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Conceptual designs of 1.5 and 3.0 T full-body magnetic resonance imaging (MRI) magnets using conduction cooled MgB2 superconductor are presented. The sizes, locations, and number of turns in the eight coil bundles are determined using optimization methods that minimize the amount of superconducting wire and produce magnetic fields with an inhomogeneity of less than 10 ppm over a 45 cm diameter spherical volume. MgB2 superconducting wire is assessed in terms of the transport, thermal, and mechanical properties for these magnet designs. Careful calculations of the normal zone propagation velocity and minimum quench energies provide support for the necessity of active quench protection instead of passive protection for medium temperature superconductors such as MgB2. A new ‘active’ protection scheme for medium Tc based MRI magnets is presented and simulations demonstrate that the magnet can be protected. Recent progress on persistent joints for multifilamentary MgB2 wire is presented. Finite difference calculations of the quench propagation and temperature rise during a quench conclude that active intervention is needed to reduce the temperature rise in the coil bundles and prevent damage to the superconductor. Comprehensive multiphysics and multiscale analytical and finite element analysis of the mechanical stress and strain in the MgB2 wire and epoxy for these designs are presented for the first time. From mechanical and thermal analysis of our designs we conclude there would be no damage to such a magnet during the manufacturing or operating stages, and that the magnet would survive various quench scenarios. This comprehensive set of magnet design considerations and analyses demonstrate the overall viability of 1.5 and 3.0 T MgB2 magnet designs. [ABSTRACT FROM AUTHOR]