Can I suggest a simple setup with the same physics as the Faraday cage, but for which the analysis is simpler?
Code:
_/\/\/\__
Vext--||--| R |--||-- ground
| |
|---||----|
"cage"
We have a capacitor-gap which I call the "cage"; a conducting path around it has a small resistance "R" (like the walls of a Faraday cage). We may confirm that this is indeed a normal Faraday cage. You can put whatever electric field you want to
outside of the cage (Vext), but since such fields are only capacitively coupled to the isolated RC loop, the field across the "cage" is zero no matter what Vext is. That's the point of a Faraday cage.
You can confirm that the "conducting path around the cage" is necessary: if you remove the R from the circuit, breaking that path, you're left with
Code:
Vext--||--| |--||-- ground
| |
|---||----|
"cage"
where the center capacitor is no longer a Faraday cage, and which DOES feel the effects of Vext. So we confirm that the Faraday cage works only when its walls are conductive.
Now we'll add a battery Vbatt, connected to the cage, such that current flows through the walls. This is Myriad's current-carrying Faraday cage, in circuit form:
Code:
___(+|-)__
| Vbatt |
| |
|_/\/\/\__|
Vext--||--| R |--||-- ground
| |
|---||----|
"cage"
Let's note the details: if (Vbatt=0) it's a normal old Faraday cage. If Vbatt is nonzero, it looks like current flows through the "wall" R but also creates a field inside the "cage". The question is: NOW, what is the effect of Vext? Does it change the field in the central capacitor? ("because the current is flowing, the Faraday cage stops screening external fields")? Does Vext have the effect of turning off the loop current ("it's like a field effect transistor, the current cannot go past the stationary charges that screen Vext")?
Analyze away. This is obviously simpler than any sort of 3D field cage you might be considering, and I hope you agree that the electromagnetic principles are the same.