First consider the sphere with charge Q which starts out physically remote from anything else. We have by Gauss:

Where S is any closed surface enclosing the sphere. The problem has spherical symmetry so if we choose a sphere of radius r for the surface S then the E field is normal to the surface and the same magnitude everywhere. This makes the integral easy to solve:

or

Now the voltage at the point of infinity is zero so we can find the voltage on the surface of the sphere by integrating E from infinity to the surface of the sphere:

or

Now once the sphere is placed inside your capacitor the plates of the capacitor force the E field to be uniform between the concentric plates. Therefore, in this area the field will have cylindrical symmetry. We use Gauss again only this time we choose a cylinder radius r, length l for our integrating surface:

or

Substituting for Q gives: