  A small dielectric object with positive permittivity may resonate when the
free-space wavelength is large in comparison with the object dimensions if the
permittivity is sufficiently high. We show that these resonances are described
by the magnetoquasistatic approximation of the Maxwell's equations in which the
normal component of the displacement current density field vanishes on the
surface of the particle. They are associated to values of permittivities and
frequencies for which source-free quasistatic magnetic fields exist, which are
connected to the eigenvalues of a magnetostatic integral operator. We present
the general physical properties of magnetoquasistatic resonances in dielectrics
with arbitrary shape. They arise from the interplay between the polarization
energy stored in the dielectric and the energy stored in the magnetic field.
Our findings improve the understanding of resonances in high-permittivity
dielectric objects and provide a powerful tool that greatly simplifies the
analysis and design of high index resonators.
