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I am new to this subject, and there's a problem that is really bugging me. If we put a charge $q$ inside a spherical cavity in a spherical conductor, even if it is off-center, the charges on the outer surface will distribute evenly, and will cause a uniform field, as if the charge $q$ was at the center.

The reasoning that is given for this is that since there is no electrical field inside the volume of the conductor, the best configuration for the outer surface charges is to distribute evenly. The field is zero since if it wasn't, then the charges would rearrange to eventually cancel it. But, if a charge is placed off-center in the cavity, I think it will surely be attracted towards the side it is closer to, and move towards it. And this would also cause the charges in the inner surface to redistribute continuously; the charge density would increase gradually in the region close to that charge. The whole assumption that the field inside the conductor is zero is based on the charges having attained static positions, but this doesn't seem to be a static situation? Where am I wrong in my reasoning?

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    $\begingroup$ I think the key is precise language. If you see “If we put a charge q inside a spherical cavity…” then it is not clear whether they mean “we put it there and let go” or “we put it there and hold it there”. You are imagining the former, which is quite reasonable. But most textbooks mean the latter. The textbook authors will tend not to realize the ambiguity because they have spent a lifetime studying and teaching the latter problem so they longer have the perspective of a new student. You might hope this would be caught by good editors or proofreaders. But textbooks are just mostly garbage. $\endgroup$ Commented 15 hours ago

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$\dots$ the charges on the outer surface will distribute evenly $\dots$ . This is because the outer surface of the conducting spherical shell is an equipotential so when bringing charge charge from infinity to the surface the work done is independent of the path taken, ie the surface charge density is constant when looked st from any perspective. The total charge induced on the outer surface of the conducting shell is equal to the charge inside the shell and there is an equal magnitude negative charge induced on the inner surface of the conducting spherical shell but in general the charge per unit area on the inner surface of the conducting spherical shell is not constant.

A charge inside the shell is attracted to the induced charge of opposite sign on the inside of the conducting spherical shell.

If the chrage inside the shell is free to move towards the inner surface of the conducting spherical shell and touch it, neutralising the induced charges of the inner surface of the conducting spherical shell.
So what you would ahve left is no charge inside the conducting spherical shell and a uniformly distributed charge on the outside of the conducting spherical shell equal in sign and magnitude to the charge which was initially inside the conducting spherical shell.

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Yes, you are correct, if its off-centre it will feel a force and be attracted towards the wall of the cavity, but in general in these situations we assume the charge is held fixed by an external agent. The centre position is a position of equilibrium (not stable though)

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(1) Independent from the position of the charge $q$ in the cavity of the conductor the induced charge at the inner surface of the conductor is $-q$. By charge conservation in the conductor, or by Gauss' law, the charge on the outer surface of the conductor must again be $q$, with a zero electric field within the conductor. This holds for any closed shape of the outer conductor.

(2) Due to the spherical symmetry of the outer conductor (and its constant potential) the charge $q$ has to be evenly distributed on the outer surface as a constant surface charge producing a constant electric field at the outer surface. The surface charge has no reason to distribute inhomogeneously. The field in the conductor underneath is zero and the outer charge doesn't "feel" the position of the inside charge.

(3) If the conductor would not be a sphere, the total outer surface charge would still be $q$ and still be independent of the position of the cavity charge, but it would be distributed unevenly on the surface (of constant potential) producing an uneven surface electric field distribution.

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