
Surprises of the Faraday Cage - leephillips
https://sinews.siam.org/DetailsPage/TabId/900/ArtMID/2243/ArticleID/757/Surprises-of-the-Faraday-Cage.aspx
======
Animats
Feynman wrote " _The method we have just developed can be used to explain why_
electrostatic _shielding by means of a screen is often just as good as with a
solid metal sheet. Except within a distance from the screen a few times the
spacing of the screen wires, the fields inside a closed screen are zero. "_

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](http://www.feynmanlectures.caltech.edu/II_07.html)
(Section 7.5) [1]
[http://ipnpr.jpl.nasa.gov/progress_report2/II/IIO.PDF](http://ipnpr.jpl.nasa.gov/progress_report2/II/IIO.PDF)

~~~
bjornsing
But the OP is writing about _electrostatic_ shielding too... What makes you
think the below section from the OP does not apply to _electrostatic_
shielding?

"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.

------
plesner
> L1. There are gaps out there.

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.

[1]:
[http://h14s.p5r.org/2012/12/bernoulli.html](http://h14s.p5r.org/2012/12/bernoulli.html)

~~~
e12e
Reminds me of Alan Kay's frequently mentioned nugget about Kepler, that he
waited for _years_ to try and fit orbits to ellipsis, because, being so close
to circles and ovals, _surely_ all the people doing this stuff before had
tried ellipsis, and found them wanting? (No, they hadn't).

~~~
Gibbon1
I've been lead to understand that some issues may have been before Tycho Brahe
no one had made accurate measurements, especially of the entire tracks across
the sky, not just 'interesting' places.

~~~
e12e
True, but according to eg (random Google hit):

[http://galileoandeinstein.physics.virginia.edu/lectures/tych...](http://galileoandeinstein.physics.virginia.edu/lectures/tycho.htm)

"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"

------
ajnin
Pretty interesting read. I'd like to know the answer to the 3rd question at
the end as well, microwave manufacturers must have known about this all along
since the doors embed a chunky piece of metal with small holes and not a
lightweight grid of thin wire as one might expect with the "traditional"
understanding of the Fraday cage effect.

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.

~~~
colechristensen
Radio propagation is quite a bit like magic, and if you're an engineer making
a microwave, you probably really don't care that much about the theoretical
underpinnings. The thought process goes something like this:

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.

~~~
DINKDINK
Your post is a great description of the phenomenon that Tableb describes in
his book Antifragility, as "Lecturing Birds how to fly" [1] Essentially the
argument is: we learn by doing and formalize that knowledge in research. Birds
don't need to be lectured on fluid mechanics(drag,lift,etc) to understand how
to fly.

[1][http://www.yourparttimehrmanager.com/lecturing-birds-to-
fly-...](http://www.yourparttimehrmanager.com/lecturing-birds-to-fly-
overreliance-on-formal-education/)

~~~
kleer001
And it seems to me a trivial step to then say that formalized knowledge
churned through the science leads us to things that are impossible for birds.

------
preinheimer
I, perhaps naively, thought it was a well understood phenomenon. With random
references in sci-fi, and a quick appearance in Enemy of The State (movie), I
thought it had all been locked down.

It's somehow inspiring to learn how much is there for us to learn.

~~~
oneloop
What does "well understood" mean exactly? Ask the average inhabitant of Hacker
News and he will tell you that "Faraday Cage the name of a phenomenon whereby
a box of metal prevents electromagnetic fields inside", or something like
that. But ask him how the strength of the electromagnetic field depends on the
wire radius and mesh size, and he might not know the answer.

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 :-)

~~~
eternauta3k
A second-year Physics student can work out RF shielding using Kirchhoff's
diffraction formula, or just looking at the wave equation in the k domain
(spatial frequencies). Electrostatic shielding is the hard bit addressed in
the article.

~~~
oneloop
You didn't really read the post, did you? Keep on winging it mate.

~~~
eternauta3k
I did. Where am I mistaken?

------
reeboo
> Intuitively, sinusoidal oscillation in one direction corresponds to
> exponential decay in the direction at right angles in the complex plane. A
> contour integral estimate of Fourier coefficients exploits this decay to
> prove exponential accuracy.

I couldn't get past this part. What is the author saying here?

~~~
oneloop
Some aspects of electromagnetism are better discussed in terms of complex
analysis. I believe the first sentence is referring to this:
[https://en.wikipedia.org/wiki/Euler%27s_formula](https://en.wikipedia.org/wiki/Euler%27s_formula)
The function e^(i x) is a (complex) sinusoidal oscillation when x is a real
number, but is a (complex) exponential funciton when x is an imaginary number.
The phrasing "in the direction at right angles in the complex plane" refers to
the fact that the real and imaginary axis are perpendicular to each other.

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](https://en.wikipedia.org/wiki/Residue_theorem)

~~~
evanpw
The residue theorem implies that you can compute some integrals on the real
line by closing up the contour in the complex plane and accounting for any
poles that you've enclosed. The trick is to choose your contour so that the
contribution to the integral of the new piece goes to zero as the contour gets
larger and larger (think of a real piece which goes from -R to R, and a
semicircle in the upper half-plane connecting those two points; as R->infty,
the real part of the integral goes from -infty to infty). One reason the new
piece may go to zero is that oscillating functions on the real line turn in to
exponential decay as you go up or down along the imaginary axis (as you point
out), so as the new piece of the contour moves up or down, its contribution to
the integral gets exponentially smaller.

(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...](http://web.williams.edu/Mathematics/sjmiller/public_html/302/coursenotes/Trapper_MethodsContourIntegrals.pdf)).

