Why does a hollow conductor does not have a electric field inside it when it is enclosing a charge? When a positive charge is enclosed in a thick hollow sphere which is a conductor, the inner surface gains a negative charge distribution and due to that the outer gains a positive charge distribution. So there should be electric fields through the walls of the sphere. But it is not possible because the potential difference in a conductor must be zero, the conductor should be a  equipotential surface. Please clear this doubt.
 A: 
Why does a hollow conductor does not have a electric field inside it when it is enclosing a charge?

You can take this as a given in electrostatics. 
You can also take this fact as derivable from your statement that "the potential difference in a conductor must be zero." The potential difference divided by the distance difference is the negative of the field value (E=-$d\phi/dx$). But negative zero is zero, so the field is zero.
If you want a further reason, you can consider what would happen if there were a free charge inside the conductor. If there were some free charge, that charge would be acted on by the field and would feel a force and would move (e.g., to the surface of the conductor). But now we have a moving charge, which means we are in the realm of electrodynamics not electrostatics. This latter comment is not really a reason, just kind of pointing out that to be internally consistent we can't have free charge moving around and still say we are doing electrostatics. 

So there should be electric fields through the walls of the sphere. 

No, you can think of the field "inside the walls" as coming from the "+q" in your picture and all the little "-" signs in your picture. These two fields exactly cancel each other inside the walls of the conductor.
A: Within the conductor, there are two electric fields that cancell out. The one due to the charge enclosed by the spherical shell and the one due to the  charge seperation on the surfaces of the metal. These two oppose each othervand hence the net electric field is zero
A: The sphere is not positively charged. If the negative charges induced by positive ones, the sphere is neutral. Anyway, if there is an electric field inside the conductor there must occur an electric current that is absent in this case.
