Before embarking on an explanation, I need to explain what has been bothering me personally, about this problem. In a hand-waving way, a gold leaf electroscope measures how ‘charged’ an object is. If an object has an excess of charge (AKA the object is charged) then when connected to an electroscope, the gold leaf will rise through electrostatic repulsion of like charges. We all know that. OK, so we charge up a hollow sphere or Faraday cage, to say 10kV above ground, and connect our electroscope to the outside of the sphere, and of course the gold leaf flips up, confirming the fact that the sphere is charged. The entire sphere is at the same potential, because it is a metallic conductor. We then place an uncharged electroscope inside the charged sphere, and connect the electroscope terminal to the inside surface of the sphere, and it registers nothing, yet the inside and outside surfaces of the sphere are both at the same potential. How so? It does not need to be an electroscope, human hair standing on end performs the same experiment. Apparently we cannot detect or measure the charge on the sphere, if the observer or experiment is placed within the sphere, at least with an electroscope.
I believe that much of the confusion (or at least my confusion) stems from a misunderstanding of what and how an electroscope actually measures. As we know, an electroscope is a single terminal device. If the gold leaf is uncharged (equal number of +ve and –ve charges), then there is no repulsive force on the leaf, and no deflection of the leaf. If we push charge into the electroscope so that the leaf is ‘charged’, then the leaf will deflect.
That said, an electroscope can also function as a crude and insensitive voltmeter, which I would argue is how it is actually being used in these discussions. How so, when an electroscope has only one input terminal, while a voltmeter is a differential device with two input terminals, that measures potential difference? As I understand the original posting, the Faraday cage is charged up to 10 kV, meaning that a 10kV power supply is connected between Earth, and the cage. A wire is then taken from the cage to an electroscope, which will deflect. The deflection is proportional to the 10 kV, and the capacitance from the electroscope to Earth. The charge that will flow into the electrometer (which is what makes it deflect) is given by Q=CV, so for a given 10kV, the electrometer deflection will be proportional to this capacitance. That is why when an electroscope is used in this way, the case is grounded, to increase this capacitance. The net effect of all this is that the electroscope is actually measuring a differential voltage between it’s input terminal and case, just as with a conventional differential voltmeter.
Now we consider the case where the electroscope is placed inside the Faraday cage, which for convenience can be a spherical conducting shell. We connect the electroscope input terminal to the inside surface of the sphere/cage, but to what do we connect the electrometer case? The only thing we can connect it to is the sphere/cage, but if we do that then there will be zero volts between the input and case as they will be connected to the same thing, and we will get zero reading. We could leave the case unconnected, but that would be even worse, decreasing the capacitance from the electroscope to the cage. It’s best to forget about using an electroscope voltmeter at all, and just bring your favourite voltmeter into the cage, but you will have no more success in detecting that the cage is charged and by how much, for the only way to do that is to measure voltage between the sphere and ground, which you fundamentally cannot do if you and your equipment are inside the cage. Getting back to the electroscope, the only way that you can push charge into the electroscope and make it deflect, is if you have a potential difference, but there are no potential differences or electric fields to be found inside the cage, so the electroscope is never going to deflect.
So what about the suggestion that we could detect the mass of excess charge on a metallic object within the sphere, that had touched the inside surface. That won’t work either, because all of the excess charge is distributed on the outside surface of the sphere/cage, so can’t be ‘seen’ or detected from inside the cage. To put that another way, as your mate outside the cage starts cranking up the 10kV power supply, at all times there are no electric fields anywhere inside the cage, so no driving force to push charge around so as to create an excess charge on anything inside the cage. Or deflect an electroscope or voltmeter inside the cage. I believe the OP is right. From within the cage, there is no way of knowing whether the cage is charged or not, or by how much.
From outside the cage it’s easy to detect if the cage is charged. If a large, electrically neutral body like Earth is conveniently nearby then use a voltmeter. If the cage is in deep space, then observe the motion of nearby charged particles.
Interesting topic, even if the link to metrology is tenuous.