Formulation of design rules for NMR imaging coil by using symbolic manipulation

A common problem in electrical technology is to design a current carrying coil that will produce a given magnetic field. For over a hundred years an equation, the Biot-Savart law, has been available that defines precisely the magnetic field at any point as a line integral along the path of the electric currents that are the sources of the field. In principle then, the design problem is straightforward - it is merely necessary to invert the Biot-Savart law and find a path whose line integral has the given values at the specified field points. However, the actual solution is not trivial and there is a continuing need for improved computational methods for relating magnetic fields to their sources. Two forms of series expansion will be considered here - the ordinary Taylor series and the expansion in spherical harmonics. Other possible expansions - such as Bessel function methods for cylindrical coordinate problems - usually involve integrals over some eigenparameter rather than discrete sums and are not directly competitive with the methods discussed here.