<title>Exact current density solutions for spherical field coils</title>

Abstract

Efficient and high performance magnetic field gradient and active shim coils have always been desirable in today's molecular imaging applications using magnetic resonance imaging (MRI). Various MR techniques, such as high spatial resolution single shot echo-planar imaging (EPI), MR diffusion imaging and MR microscopy as well as cardiac imaging, do require efficient and high performance coils in order to be feasible for imaging small animals. Exact solutions for current density on a spherical surface were explored for generating various orders of field harmonics within a spherical volume. Such compact expressions can be very useful for further coil optimization. After carrying through tedious mathematical expansions and algebraic reductions, exact surface current solutions for spherical field coils have been obtained for various zonal orders and validated. In the paper, the current density on a spherical surface is expanded in associated Legendre polynomial series of order 1, the resulting vector potential of magnetic field was evaluated, and the analytical expressions for both magnetic field and stored magnetic energy were obtained. Then, the continuous current density expressions for various orders of harmonics were identified analytically. Applying the stream function technique, the discrete current wire model can be generated from that of the continuous. The analytical predictions were in an excellent agreement with numerical results. These exact analytical solutions offer a useful mathematical relationship for future magnetic field coil design involving spherical geometry.