I am aware that the (net) electric field within an empty cavity inside a conductor is 0 regardless of the charge on the conductor and the presence of any external fields outside the conductor. I have seen this described as contents of the cavity being “sheilded” from external influence. However, I feel like the word “shielding” would be meaningful only if the cavity actually contained some charge that could interact with the external field in the first place. To that end, I would like to know whether there is a difference in the electric field due to all sources excluding the charge within the cavity, $\vec{E}_\text{other}$, and thus a difference in the force on the charge, $\vec{F}=q\vec{E}_\text{other}$, depending on the presence of charge outside the conductor.
I have come up with the following argument but I am not convinced of its correctness:
Suppose that an external charge distribution $Q$ is present outside a (possibly charged) and there is an empty cavity inside it. Then the electric field inside inside the cavity is $0$. Now add the charge $q$ by infinitismal amounts inside the cavity. At each step, there the charge added and the charge subsequently induced on the innerwall remain unaffected by any other fields (as they add to $0 $ within the cavity and conductor) and thus distribute themselves as they would in the case with no external fields, which means that the charge in the cavity and the innerwalll produce no field and thus force on external sources of charge once the transient phase is over. Moreover, these sources remain unaffected even in the transient phase due to the infinitesimal nature of the charge added in each step.
This determines a valid solution of $\vec{E}$ inside the cavity which is bounded by a conductor with a fixed amount of charge which is simply the vector sum of the field due to the charge and the innerwalls. By the uniqueness theorem, this is the only solution. Therefore, the net electric field inside the cavity and thus the force on the charge within the cavity (produced by $\vec{E}_\text{net}-\vec{E}_\text{charge}$ is identical to the case where there are no external sources of charge.
