Cold collisionless plasma dielectric response (${\epsilon }_r=1-{\omega }_p^2/{\omega }^2$)

In an unmagnetized plasma, an electron displacement driven by $\mathbf{E}{e}^{-i\omega t}$ obeys $-m_e{\omega }^2{\mathbf{x}}_e=-e\mathbf{E}$. Its dipole moment is $-e{\mathbf{x}}_e$, giving electric polarization $\mathbf{P}=-n{e}^2\mathbf{E}/(m_e{\omega }^2)$. With $\mathbf{D}={\epsilon }_0\mathbf{E}+\mathbf{P}$, this yields the displayed relative permittivity and ${\omega }_p^2=n{e}^2/(m_e{\epsilon }_0)$. Ions contribute a further negative term smaller by $Z{m}_e/m_i$ for one ion species of charge $Ze$. Neglecting it requires heavy ions at electron-wave frequencies; collisions, background magnetization and pressure dispersion are excluded. The transverse electromagnetic wave relation is $c^2{k}^2={\omega }^2-{\omega }_p^2$.