The physical meaning of the magnetic scalar potential, and how to use it to design an electromagnet

The magnetic scalar potential can be used to design electromagnets accurately and efficiently. I will describe a practical construction algorithm: the prescribed field in a “Target” region (constrained only by Maxwell’s equations) specifies boundary conditions which uniquely determine the potential in the surrounding field “Return” region. As required by the physical representation of the magnetic scalar potential, the conducting elements of the resulting coil are directed along equipotentials on the surface of each region, at equal increments of the potential. I give some examples and comments on the limits of precision of the constructed field.