Imagine an uncharged solid spherical conductor. Inside this spherical conductor, there is a cavity of a weird shape carved out of it. And somewhere inside this cavity, there is a charge +q (or rather, a charge distribution of total charge +q). The charge +q induces an opposite charge -q on the wall of the cavity of the conductor, which distributes itself in a such way that its electric field cancels out that of +q for all points exterior to the cavity. Since the conductor carries no charge, this leaves +q to distribute itself uniformly over the surface of the sphere. How can we detect or figure out the charge distribution in the cavity? Since the charge distribution on the outer surface is uniform, measuring the electric field won't work. Ultrasound and x-ray have been suggested. But I suppose they are meant for detecting mass distribution because I don't see how ultrasound and x-ray can detect a charge distribution (within a cavity of a spherical conductor). I suppose mass distribution and charge distribution are not necessarily related. Ultrasound and x-ray are probably good for detecting the size and the location of the cavity, but they may not be good for detecting the charge distribution in it. I thought about perturbing or varying the ambient electric field in which the sphere is placed, thinking that a sphere with a cavity would react differently from another one that has a different charge distribution in its cavity. By reacting differently, I mean the surface charges on the spheres would move differently. And hence, by observing their motion or re-distribution during a perturbation of the ambient electric field, we can, in principle, deduce the contents or structures underneath. But on second thought, I think the spheres won't react differently since the surface charge of the cavity cancels the effect of the charge in it completely. Before we try to figure out a method of detection, it makes more sense if we first answer this question: Does the requirement of a uniform charge distribution on the outer surface constitute a no-go theorem? In other words, does physics say it's impossible to detect the charge distribution within the cavity (unless you cut open the sphere)? Lastly, for emphasis: Is it a no-go theorem? Is it a no-go theorem? Is it a no-go theorem?