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Optimal current shell approximation for solenoids of rectangular cross-section

The manipulation of magnetic objects using variable magnetic fields is a growing field of study with a variety of applications. Many magnetic manipulation systems use multiple electromagnets to generate magnetic fields. To control objects in real time, it is necessary to have a computationally efficient method of calculating the field produced by each solenoid anywhere in space. This paper presents a procedure to replace a real cylindrical solenoid of rectangular cross section with infinitely thin shells and rings that generate an equivalent magnetic field. The best approximation for a real solenoid is determined by its physical characteristics. The field produced by these idealized shapes can be calculated expediently using elliptic integrals as can the field gradient and higher-order derivatives. We find that for most real solenoids, the error in the magnetic field approximation is at most 2.5% at a 50% offset and in most cases is much less.