# Charge on a conductor's surface

I take a charged conductor completely insulated. The charge is distributed over the surface, maintaining the surface at a given potential.

The charge distribution that gives this potential is unique?

Yes it is. Not only is it unique, but you can also calculate it using the boundary conditions for the normal electric field component: $$\hat n .(\mathbf D_{1}-\mathbf D_{2}) = \sigma$$ In the conductor $\mathbf D_2=\epsilon_0 \mathbf E_2=0$ and just outside the conductor $\mathbf D_1=\epsilon_0\mathbf E_1 =- \epsilon_0 \boldsymbol\nabla \phi$ . Thus, the surface charge density is: $$\sigma=-\epsilon_0 \ \hat n .\boldsymbol\nabla \phi \ |_{boundary}$$ So the charge distribution $\sigma(\mathbf x)$ that gives the potential $\phi(\mathbf x)$, is unique because you can calculate it from $\phi$ uniquely. However, your question is if the charge distribution is unique given just the potential on the surface $\phi_0$, not the complete potential distribution $\phi(\mathbf x)$. The answer to that is also yes, because the complete potential distribution can be calculated from just the potential on the surface using the laplace equation (or the poisson equation if there are other charges around), and the solution to the laplace equation is unique for every given set of boundary conditions.