The 3D Vector Potential, Magnetic Field and Stored Energy in a Thin cos2 theta Coil Array

The vector potential and the magnetic field have been derived for an arrays of quadrupole magnets with thin Cos(2{theta}) current sheet placed at r = R.{sup bc}. The field strength of each coil within the array, varies purely as a Fourier sinusoidal series of the longidutinal coordinate z in proportion to {omega}{sub m}z, where {omega}{sub m} = (2m-1){pi}/L, L denotes the half-period, and m = 1,2,3 etc. The analysis is based on the expansion of the vector potential in the region external to the windings of a linear 3D quad, and a revision of that expansion by the application of the 'Addition Theorem' from that around the coil center to that around any arbitrary point in space.