  We analyze microwave cavity perturbation methods, and show that the technique
is an excellent, precision method to study the dynamic magnetic and dielectric
response in the $GHz$ frequency range. Using superconducting cavities, we
obtain exceptionally high precision and sensitivity for measurements of
relative changes. A dynamic electromagnetic susceptibility
$\tilde{\zeta}(T)=\zeta ^{\prime}+i\zeta ^{\prime \prime}$ is introduced, which
is obtained from the measured parameters: the shift of cavity resonant
frequency $\delta f$ and quality factor $Q$. We focus on the case of a
spherical sample placed at the center of a cylindrical cavity resonant in the
$TE_{011}$ mode. Depending on the sample characteristics, the magnetic
permeability $\tilde{\mu}$, the dielectric permittivity $\tilde{\epsilon}$ and
the complex conductivity $\tilde{\sigma}$ can be extracted from
$\tilde{\zeta}_{H}$. A full spherical wave analysis of the cavity perturbation
is given. This analysis has led to the observation of new phenomena in novel
low dimensional materials.
