Some queries on uniqueness theorem

[VishweshM]

Consider a solid conductor with a cavity inside. Place a charge well inside the cavity. The induced charge on the cavity wall and the compensating charge on the outer surface of the conductor will be distributed in a unique way. How does this follow from the Uniqueness Theorem of EM? David Griffith claims this but never gets around to explain it in his book. Any thoughts?

 

[Jano L.]

All densities of charge are determined by the normal component of the electric field near the surface:
$$\sigma = \epsilon_0 E_n$$

The electric field in turn is determined by the electrostatic potential $\varphi(\mathbf x)$. This potential is a solution of Poisson's equation
$$\Delta \varphi = -\frac{\rho}{\epsilon_0},$$
given the known charge inside the cavity and the boundary condition that potential is constant throughout the metal.

The uniqueness theorem states that the solution of this equation and conditions is unique. Since the potential determines everything, the surface charges are unique too.