On the magnetic field of rectangular coils

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Abstract
A derivation of the magnetic field, created by rectangular single-turn coils, is achieved with the aid of magnetic scalar potential. The transformation properties of spherical harmonics under rotation permit one to analyze the magnetic field of a system of several coils. Application to a four, rectangular coil system, generating a magnetic field gradient, leads to the optimal dimensions of each element of the system: h=1.732, b=2.486, when a=1 and β=45°, where a represents the half-width of the rectangle, b its half-length, h the distance between the origin of the reference frame and the center of each coil, and β is the angle between the directions of the normals to the planes of two neighboring coils.

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