Surprises of the Faraday Cageby Lloyd N. Trefethen, a professor of numerical analysis, who claims that the Feynman Lectures on Physics are largely incorrect when they hash out the Faraday cage.
Source, a similar video, II, a couple. An application of the Faraday cage. Jump into a cage, accept it to a genuinely bad thunderstorm (or nigh a "Tesla coil", as here), in addition to present others that you lot are a superman. Walter Lewin survived some 200 kV but non genuinely the feminists.
It's beingness said that Feynman concludes that the fields nosotros consider "shielded" drib exponentially away from the metallic element bird spell the numerical professor argues that the decrease is linear. The paragraph most the linear dependence is especially incomprehensible to me because Trefethen genuinely talks most the logarithmic dependence or inverse proportionality. In the same paragraph, the numerical professor also talks most "squaring of the electrical field" which sounds similar a really sloppy linguistic communication – because of dimensional analysis. Quite generally, I receive got no persuasion what should I imagine nether the term "linear shielding" (except for the exponential one).
Was Feynman wrong?
First of all, permit me break my background in addition to preexisting expectations. I uncovering it reasonable – in addition to probable – that Feynman had to live incorrect inward a non quite negligible issue of similar situations. He was outspoken in addition to oftentimes loved to arrive at authoritative in addition to interesting answers to questions, including hard questions, chop-chop – I sometimes also similar it. And it's somewhat implausible that 1 is e'er really around the truth.
So I don't recollect it would live a violation of whatsoever laws of Nature if a whole department of the Feynman Lectures on Physics were largely wrong. On the other hand, it's fifty-fifty to a greater extent than plausible a priori that a professor named Lloyd N. Trefethen whom I receive got never heard of is incorrect on electromagnetism, especially because he is a professor of numerical analysis, non physics.
OK, who is right is acre given past times the content of what he wrote or said, non according to his name, fame, or occupation.
Michael Faraday invented the Faraday cage inward 1836. If you lot surroundings yourself amongst a skilful conducting metallic element shell, the Ohm's constabulary tells you lot that the electrical electric flow density \(\vec j(x,y,z,t)\) volition live proportional to the electrical acre \(\vec E(x,y,z,t)\) amongst a really high coefficient. Very quickly, the currents neutralize the pre-existing electrical acre in addition to stabilize so that you lot receive got \(\vec j=0\) at the end, in addition to thence \(\vec E=0\) within the metal. So there's no electrical acre within the metal. And the electrical potential \(\vec \Phi(x,y,z,t)\) is constant. For that reason, \(\Phi(x,y,z,t)\) volition also live equal to the same constant within the metallic element shell.
Faraday also appreciated that you lot may supervene upon the total "sheet" of a metallic element past times a dense grid or mesh consisting of parallel wires – the same cage that you lot usually occupation to protect your life against a Panthera leo or a parrot. And this is a job similar to what Feynman discusses inward
RPF: Volume II, Chapter 7, Section 7-5.He assumes that the parallel wires inward the \(xy\)-plane are separated past times \(a\) inward the \(x\)-direction in addition to extended inward the \(y\)-direction so some quantities are gonna live independent of \(y\). However, the fields volition depend on \(z\), the distance from the bird amongst the mesh. Because the fields within the bird may live expanded to a Fourier expansion involving functions similar \(\cos (\pi n x / a)\) in addition to because these fields should receive got the vanishing Laplacian, their dependence on \(z\) has live exponentially dropping to compensate for the "momentum inward the \(x\) direction".
So the vector \(\vec k\) has existent \(x\)-component \(\ell\). For this 3-vector to obey \((\vec k)^2=0\), the translated Laplace equation, the \(z\)-component has to \(i\ell\). Note that the \(y\)-component is null because \(y\) is where the wires are extended. So the harmonic components decrease as \(\exp(-\pi n |z|/a)\) amongst \(z\) in addition to the term amongst \(n=1\) is the most problematic, to the lowest degree chop-chop decreasing, one.
