Dielectric constant of a gaseous medium

In Sect. 9.5, we discussed a rather crude model of an atom interacting with an electromagnetic wave. According to this model, the dipole moment ${\bf p}$ of the atom induced by the wave electric field ${\bf E}$ is given by

$${\bf p} = \frac{e^2}{m_e (\omega_0^{ 2}-\omega^2)} {\bf E},$$ (1147)

where $\omega_0$ is the natural frequency of the atom (i.e., the frequency of one of the atom's spectral lines), and $\omega$ the frequency of the incident radiation. Suppose that there are $n$ atoms per unit volume. It follows that the induced dipole moment per unit volume of the assemblage of atoms takes the form

$${\bf P} = \frac{n e^2}{m_e (\omega_0^{ 2}-\omega^2)} {\bf E}.$$ (1148)

Finally, a comparison with Eq. (1139) yields the following expression for the dielectric constant of the collection of atoms,

$$\epsilon = 1 + \frac{n e^2}{\epsilon_0 m_e (\omega_0^{ 2}-\omega^2)}.$$ (1149)

The above formula works fairly well for dilute gases, although it is, of course, necessary to sum over all species and all important spectral lines.

Note that, in general, the dielectric ``constant'' of a gaseous medium (as far as electromagnetic radiation is concerned) is a function of the wave frequency, $\omega$. Since the effective wave speed through the medium is $c/\sqrt{\epsilon}$, it follows that waves of different frequencies traveling through a gaesous medium do so at different speeds. This phenomenon is called dispersion, since it can be shown to cause short wave-pulses to spread out as they propagate through the medium. At low frequencies ( $\omega \ll \omega_0$), however, our expression for $\epsilon$ becomes frequency independent, so there is no dispersion of low frequency waves by a gaseous medium.