Insulators — Question 4

Question 4

A microscopic polarization model predicts the dielectric constant

A dilute dielectric contains N polarizable units per volume, each with dipole $p=\alpha E$ along the macroscopic field. Neglect local-field corrections, so $P=N\alpha E$ and $D=\varepsilon_0E+P$.

1. Derive relative permittivity.

2. For $N=2.00\times10^{28}\,\mathrm{m^{-3}}$, $\alpha=8.854\times10^{-40}\,\mathrm{C\,m^2/V}$, and $\varepsilon_0=8.854\times10^{-12}\,\mathrm{F/m}$, find $\varepsilon_r$ and P at $E=10^5\,\mathrm{V/m}$.

Question 4 - Solution

A microscopic polarization model predicts the dielectric constant

(a) Method and calculation

$$D=(\varepsilon_0+N\alpha)E=\varepsilon_0\varepsilon_rE.$$$$\boxed{\varepsilon_r=1+N\alpha/\varepsilon_0}.$$

(b) Interpretation and further calculation

$$\boxed{\varepsilon_r=1+2.00=3.00}.$$$$\boxed{P=N\alpha E=1.7708\times10^{-6}\,\mathrm{C/m^2}}.$$

Physical check

Polarization displaces bound charges without requiring sustained conduction. The neglect of local fields is a supplied model assumption, especially restrictive at high density.