A small polarizable and magnetizable object inserted into a resonant cavity produces a shift in each resonance frequency by an amount δω/ω. Derive an expression for this frequency shift in terms of the time-harmonic dipole moments p(t) and m(t ) induced in the object by the relevant cavity mode and the values of the unperturbed mode fields E₀ and B₀ evaluated at the position of the object. Do this using the Boltzmann-Ehrenfest adiabatic theorem, which states that any adiabatic (quasistatic) change in the state of any oscillating system has no effect on the product of the oscillation period T and the time-averaged total energy (U_EM).