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In reading about Faraday cages, I have found that most "real world" designs incorporate multiple layers of mesh. For example, this article from the National High Magnetic Field Lab discusses several different shielded rooms built of varying layers.

In the MagLab's High B/T Facility at the University of Florida's Microkelvin Laboratory in Gainesville, for example, all electromagnetic waves must be excluded. The magnet there is in a "tempest" quality shielded room, featuring walls made of layers of copper and welded steel that absorb the entire spectrum of EM radiation.

My presumption is that each layer is tuned to or built for a particular range of frequencies. It's not clear to me if the tuning is due to thickness requirements for the layer and therefore a cost factor, or if there are other factors driving the design of each layer.

My question then, is can (or should) a Faraday cage be treated as a notch filter as opposed to a high-pass filter? If so, how do you determine the lower end of the notch?

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First, here's a great answer to a similar question on the physics.SE. I will attempt to summarize it a bit and tune it to your question, but I think that all of your answers are already in that thread if you don't wish to read it from me.

I don't think that the mesh is tuned to particular frequencies for EM waves. It's just cheaper than building walls out of thick sheet metal. The thickness of the resulting mesh must be thick enough to attenuate the frequency (see Skin Effect), and the holes in the material must be smaller than the wavelength of signal. Overlapping layers of mesh provides the right thickness, and the randomness of one layer of mesh on top of another should eliminate the holes for most applicable wavelengths, making it a high-pass filter.

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I would say a Faraday cage (at it's simplest) is more like a high pass filter.
And more specifically one that blocks only Electric fields.
The metal box we put our electronics in is a Faraday shield.
Blocking magnetic fields is harder, it takes more layers, of special materials.
(I didn't read NHMFL artcile, I'm at home w/ limited download, I'll read it tomorrow and edit my answer)

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