electric field inside the conductor

Claim : For a conductor at electrostatic equilibrium the electric field anywhere beneath the surface of a charged conductor is zero.

Logical Proof: If an electric field did exist beneath the surface of a conductor (and inside of it), then the electric field would exert a force on all electrons that were present there. This net force would begin to accelerate and move these electrons.

But as we know objects at electrostatic equilibrium have no further motion of charge about the surface. (ie., there doesn't exists any current along the surface of conductor) So if this were to occur, then the original claim that the object was at electrostatic equilibrium would be a false claim. If the electrons within a conductor have assumed an equilibrium state, then the net force upon those electrons is zero. And Hence, the charge exists solely upon its outer surface.

Other Proof for Spherical Conductor

Take a Spherical Gaussian surface of radius r < R (the radius of conductor). Since ∫ E . ds = 0 by above logic (Or a current would exist) , Hence Q_enclosed = 0

Electric field inside a conductor is 0 only due to a unique property called free electrons.

Consider that at a particular instant there in some electric field in a conductor.
Now this electric field will exert a force on the electrons and will cause redistribution of electrons.
Now as time passes elec. will continue moving and after some time such a situation will arise that
the redistribution of charges will exert an opp. force on the electrons such that they no longer move.
If not,then they will continue to move unless such a situation arises.

Hence at the end net electric field(more accurately force) inside a conductor will be 0.

but electric field can be generated inside a conductor by bringing a charge near it (or in other words inducing it)

look, charges according to their property tries to attain stability in terms of Potential and also forces..
now as these charges are free to roam around at the surface of your sphere.. they will thus automatically arrange themselves so that the electric field among them results to be zero...

Its a very interesting question of course and the actual ans. too is very interesting..
Actually what happens is that

when an external field is there the charges distribute inside the conductors in this way..
producing a induced field which is opp. site to that of external field and same in magnitude..
so the E_ext.=-E_ind.which cancels out and make the field zero.