3D magnetic optimization of dipole ends with rectangular cross-sections

In particle accelerators, the bending magnets have to be wound with flared ends. This particular shape induces an inhomogeneity in the magnetic field in the ends, which degrades the particle beam stability. This paper proposes a 3D modeling for conductors with rectangular cross-sections. The coil is first divided into blocks (parallelepipeds and arcs). Then the magnetic field is evaluated in every point of space by means of analytical formulas which allow an accurate computation. This method is also faster than numerical integrations of Biot and Savart law. Some optimization examples are finally presented and show that it is possible to minimize, with a few degrees of freedom only, the first field harmonics, integrated along the beam tube.