The electromagnetic flow pattern in one-and two-dimensional gratings of large nonconducting circular cylinders in an external uniform field is studied. It is shown that the cylinders can be replaced by suitable arrangement of pairs of dipoles opposite in sense and placed on an axis perpendicular to the external field. The relation between the spacing of the dipoles and the cylinder radius R yields the complex potential as a function of R, with the imaginary part used to describe the flow pattern. Solutions for unit cells of square, rectangular, and infinite-strip geometry are discussed, with details in tables and plots. In the same context, a uniform field obliquely oriented toward the grating is considered, as is the flux pattern.