  We present a modal approach to compute the Purcell factor in Mie resonators
exhibiting both electric and magnetic resonances. The analytic expressions of
the normal modes are used to calculate the effective volumes. We show that
important features of the effective volume can be predicted thanks to the
translation-addition coefficients of a displaced dipole. Using our formalism,
it is easy to see that, in general, the Purcell factor of Mie resonators is not
dominated by a single mode, but rather by a large superposition. Finally we
consider a silicon resonator homogeneously doped with electric dipolar
emitters, and we show that the average electric Purcell factor dominates over
the magnetic one.
