Elliptical and Circular Current Sheets to Produce a Prescribed Internal Field

Abstract

Any desired two-dimensional magnetic field, which is compatible with Maxwell's equations in empty space, can be produced inside a given elliptic or circular cylinder by a distribution of currents flowing along the elements of the cylinder. Formulas are given for the required current distribution, the resulting external field, and for the total field energy in the internal and external regions. It is further possible to eliminate the external or stray field completely while producing the prescribed inner field by adding a second confocal elliptic (or coaxial circular) current sheet. Formulas are given for the required current distributions. To facilitate their use the formulas are here presented unencumbered by derivations but in a form which makes the correspondence between the elliptic and circular cases, as well as the specialization to median plane fields, appear as simply as possible.