A uniform electric field $E_{0} \hat{z}$ is produced by fixed charges located at a large distance from a particular spherical region in the field. Find a distribution of charge density and dipole layer density on the surface of the spherical region that will make the field zero in the interior of this region without changing it in the exterior.
The potential due to the charge distributions of the sphere should be $E_{0}z$ inside the sphere (to cancel out the potential $-E_0 z$ created by the uniform field) and a constant outside the sphere (so as to not change the field). Laplace's equation gives the volume charge density as zero (second derivative of $z$ is zero) so we only have surface charge. Using $\partial \phi/\partial r = \sigma/\epsilon_0$ at the boundary with $\phi = E_0 r \cos \theta$ gives $\sigma = -\epsilon_0 E_0 \cos \theta$; this is the surface charge density required to have a uniform field inside the sphere. The problem is, I'm not sure how the dipole layer enters into this. It must have something to do with the second part, which is to make sure that the field outside the sphere is unchanged, which doesn't seem to be true.