It is not necessary to specify a hollow superconducting sphere in order to plainly address the the use of a Faraday cage to attenuate the emission of electromagnetic signals originating within the enclosure. Also, we'll just assume a metal foil or panel rather than a metal screen material for the walls (and floor and ceiling) of the enclosure, since the reasons for using screen are unrelated, e.g., simplified ventilation, light weight and visibility advantage over solid panels. I will refer to the cage as a shielded enclosure or shielded room since that is typically where the application lies, i.e., the reduction of EMI/RFI emission for security applications (for example EMSEC in the NSA TEMPEST specifications). I think this model is more applicable than attempting to visualize typically static charge redistribution.
There is an excellent treatment of the problem at Architectural Electromagnetic Shielding Handbook by Leland H. Hemming – 2000.
We will assume the emitting source is placed at a reasonable distance from the wall of the enclosure (inside) such that we are concerned with far field radiation (radiation that has escaped the antenna; 1/6 wavelength is often used as an approximate distance for predominance of radiation field in shielding applications) propagating outwards towards the enclosing shield wall.
The attenuation provided by the shield results from three mechanisms: (1) reflection of an electromagnetic wave when it encounters an impedance disontinuity, e.g., the air to metal impedance discontinuity as the wave encounters the shield (2) absorption within the shield material of portions of the wave energy not reflected as it transfers some energy in heating the shield and (3) possible additional reflections within the shield and at the impedance discontinuity as any remnant of the wave encounters another metal (typical shield material) air boundary. The diagram below is for a wave approaching from outside; simply reverse the label, i.e., let "inside of enclosure" in the diagram be "outside of enclosure."
Electromagnetic radiation is primarily shielded by reflection via mobile charge carriers (electrons or holes) which interact with the radiation. High conductivity of the shield is not required, e.g. on the order of 1 ohm is usually sufficient. Electrical conductivity is not the criterion for shielding (though conductivity enhances shielding) though, since that would require connectivity in the conduction path, e.g., to ground.
Metals, the most common EM shielding material, function mainly by reflection via their free electrons. The secondary mechanism of shielding is absorption via electric and/or magnetic dipoles which interact with the incident EM field. Materials for EMI shielding
The reflection loss depends on the impedance of the shield and the wave impedance. Metals have a much lower surface impedance than the high-impedance electric field and therefore reflect that energy well. The lower impedance magnetic field is comparable in impedance to the metal surface so attenuation of magnetic field is primarily through absorption in the shield. There is typically not an excessive amount of shield thickness required to achieve that end. For example, a Series 81 Shielded Room uses 28-gauge galvanized steel (about 0.0187 inch thick) to exceed NSA 65-694-106/CID 09.12 shielding effectiveness.Series 81 Shielded Room Datasheet (The sum of reflection, absorption or multiple reflections in dB is the shielding effectiveness and the absorption loss is proportional to the thickness of the shield.)
If you would like to see a more mathematical treatment of the behavior of electromagnetic waves as they encounter a boundary, MIT has some useful work here:
Reflection & Transmission of EM Waves