Why do 2 charged conductors gain same potential when they are connected via a wire.Can somebody also give a mathematical proof of it?
A charged conductor is equipotential. Reason:
This is because an external electric field causes current to flow inside it, making electrons move , causing seperation of charges, and setting up an electric field inside it which opposes the external electric field. The charge will keep on flowing until, the new electric field grows and equals(in magnitude) the external applied field, such that carriers don't experience any net force. So the net electric field inside a conductor is zero. Since electric field is zero, this implies that each point inside conductor is at same potential, or the conductor is equipotential.
So, when we join two conductors using a wire, we can consider a single conductor, consisting of two conductor bodies and a wire. The charges will flow accordingly such that net charge on system is conserved,and the system will become equipotential.
Charges flow when there is a potential difference. One conductor is at a higher potential than the other.
The fact that the flow arises does not require a mathematical proof because it is pretty obvious from the conditions set for the problem that the electron's response to an electric field is to start moving or constitute a flow that we call current.
As to why that flow arises, it is because of the electric field. There is an electric field existing between the two conductors and that difference in energy that it provides the electrons with causes it to flow because there is a non-zero force that is acting.