The following demonstration solves Laplace’s equation for arbitrary boundary conditions on various geometries. A certain sinusoidal boundary condition produces a solution in the shape of a “Pringle” chip.
Here is a summary of how to calculate one quantity from another in magneto-statics. The hardest conversion to do is to get A from B because there is no easy recipe for undoing a curl. See, for example, Section 2.5.
A dielectric ball placed in a uniform external electric field is an important example that will appear again in the discussion of magnetic fields in matter. The result is a uniform electric field inside the ball, which contributes to the external field as a pure dipole. It is described in Section 3.4.