Question

The Clausius–Mossotti equation relates

• A. Polarization and temperature
• B. Dielectric constant and polarizability
• C. Conductivity and permittivity
• D. Stress and strain

Hint:

Remember that the Clausius–Mossotti equation bridges the macroscopic world (\(\varepsilon_r\), measured in a lab using a capacitor) with the atomic world (\(\alpha\), calculating how easily electron clouds distort).

Answer 1

The Correct Option is B

Concept: The Clausius–Mossotti equation connects a macroscopic measurable dielectric parameter (the relative permittivity or dielectric constant, \(\varepsilon_r\)) of a material to the microscopic property of its constituent atoms or molecules (the electronic or ionic polarizability, \(\alpha\)). The equation is conventionally written as: \[ \frac{\varepsilon_r - 1}{\varepsilon_r + 2} = \frac{N\alpha}{3\varepsilon_0} \] Where:
• \(\varepsilon_r\) is the relative dielectric constant of the material.
• \(N\) is the number of atoms or molecules per unit volume (number density).
• \(\alpha\) represents the total polarizability of an individual atom/molecule.
• \(\varepsilon_0\) is the permittivity of free space (\(8.854 \times 10^{-12} \text{ F/m}\)).

Step 1: Derivation background and physical significance.
When a dielectric medium is placed in an external electric field, the local field (\(E_{\text{local}}\)) experienced by an individual atom inside the material is different from the applied macroscopic electric field (\(E\)). According to the Lorentz field derivation for a spherical cavity: \[ E_{\text{local}} = E + \frac{P}{3\varepsilon_0} \] Where \(P\) is the macroscopic polarization of the medium, defined by the total dipole moment per unit volume: \[ P = N p_{\text{induced}} = N \alpha E_{\text{local}} \] Substituting the expression for \(E_{\text{local}}\) gives: \[ P = N \alpha \left( E + \frac{P}{3\varepsilon_0} \right) \] We also know from standard electromagnetic relationships that polarization relates to the macroscopic dielectric constant via: \[ P = \varepsilon_0 (\varepsilon_r - 1) E \] By equating these two expressions and solving algebraically to eliminate \(P\) and \(E\), we get: \[ \varepsilon_0 (\varepsilon_r - 1) E = N \alpha E \left( 1 + \frac{\varepsilon_r - 1}{3} \right) \quad \Rightarrow \quad \varepsilon_0 (\varepsilon_r - 1) = N \alpha \left( \frac{\varepsilon_r + 2}{3} \right) \] Rearranging terms yields the Clausius–Mossotti relation: \[ \frac{\varepsilon_r - 1}{\varepsilon_r + 2} = \frac{N\alpha}{3\varepsilon_0} \]

Step 2: Identifying the correct statement option.
Looking closely at the terms inside the expression, the left side is a function of the dielectric constant (\(\varepsilon_r\)), and the right side depends linearly on the microscopic polarizability (\(\alpha\)). This directly maps to option (B).