How conductors behave in the presence of electrostatic field

We know that conductors like copper , silver, aluminium etc. , have very large number of free and movable charge carriers , usually one free electron per atom. These free electrons are not bound to its atom and moves freely in the space between the atoms. These free electrons can move under the action of electric field present inside the conductor.
Consider an arbitrary shaped conductor placed inside an electric field such that the field in the conductor is directed from left to right. As a result of this electric field positive charge in the conductor moves from left to right and negative charge moves from right to left. As a result there is a surplus negative charge on the left side of the conductor and a surplus positive charge on the right side of the conductor. This induced surplus chare on both the sides of the conductor acts as a source of an induced electric field which is directed from right to left i.e., in the direction opposite to the initial electric field.
Now with the increase in the amount of induced electric charges, magnitude of induced electric field also increases which cancels out the original electric field having direction opposite to it. This results in a progressive decrease in total field inside the conductor. In the end induced electric field cancels out all the initial electric field thus reaching an electrostatic equilibrium where there is zero electric field at each and every point inside the conductor. Hence we can conclude thet,
E=0 inside the conductor
Now if we apply Gauss's law to any arbitrary surface inside the conductor then total charge enclosed by the gaussian surface equals zero as vector E=0 at all points inside the gaussian surface. From this we conclude that
Al the excess charge (if any) is distributed on the surface of the conductor
We have established the fact that there is no E inside the conductor so tangential component of E is zero on the surface of the conductor hence the potential difference between any two points on the surface of the conductor would also be zero. This indicates that the surface of conductor in electrostatics is equipotential one. Since there is no E inside the conductor so all the points in the conductor are at the same potential.
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