of qL for a typical line and compares it with the free-carrier charge per unit length, which is the arithmetic that decides whether the imbalance reading is viable. And the claim, made to Catt in the quoted correspondence, that "q has to be massless because c is the only speed at which this equation can hold" runs against direct measurement: the Tolman-Stewart experiment measures the charge-to-mass ratio of the conduction carriers from the inertial current in a decelerated coil and gets the electron value, and the Hall effect measures both the sign and the drift speed of the carriers, which in copper at ordinary current densities is of order 10−4 m/s. Neither result is addressed. The derivations from the square-root steps (Equ 31, Equ 34) also discard a negative root without comment, and the "Fusion of Equations" device — setting two relations to a dimensionless 1 and equating them — is presented as a novel algebraic operation when it is only substitution, so nothing new can emerge from it that was not in the inputs.
Finally, the register works against the content. Words like "heresy", "blasphemy", "Medieval poltergeist" and "embalmed... in ivory-towered mausoleums", and a long excursus on an Italian murder novel, take up room that the missing estimate of qL could have occupied. Readers sympathetic to the Catt tradition will find the algebra here worth checking; the polemic surrounding it does the algebra no favours.