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 because an external electric field causes current to flow inside it, making electrons move, causing separation 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 conducting 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.
When two conducting bodies are connected via a conducting wire, the three objects become a single conducting body.
If two areas of a conducting body are at different potentials, say $V_1$ and $V_2$, then the potential difference $V$ between them will be $V_1-V_2$. From Ohm's law, $I=V/R$. Since the body is conducting, $R$ is everywhere small. The potential difference $V$ therefore causes a finite current $I$ to flow. This current comprises electrons with negative charge, which flow towards the more positive potential so the potential difference reduces. Once the potential is evened throughout the connected bodies, the potential difference is everywhere zero and the current stops.
In a typical conductor the current flow propagates at around 0.6 time the speed of light (though the individual electrons move only slowly), so for most purposes the evening-out of local potentials is instantaneous.
Thus any conducting body is effectively equipotential.