The physical meaning of the magnetic scalar potential and its use in the design of hermetic electromagnetic coils

The magnetic scalar potential U(r↔) has an intuitive physical interpretation, representing the electric current I = ΔU, which must be directed between any pair of isocontours along the boundary of a region, differing in potential by ΔU, in order to generate the corresponding magnetic intensity H↔ =-Δ↔ - U inside the region with no tangential component outside the boundary. This physical significance is exploited to invert the design process of hermetic (fringeless) electromagnetic coils from standard iterative techniques (calculating the magnetic field of refined current distributions) to a procedure for calculating the exact surface currents required to generate a given field configuration. A practical construction algorithm is given to produce the prescribed field in a "Target"region (constrained only by Maxwell's equations) while confining the fringe field to a specified hermetic "Return"region. Example coils are analyzed along with a discussion of the limits of precision of the constructed field.