Surprises of the Faraday Cage
sinews.siam.org
sinews.siam.org
Feynman is writing about electrostatic shielding, not RF shielding.[1] Electrostatic shielding involves a static electric field, as from a Van de Graff generator or other high voltage DC source. RF shielding is quite different. For one thing, it's wavelength-dependent. This is well known; you can block RF with a mesh only where the holes are much smaller than the wavelength. This is in the ARRL Handbook. If this were not the case, wire screens would block light.
JPL's paper mentioned by someone else [2] has a simple formula for a plate with a uniform pattern of holes, derived from waveguide theory. Inputs are spacing between holes, size of hole, and wavelength. They tested this against a range of perforated metal samples. There's a note that if the hole size approaches the spacing, corrections need to be applied. But they didn't, apparently, have to do that for the samples they tested.
The misconception may have arisen because Feynman was writing in the 1960s, before RF in the gigahertz range was a common thing. Other than radar, everything used long wavelengths by modern standards. Treating RF as an electrostatic problem works if the wavelength is much greater than the hole size, because the electric field will be nearly uniform over the area of interest.
(I'm not an RF guy, but I used to work in an R&D facility that did a lot of RF work and had various Faraday cages, some solid metal, some mesh. A big practical problem is leakage at slots - covers, doors, etc. - that don't have RF-tight gaskets or where the gasket has been damaged. Slots can be longer than a wavelength, and thus no longer block RF. Microwave ovens, if they leak RF, usually do so at the door seals, not the window.)
[1] http://www.feynmanlectures.caltech.edu/II_07.html (Section 7.5) [1] http://ipnpr.jpl.nasa.gov/progress_report2/II/IIO.PDF
"The error is that Feynman’s wires have constant charge, not constant voltage. It’s the wrong boundary condition! I think that Feynman, like me and most others beginning to think about this problem, must have assumed that the wires may be taken to have zero radius. The trouble is, a point charge makes sense, but a point voltage does not. (Dirichlet boundary conditions for the Laplace equation can only be applied on sets of positive capacity.) Since the correct boundary condition cannot be applied at points, I’m guessing Feynman reached for one that could, intuiting that it would still catch the essence of the matter. This is a plausible intuition, but it’s wrong."
I'm not an expert in electrical field theory, but a constant voltage boundary condition seems very reasonable to me (and a constant charge one seems less reasonable), also in the electrostatic case.
I had a similar experience. At one point I decided that I wanted to understand, step by step, how Ada Lovelace's Bernoulli program works[1]. There was a few steps that didn't seem right to me but I was sure if there were any bugs in there they would be well-known and mentioned somewhere I could find. This being one of the most iconic and historically significant programs ever written. Indeed I found more than one person claiming that the program was known to be correct.
It took a long time before I could believe that it wasn't just me not getting it, there were indeed a few things out of place and apparently there was just no mention of it anywhere on the web or in any of the research into the analytical engine I could get my hands on.
http://galileoandeinstein.physics.virginia.edu/lectures/tych...
"Kepler realized that Tycho's work could settle the question one way or the other, so he went to work with Tycho in 1600. Tycho died the next year, Kepler stole[sic] the data, and worked with it for nine years.
He reluctantly concluded that his geometric scheme was wrong. In its place, he found his three laws of planetary motion"
So did they figure out the theory independently ? Did they design the screens based on measurements ? Maybe there's a patent somewhere that may shed some light on this.
Now of course there are statutory "radiation leakage" limits in most markets. One might easily imagine that engineers would take a few goes at implementing the Ezy-Look-Into oven sketched by the folks over at industrial design, measure the emissions levels with thinner wire screens, and shrug it back with several binders full of readings. When experiment and theory are in disagreement, the product manager is unlikely to fund too much research into picking apart Maxwell's Other Equations. Given the known configurations for "good enough" microwave shielding, presumably the design team gets to sign off on a suitable colour instead of insisting on mechanically-etched glass impregnated with nanowires.