~~~
oneloop
I know what it is, how to use it and all that. I just don't know how to make
sense of the specific phrase that the parent asked about.

------
adrianratnapala
I'm still reading the article, so perhaps I am missing something. But it seems
to me that Feynman was perfectly entitled to think in terms of point (really
line) charges.

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.

~~~
oneloop
Let me tell you how I understand this.

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 :-)

~~~
adrianratnapala
It's absolutely true that fat wires will behave at least a little differently
from the point-like wires.

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.

~~~
stouset
Perhaps I read this wrong, but isn't the opposite story the case? As wires
become more point-like, the effective shielding drops to zero. However, when
wires become _precisely_ points, shielding becomes perfect?

~~~
adrianratnapala
It dosen't drop to zero. It is worse than for fat wires -- but the maths is
easier.

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.

------
MichailP
Analysis of such problems is usually done using numerical electromagnetic
tools, based on finite element method (FEM), finite difference time domain
method (FDTD), method of moments (MOM) or hybrid methods. Use of such tools is
not a straightforward task, and it takes a lot of effort to create accurate
models. Numerical methods are used because apart from very few problems almost
all problems of practical interest (such as microwave ovens mentioned in the
article) can only be solved numerically.

------
gtrubetskoy
Another example in science that you'd think is well understood, but isn't is
water. We still don't know why and how clouds form or why ice is slippery, we
don't even know whether liquid water is strictly speaking a liquid and not a
crystal some of the time.

~~~
chadgeidel
We don't know why ice is slippery? I thought that was understood as
hydroplaning (small amounts of ice melt and then you have a low-friction
scenario where water separates the two surfaces). I've read that when ice is
cold enough and the other surface (a rubber tire for example) is also below
freezing point there is good traction.

Is there a description of this that a layman such as myself with a college
degree in mathematics (heavy undergraduate physics) could understand?

~~~
nkurz
It may be too basic for you, but I liked this 2005 overview from Physics
Today:
[http://www.chemistry.northwestern.edu/documents/about/Rosenb...](http://www.chemistry.northwestern.edu/documents/about/RosenbergPhysicsToday.pdf)

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".

~~~
chadgeidel
Thank you!

------
KarthikaCohen
There will be a live chat with the author of the article, Nick Trefethen, on
Thursday, August 18 from 12 noon - 1:00 p.m. ET through the comments page on
the SIAM News site:
[https://sinews.siam.org/DetailsPage/TabId/900/ArtMID/2243/Ar...](https://sinews.siam.org/DetailsPage/TabId/900/ArtMID/2243/ArticleID/757/Surprises-
of-the-Faraday-Cage.aspx) Be sure to join us and ask any questions of the
author regarding this topic!

------
Panoramix
So, intuitively speaking, why does the field decreases with increasing wire
radius?

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)

~~~
oneloop
epsilon log(r) is the "shielding". I suspect that the field inside will be
attenuated by a factor of 1/(epsilon log(r)).

~~~
Panoramix
epsilon is the gap, so either way it should be on opposite side of the log(r).
Smaller gap, stronger shielding but smaller radius weaker shielding. Am I
going nuts?

~~~
oneloop
Hm I guess you're right. I don't know what to tell you. I guess we'd have to
have a look at the exact definitions they use in the paper :-)

------
Wildgoose
I distinctly remember a conversation with a friend when we were both working
together around 1989 which was about the propagation of microwaves and the
shielding in microwave ovens. He was an Electronics graduate from Durham
University and he said that it wasn't properly understood.

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!

------
danielmorozoff
This is a fantastic paper. Truly shows a deeper intuition can be gained if a
rigorous mathematical description is made. Upon reading it, it seems so
obvious how treating wires as point charges would give flaws, and the surprise
that the radius is more important than the interwire distance, after seeing
the results make sense. Remarkable work.

------
ChuckMcM
I'm just saying _" Intuitively, sinusoidal oscillation in one direction
corresponds to exponential decay in the direction at right angles in the
complex plane. A contour integral estimate of Fourier coefficients exploits
this decay to prove exponential accuracy."_

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.

------
lifeisstillgood
So, is there a good foundational course on RF? It is more and more important
but I have almost no intuition (nor it seems the maths) to approach it beyond
"ohh look, three bars of strength, I can rely on IP now"

~~~
aidenn0
RF is a really big field and it depends on whether you want to approach it
from a theoretical or practical direction.

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...](https://en.wikipedia.org/wiki/Maxwell%27s_equations#Formulation_in_SI_units)

------
IshKebab
I tried to build a faraday cage out of a big tube of aluminium (~10mm thick)
with similar caps. Machined so very thin gaps, and I even taped them up with
copper tape. The weird thing is it doesn't work. I put a BLE transmitter in
there and it reduced the signal strength by maybe 40 dB, but I was expecting
perfect containment.

Can anyone explain that?

------
madengr
The Faraday cage shields perfectly against electrostatic fields, not
electromagnetic fields. That's where the skin depth is important.

~~~
oneloop
Very not static

[https://upload.wikimedia.org/wikipedia/commons/c/c0/Cage_de_...](https://upload.wikimedia.org/wikipedia/commons/c/c0/Cage_de_Faraday.jpg)

~~~
madengr
I suppose we have a different notion of shielding. If you look at this page,
you'll see the shielding effectiveness dropping with frequency, hence the skin
depth.

[http://www.ets-lindgren.com/iSeries-71](http://www.ets-
lindgren.com/iSeries-71)

------
andyidsinga
next time someone suggests a faraday cage for rf work, I think I'll suggest a
maxwell array instead :)

