Clausius - Mossotti Equation

If $N$ be the number of molecules per unit volume and α the molecular polarizability then

Total polarization $\quad \mathrm{P}=\mathrm{N} \alpha \mathrm{E}_{\mathrm{int}}$

eqn, (10)

$\therefore \quad E_{\text {int }}=\frac{\mathrm{P}}{\mathrm{N} \alpha}$

Further, we know that

$\begin{aligned} \mathrm{D} & =\varepsilon \mathrm{E}=\varepsilon_{0} \mathrm{E}+\mathrm{P} \\ \left(\varepsilon-\varepsilon_{0}\right) \mathrm{E} & =\mathrm{P} \\ \mathrm{E} & =\frac{\mathrm{P}}{\varepsilon-\varepsilon_{0}} \end{aligned}$

Lorentz field is given by

$\mathrm{E}_{\mathrm{int}}=\mathrm{E}+\frac{\mathrm{P}}{3 \varepsilon_{0}}$

Substituting the value for $E$ , we have

$\begin{aligned} & \mathrm{E}_{\text {int }}=\frac{P}{\varepsilon-\varepsilon_{0}}+\frac{P}{3 \varepsilon_{0}} \\ & \mathrm{E}_{\text {int }}=P\left[\frac{3 \varepsilon_{0}+\varepsilon-\varepsilon_{0}}{3 \varepsilon_{0}\left(\varepsilon- varepsilon_{0}\right)}\right] \\ & \mathrm{E}_{\text {int }}=\frac{P}{3 \varepsilon_{0}}\left[\frac{2 \varepsilon_{0}+\varepsilon}{\left(\varepsilon-\varepsilon_{0}\right)}\right] \end{aligned}$ eqn, (13)

Substituting eqn, (10) in (13), we have

$\begin{aligned} & \frac{P}{N \alpha}=\frac{P}{3 \varepsilon_{0}}\left(\frac{\varepsilon+2 \varepsilon_{0}}{\varepsilon-\varepsilon_{0}}\right) \\ & \frac{N \alpha}{3 \varepsilon_{0}}=\frac{\varepsilon-\varepsilon_{0}}{\varepsilon+2 \varepsilon_{0}} \end{aligned}$

$=\frac{\left(\frac{\varepsilon}{\varepsilon_{0}}\right)-1}{\left(\frac{\varepsilon}{\varepsilon_{0}}\right)+2}$

$\frac{\mathrm{N} \alpha}{3 \varepsilon_{0}}=\frac{\varepsilon_{\mathrm{r}}-1}{\varepsilon_{\mathrm{r}}+2} \quad \ldots(14) \quad\left[\therefore \varepsilon_{\mathrm{r}}=\frac{\varepsilon}{\varepsilon_{0}}\right]$

The above equation is Clausius – Mosotti relation, which relates the dielectric constant of the material and polarizability. Thus, it relates macroscopic quantity dielectric constant with microscopic quantity polarizability.

Read More Topics

Internal field or local field

Frequency and temperature dependance of polarization

Classification of dielectric materials