Electrostatics in conductors A spherical metal body has a spherical cavity and a charge q is placed at the centre of the cavity and an outside charge (out of the metal sphere )if the inside charge is shifted to the other position, why does the electric potential at the centre of the conductor due  to charge present on outer surface of the conductor doesn't changes? 
 A: It is because the condition of electrostatic equilibrium alone fixes the boundary conditions for the differential equation the electric field (or the electric potential) must satisfy, and this fixes the solution of Poisson's equation.
If you have a conducting sphere, the potential must be constant in its border, to guarantee electrostatic equilibrium. Suppose you know the electric potential $\varphi_1$ inside the sphere with no outside charges. So, inside the sphere you have $\pmb{\nabla}^2 \varphi_1 = \rho/\epsilon_0$, and also $\varphi_1 = V$ at the border of the sphere.
If you put any charge close to the system, you still have electrostatic equilibrium at the conductor. All that can do is change the potential at the border of the sphere. So you must have $\varphi_2 = V'$ at the border, but still $\pmb{\nabla}^2 \varphi_2 = \rho/\epsilon_0$ inside. Notice that $\varphi_2 = \varphi_1 + V' - V$ solves this new equation, and since $\varphi_2$ and $\varphi_1$ only change by a constant, the electric field is the same in both systems.
