An integral equation for the design of magnetic field coils

The use of the calculus of variations to design magnet coils that have a minimum stored energy was investigated. The problem of computing the winding density along a cylindrical surface that will minimize the stored energy, while at the same time produce a magnetic field with certain desired characteristics is considered. These desired characteristics are prescribed as constraints on various coefficients in the spherical harmonic expansion of the magnetic field. They are introduced into the minimization problem by the use of Lagrange multipliers.