The quasistatic equations that describe the scattering and absorbance cross-sections for very small spherical nanoparticles are:
where  is the wavenumber of the electric field,  is the radius of the particle,  is the relative permittivity of the dielectric medium and   is the relative permittivity of the nanoparticle defined by
also known as the Drude Model for free electrons where  is the plasma frequency,  is the relaxation frequency of the charge carries, and  is the frequency of the electromagnetic radiation. This equation is the result of solving the differential equation for a harmonic oscillator with a driving force proportional to the electric field that the particle is subjected to. For a more thorough derivation, see surface plasmon.
It logically follows that the resonance conditions for these equations is reached when the denominator is around zero such that
When this condition is fulfilled the cross-sections are at their maximum.
These cross-sections are for single, spherical particles. The equations change when particles are non-spherical, or are coupled to 1 or more other nanoparticles, such as when their geometry changes. This principle is important for several applications.
Rigorous electrodynamic analysis of plasma oscillations in a spherical metal nanoparticle of a finite size was performed in.