We present an analysis of electromagnetic oscillations in a spherical conducting cavity filled concentrically with either dielectric or vacuum layers. The fields are given analytically, and the resonant frequency is determined numerically. An important special case of a spherical conducting cavity with a smaller dielectric sphere at its center is treated in more detail. By numerically integrating the equations of motion we demonstrate that the transverse electric oscillations in such cavity can be used to accelerate strongly relativistic electrons. The electrons trajectory is assumed to be nearly tangential to the dielectric sphere. We demonstrate that the interaction of such electrons with the oscillating magnetic field deflects their trajectory from a straight line only slightly. The Q factor of such a resonator only depends on losses in the dielectric. For existing ultra low loss dielectrics, Q can be three orders of magnitude better than obtained in existing cylindrical cavities.
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