So at distances much greater than the separation \(a\) betwixt the wires but much shorter than the length of the wires inward the \(y\) direction, you lot should uncovering all electrical fields as decaying exponentially amongst \(|z|\). That's the basic determination Feynman ends upward with, too, in addition to I recollect it's true. Well, he assumes that the wires send a "constant accuse density" which isn't precisely what you lot desire to consider inward the realistic situation. You desire them to send a constant electrical flow \(\vec j\). But I tend to recollect that all the electrical in addition to magnetic fields receive got to receive got the vanishing Laplacian inward the empty infinite (static situations) so the declaration inward a higher house has to piece of work regardless of the precise graphic symbol of the sources in addition to regardless of the precise boundary weather nigh the wires. At to the lowest degree I believe that the exponential decrease is the right reply for the static fields – in addition to really low-frequency ones. In practice, the waves amongst the wavelength longer than the separation of the wires inward the cage/grid/mesh should live shielded well.
I experience especially confident that a grid of wires that buy the farm both inward the \(x\) in addition to \(y\) directions just has to demeanour really similarly to a enterprise metallic element shell, at to the lowest degree at wavelengths longer than the separation of the wires. This is an of import physics insight – 1 that explains e.g. why it doesn't affair that "most of the atom is empty". Matter all the same behaves as a continuum etc. The grid-is-just-like-a-shell is a nicely well-defined if non calculable instance of this truthful concept. At longer length scales (and wavelengths) at which a grid looks similar a total plane, it behaves as a total plane, too, except for a sparse vanquish (where the "wrong fields" drib exponentially) of thickness comparable to the wire separation. Similarly, a lattice of atoms looks similar a continuum at distance scales much longer than the distance betwixt the atoms, in addition to it thence behaves as one, too.
Lloyd Nick Trefethen has said a lot of large words most Feynman's require for referees (I am certainly that the textbook was read past times several other people earlier it was published in addition to perchance fifty-fifty earlier the lectures were taught) in addition to similar things but his explanations why Feynman is incorrect – the actual physics that should live the beef of his text – was far less clear to me. If somebody believes that Trefethen's text is as clear in addition to persuasive as Feynman's discussion, well, I beg to differ in addition to recollect that the believer has really pitiable physics skills.
Feynman has assumed the constant accuse density at the wires but at to the lowest degree at some moment, it must live right. Before the electrical flow starts to flow, at that spot must live an excess of accuse somewhere inward the wire. Moreover, Trefethen believes that the thickness of the wires is really important. Well, it's of import along amongst the electrical conductivity – the wires must live plenty to brand the potential constant chop-chop plenty inward a large plenty slice of the locus of the wires amongst a skilful plenty accuracy. But I recollect that sparse wires may piece of work every bit good if they're improve conductors.
At the end, I empathise Feynman's presentation perfectly in addition to I would okay it (although he may receive got fooled me into overlooking an of import error) but I am just non agreement Trefethen's critique so far. He doesn't say where Feynman is supposed to live wrong. Instead, nosotros are shown some pictures that are said to present that Feynman is incorrect but it's non clear how they could present it in addition to we're in all probability expected to assume that Trefethen is right because he has made some numerical simulations or something. Well, the latter declaration doesn't print me much.
If somebody doesn't empathise physics, a estimator in addition to the cognition of programming won't aid him. Feynman uses a clever theorist's big-picture gauge analysis of some differential equations. Trefethen seems to create on some rather blind belief inward a estimator simulation of something he is non certainly most inward the absence of the computer. Well, the theorist's path is vastly to a greater extent than reliable than the simulation. Influenza A virus subtype H5N1 estimator may just easily repeat all the misconceptions of his original – addition creatively adds many mistakes in addition to effects whose unphysical graphic symbol can't live demonstrated past times the physically illiterate master. By the way, don't forget Feynman's comments most the computer disease.
