Dual energy methods in electromagnetism using tubes and slices

Geometrical factors play an important role in the calculation of electromagnetic fields. They are of two kinds which are closely related: there is the geometry of the physical device or system and there is the inherent geometry of the field. The authors explore the second type of geometrical problem with the help of the notation of differential forms. It is shown that vector fields have two properties which combine to define the energy distribution. The two properties lead to two variational principles from which the equilibrium energy can be obtained. In numerical calculations, the two values of the energy give lower and upper bounds to the correct value. A subdivision into tubes and slices provides the most economical way of using the geometrical properties of the field