How would I evaluate the surface charge density on the inner and outer surface of a neutral, spherical, conducting shell which has an off-centre charge $q$ inside? I believe that we can not use the method of image charges since even though we know the potential of the shell is constant we do not know its value; it is not even fixed (unlike a grounded shell).
[Physics] Surface charge density for an off-centre charge in a spherical shell
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Best Answer
In the case of a grounded conducting shell, it is well known that the method of images can be used to calculate how the total charge $-q$ on the inner surface is distributed. The solution is given in the wikipedia link above.
Now, you ask what happens if the the potential of the shell is fixed, but not necessarily zero. The electric field inside the conductor must still be zero. Using Gauss's law and a surface that is inside the conductor we know that there then must still be a charge $-q$ distributed over the inner surface in some way. To conserve charge there must now also be a charge $+q$ distributed over the outer surface.
We know that the electric field lines leaving the outer surface of a conductor must be perpendicular to that surface. But for a spherical outer surface, the only way this can be arranged and keep the outer surface as an equipotential is if the charge $+q$ is distributed uniformly over that surface. This spherically symmetric arrangement of charge contributes no net electric field inside the conducting shell or in its interior. i.e. the electric field outside the conducting shell will be exactly equivalent to that of a positive charge at the centre of the shell. Furthermore, as the contribution of this charge to the field inside the shell is zero, then it cannot alter the electric field deduced using the method of images for a grounded shell.
As the electric field at the inner surface of the shell is unchanged, then the surface distribution of charge must also be unchanged.
Therefore nothing changes about the inner shell charge surface distribution if the shell is not grounded.