electromagnetism - Does a Faraday cage block outgoing signals?

I see lots of discussions about how a Faraday cage can block EM signals, but almost no one addresses the possible difference between incoming signals versus outgoing signals. It seems to me that those are different situations.

For sake of argument, lets say the cage is a hollow superconducting sphere.

I can see how EM fields outside the sphere could re-distribute the charge on the surface of the sphere in a way that there would be no EM field inside the sphere.

However, in the simple case where a point charge is placed in the center of the sphere, the charge on the surface redistributes in such a way that the EM field outside the sphere is as if the sphere was not there.

Similarly, it seems like if the charge inside the sphere were rapidly moved up and down along the z-axis, it would create an EM wave outside the sphere as if the sphere was not there.

Am I missing something here?

**Edit: I used the term "Faraday Cage" because it is in common use and it makes this discussion easier for others to find. But I am asking about a highly idealized case of a superconducting spherical shell with a charge inside but not connected to the shell. For me it provides a good starting point for understanding real Faraday cages.

By the way: I was once told by a physics professor that this is one of the few things Feynman gets wrong in the Feynman lectures. Which shows how easy it is to get this wrong. Either a physics professor at a major university gets this wrong or Feynman.

Answer

An "RF cage" is commonly used to keep signals IN as well as OUT (see for example the mesh door of a microwave oven.)

The short answer is - reciprocity says "if it works in one direction, it works in the opposite direction". The induced charges on the sphere (in the case of a charge inside it) are just enough to cancel the field outside exactly - because that's how superconductivity works. The E field must be normal to the surface at every point. The charge redistributes to make it so. Any field due to the charge inside is exactly cancelled by the induced charge on the surface. Note - this does require the cage to be grounded: that is especially important at low frequencies (where the wave length of the radiation is long compared to the size of the cage). Grounding ensures there is no net charge on the sphere-plus-contents, and should eliminate your concern.

So yes, a Faraday cage blocks outgoing signals.