  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 electron's 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.