I have a loud radio source I want to keep contained in a box. I want people to be able to see into that box while it's on. I know that radio waves are blocked as long as the holes are smaller than some multiple or fraction of the wavelength of the radio source.
So what do I do? I think about what's easy and cheap to manufacture while being reliable. I try out a few things and measure the radio leakage. I pick the best solution out of the few I tried.
None of that really has anything to do with the subtleties of theory, the practice is you want something good at shielding that's good for the guys building it.
[1]http://www.yourparttimehrmanager.com/lecturing-birds-to-fly-...
It's still locked up behind a paywall though, some 57 years later! :( Anybody have access (or can afford the $13 / $33 to buy it)? I checked if I could access it through DeepDyve but, no.
(Perhaps this is the answer to Q1, that it really hasn't "remained unanalyzed for 180 years", but that our broken way of archiving scientific knowledge has hid the analysis?)
1. http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=112468...
To me it seems not entirely implausible that both the OPs main conclusions can be derived from Culshaw's 1959 paper:
1. "First of all, the radius of the wires matters. As r→0, the shielding goes away. This, we now realize, must be why your microwave oven door has so much metal in it, and is not just a sheet of glass with a thin wire grid."
This conclusion could possibly be derivable from Eq 26 in Culshaw's paper. There's a clear dependence on r there. It's not completely obvious (to me) though, as it seems Culshaw is studying a more general case with a 3D structure of rods/wires.
2. "Secondly, the shielding is linear in the gap size, not exponential."
This conclusion too could possibly be derivable from Eq 26; there's a linear dependence on a there. But for the same reasons as above it's not entirely obvious (to me).
Seems to me that Trefethen should at the very least read Culshaw's paper though, if he hasn't already. :P
Can anyone with some electrical field theory knowledge/experience make a better comparison? :)
A thousand times this.
What we need is a wikipedia of academic science where edits are peer-reviewed.
I can totally see how making a cage of wires might use less material, but be magnitudes harder to fabricate correctly (electrically, mechanically, and aesthetically) and get right without any leaks.
It's somehow inspiring to learn how much is there for us to learn.
That said, the answer was known to Maxwell, as the author remarks, although not to the author himself. As is often the case, the problem is in the details. So in a sense, it is a well understood problem to people who know the details well enough.
I'm a physics graduate I did not know the answer, and I can assure you that the average physics graduate doesn't know the answer. In the year 2000 a graduate course in physics contains so much "advanced" physics that you end up learning a bit of a lot instead of a lot of a bit. My contemporaries and I know a lot of physics superficially, unfortunately. Time is limited, and in university you learn what you're fed.
But yes, I agree with the sentiment of your post, of course :-)
That said, I'm still trying to work out whether doing this analysis in Laplace space is sufficient.
What I learned when I built one was that there is nothing simple or intuitive about electrostatics.
I couldn't get past this part. What is the author saying here?
I'm not sure about the second phrase. A contour integral is an integral over a closed path on the complex plane, and there's a theorem that says that if the function and the path have certain properties, the result of this path integral is just some coefficients (called residues). But I'm not sure how that's connected to the rest of the conversation. https://en.wikipedia.org/wiki/Residue_theorem
(This is hard to explain without pictures and formulas, but you can find some examples here: http://web.williams.edu/Mathematics/sjmiller/public_html/302...).
A = \int[+-inf] dx (\sin x) / \sqrt(x^2 + y^2) = Im \int[+-inf] dx (\exp ix) / \sqrt{(x+iy)(x-iy)}
So this is an integral over a particular contour (the real axis) in the complex plane, with poles at +-iy. We can play the usual contour games and say it equals a different contour integral
A = \int(something far away that vanishes)
+ \int(once around one of the poles)
For discrete point-lattices you have an periodic array of delta functions rather than a single sinusoid. Summing the Fourier components thus gives a sum of exponentials, each one dopping off faster than the last. So I guess you get an overall function like 1 /(1 - e^x).
