What does the Neumann boundary condition imply for the electric flux lines? If I prescribe a Neumann boundary condition in an electrostatic problem, that means that I am prescribing a surface charge density, and in the way I understand it that means that the electric flux lines have to be normal to this surface. Is this right?
 A: Neumann boundary conditions is a general mathematical term for the conditions on the derivatives of a function. It is applicable in this context in the sense that we are talking about the Laplace/Poisson equation for the electric potential, $\varphi$, although the conditions are actually on the electric field strength, $\mathbf{E}=-\nabla\cdot\varphi$, and the electric displacement field $\mathbf{D}$ (see here):
$$\mathbf{n}_{12}\times (\mathbf{E}_2 - \mathbf{E}_1) = 0,\\
\mathbf{n}_{12}\cdot (\mathbf{D}_2 - \mathbf{D}_1) = \sigma_s,$$
where $\mathbf{n}_{12}$ is the normal to the surface. The first condition impose continuity of the component of the electric field parallel to the surface, whereas the second means that its normal component $\mathbf{E}_i = \mathbf{D}_i/\epsilon_i$ changes by a jump. Thus, the direction of the electric field lines, which is the direction of $\mathbf{E}$ changes.
Note that the electric field lines do not have to be perpendicular to the surface! This is the case for metals, since the electric field inside a metal should be zero ($\mathbf{E} = \mathbf{D} = 0$), but not for dielectrics.
