Vector potential and stored energy of a quadrupole magnet array

The vector potential, magnetic field and stored energy of a quadrupole magnet array are derived. Each magnet within the array is a current sheet with a current density proportional to the azimuthal angle 2θ and the longitudinal periodicity (2m-1)π/L. Individual quadrupoles within the array are oriented in a way that maximizes the field gradient. The array does not have to be of equal spacing and can be of a finite size, however when the array is equally spaced and is of infinite size the solution can be simplified, We note that whereas, in a single quadrupole magnet with a current density proportional to cos2θ the gradient is pure, such purity is not preserved in a quadrupole array