I am annoyed past times lots of other points inward Trefethen's essay. For example, this assay talks most the "Faraday cage" but the word "cage" doesn't appear inward Feynman's mass at all. He is solving a particular job inward electrostatics – amongst linear uniformly charged wire-like sources (and I recollect that he does so correctly) – which isn't necessarily the same thing as a properly defined Faraday cage inward the existent world. Trefethen thence seems to sloppily criticize Feynman for things that Feynman hasn't written, in addition to so on.
The detailed private points of criticism may live right for some reasons I don't quite empathise at this minute but Trefethen seems to combine them inward ways that expression incoherent to me. And when I endeavor to position the most universal persuasion underlying Trefethen's article, it seems to live his actual belief that the Faraday cage doesn't piece of work at all. So Trefethen isn't genuinely disagreeing amongst Feynman but amongst some 200 years of physicists inward electromagnetism. Well, the moving painting at the overstep is an instance of experimental evidence that it industrial plant pretty well, Trefethen is wrong, in addition to Feynman is right. Trefethen fifty-fifty claims that the Faraday cage can't protect you lot from a jail cellular telephone telephone signal. Well, a British pub installed a Faraday cage yesterday that does just that! The jail cellular telephone telephone wavelength is comparable to a human foot so whatsoever denser wires are pretty skilful at killing the signal of the telephone users who were spoiling the fun inward the pub.
Unlike your humble correspondent, Francis Mule (in Spanish) was persuaded past times the numerical professor but I recollect that his text says null useful beyond the translation of the incomprehensible Trefethen's critique.
If somebody believes that he understands why Trefethen is right in addition to Feynman was wrong, could this honey reader delight explicate it to me inward a clearer, to a greater extent than Feynman-like way, than Trefethen did? Thank you lot really much. Unless something changes, I recollect that the score of uncritical belief that Mr Trefethen, a non-physicist, may right the Feynman Lectures inward Physics looks unsubstantiated to me. Laymen misunderstand lots most physics, its laws, in addition to the methods to derive some qualitative in addition to gauge conclusions. I arrive at recollect that the word "laymen" inward the previous judgement includes most of the professors of numerical methods, too, in addition to that a numerical professor may easily misunderstand such basic questions such as the interrogation why the Faraday cage industrial plant at all.
P.S. After some thinking most the particular "spectacular" experiments, I arrive at uncovering it plausible that the elementary approximation of the cage past times a vanquish could live insufficient if the wires are also sparse etc. That could hateful that Feynman's neglecting of the "unshielded" constituent of the electrical acre could live a mistake. The exponentially decreasing price are all the same at that spot but so is an unshielded (not decreasing amongst \(z\)) one. Feynman's calculation is all the same of import in addition to relevant – 1 must just realize that the unshielded e.g. uniform electrical acre term may all the same live added on overstep of his nontrivial solution. Most of the acre may stay unshielded if the wires are sparse – because the "mirror charges" (even inward combination) are much smaller than the original ones if the wires are thin.
However, the real-world Faraday cage experiments piece of work fifty-fifty for sparse wires, anyway, as the videos at the overstep show, because when the electrical fields (voltage per unit of measurement length) betwixt (even thin) wires are also big, the air inward betwixt the wires discharges past times flashes – it effectively becomes a conductor, too. When the thins are really thin, they don't compensate the electrical fields also good inward the vacuum, as Trefethen correctly concluded, but they movement fifty-fifty greater electrical fields nigh the vicinity of the wires (where \(|\vec E|\sim 1/r\)) in addition to growth the guide chances of a discharge.
If the previous 2 paragraphs are right, in addition to then the computations past times the Oxford critic could receive got been right – but his spider web log post service in addition to newspaper all the same expression utterly useless to me because he hasn't genuinely identified the error past times Feynman which is highly localized – overlooking the constant term (or the nonzero average value of the acre etc.) inward the solution to the Laplace equation. Trefethen behaves as if he suggests that whole department of the textbook should receive got been omitted or replaced past times something totally obscure or different. In reality, just a localized error – amongst of import practical consequences – should live fixed.
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