= Cold collisionless plasma dielectric response
{title2=$\varepsilon_r=1-\omega_p^2/\omega^2$}
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> $\mathbf P=-ne^2\mathbf E/(m_e\omega^2)$. With $\mathbf D=\varepsilon_0\mathbf E+\mathbf P$, this yields the displayed <relative permittivity> and $\omega_p^2=ne^2/(m_e\varepsilon_0)$. Ions contribute a further negative term smaller by $Zm_e/m_i$ 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 $c^2k^2=\omega^2-\omega_p^2$.
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