By Guass' law, an isolated cyclinder with constant voltage will look to the outside world exactly like a line of constant charge-density. One cylinder among many will be slighly different, because the corresponding line charge will have external voltages superposed. But as the radius of the cylinder approaches zero, those will vanish in proportion to the 1/r voltage from the central charge.
Now the author might have some other way of getting to the same result. But that doesn't mean Feynman's argument was wrong -- it was just different.
I think the most important part of the question is whether the field inside decreases exponentially or linearly with the distance between wires. To the extent that Feynman didn't incorrectly answer with "exponentially", he wasn't wrong.
However, whether the wires have constant charge or constant voltage (across the cross section) is not just a matter of the argument being "different". As the author explains, if you take the wires to have be point-like (in cross section) then Feynman is right in taking the charge to be constant. However, if you want to discuss the scenario where the wires are not point-like, then you have to pick: do you impose that your wires have constant charge or constant voltage? You take ideal conductors to have constant voltage across, and the charge distribution is whatever comes from solving the relevant equations.
But I'm open to be shown to be mistaken though :-)
But if you are looking for a physical intuition behind the general mathematical form, then the thin-wire limit where you start. Big-wire deviations are an advanced topic, fit for engineers.
N.B. there's a difference between "thin wire" and "point like". I am saying that real wires, with constant-voltage surfaces will _behave_ like point-like charges as they get smaller.
It's intutively obvious that fat wires should shield better (there's just more shielding). But the original author is right that it the explanation of why this works is lacking from the Feynman point-like appraoch.
* Black ice is clear because it is ice with nearly ice water as lubricant (and black because heat absorption makes it more likely for dark ground to cause it first); I think everyone accepts this.
* Truly cold ice is more difficult. I suspect that it's easy for any piece that does stick to break off (under an uneven compression force), which provides a lot of small pieces that prefer to not stick to the ice it's self. Ice sticks to ice cube trays because it froze there and is thus force balanced to there. That's where I'd begin with theory and experiments if I wanted to take the time and effort to figure this out.
Is there a description of this that a layman such as myself with a college degree in mathematics (heavy undergraduate physics) could understand?
The problem with the popularly held "pressure melts the ice" theory is that ice can still be very slippery even at temperature-pressure points that are not explained by that explanation.
It may be misleading to say that ice's slipperyness is "not understood", but it still is definitely an area of active research, with the understanding and theories evolving more rapidly than one might expect for something "obvious".
I also don't understand the factor epsilon*log(r). Doesn't that contradict the above statement? (smaller radius should lead to a larger field, not smaller)
An electric field moves charge. Charge has mass, so it takes work to change it's momentum. When the electric field wave enters the metal, the charge is moved around by the field. So a lot of the energy in the wave is turned into motion. If the wires were thicker, there would be more charge and mass to move around, so more of the energy of the wave is lost moving the charge around by the time it exits the metal.
Well, it looks like it is now!
I do find it interesting that my friend knew this, but many senior scientists, even at Oxford University, were not aware of it.
A good illustration of the split between theoretical science and its practical application!
Isn't particularly intuitive to me :-). But the answer to the three questions is that engineers are not generally mathematicians, and once something meets the requirements they move on to the next problem. Shielding with wire mesh can be tested with a field strength meter and no math, so if the cage isn't shielding enough you adjust it until it does, and then move on.
I have a physics background, and all I can say about EM from that point of view is that learning EM without basic vector calculus is a bit like learning mechanics without basic integral calculus: it's more complicated than it would be with the more advanced math background.
One simple example is just maxwell's equations; compare the two forms in this table from wikipedia[1]. With about one semester extra of college math you can use the form on the right rather than the left.
1: https://en.wikipedia.org/wiki/Maxwell%27s_equations#Formulat...
Can anyone explain that?