In a conductor, all the charges must lie on the surfaces. In addition to the charge q on the inner sphere, note that induced positive and negative charges can strategically be distributed on the surfaces of the outer conduction shell. Which results in no net charge on the shell.
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Please see the attached file.
(a) Here the charge density σ is the charge per unit area. As the charge given to a conductor always remains on the outer surface and on a sphere it is uniformly divided on the surface, the charge on the sphere is q and the surface area is 4 π R2 hence the density on the sphere of radius R is given by
This charge will induce equal and opposite charge -q on the inner surface of the outer shell and the corresponding free +q charge will come to the outer surface of the shell.
[in terms of flux we can say that there will be no electric field inside a conductor and hence no flux inside metallic shell, all the flux originating from + q must be terminated on the inner surface ...
For a system of concentric sphere and shell with a given charge on the sphere the charge densities, field and the potentials are calculated at different points. With that the coulomb force on a charge due to system of point charges calculated.