If I have an electrical point dipole inside a grounded sphericall shell, what is the electric potential outside of the sphere? In particular, I am confused about how the distribution of charge will take place, and how it will affect the outside. 
 It seems to me that charges induced by the dipole (positive and negative on both  extremes of the sphere) will also produce an electrical field on the outside. Is this correct even for a grounded sphere? 
 A: Starting with an ungrounded spherical shell, we can determine the electrical field outside the shell using Gauss's law: 

The net electric flux through any hypothetical closed surface is equal
  to  1 / ε  times the net electric charge within that closed surface.
  Source: Wikipedia.

Since the dipole, as a whole, is neutral, we can state that the total flux of the field coming out of the sphere is going to be zero. Moreover, in the absence of other charges in the vicinity, we can state that the field everywhere around the sphere and therefore the potential on the sphere will be zero. 
This is because the direction of the field on the surface of the sphere would determine the sign of its potential, but, since the potential has to be the same everywhere on the sphere, the sign of the field would have to be the same as well. But if the sign of the field is the same everywhere and the total flux of the field through the sphere is zero, we have to conclude that the field everywhere has to be zero. 
This state is achieved by the redistribution of the charges on the inside surface of the sphere in such a way that they cancel the effect of the dipole charges. 
Since the potential on the sphere is zero to start with, grounding it would not change anything. If the net charge inside the sphere was not zero to start with and the sphere had some initial potential and field, grounding it would make its potential zero and kill the field.  
