Capacity of shrinking condensers in the plane

Two Theorems are proved, regarding the capacity of a plane condenser having one circular plate of radius increasing to one, and a closed plate on the unit circle having fixed (small) positive capacity. (1) The capacity of the condenser tends to infinity as the radius of the circular condenser tends to one. (2) There can be no asymptotic estimate of the blow-up in (1). The proof relies on the discretization of the problem and on some novel results in discrete potential theory.