Pre-Quantum Electrodynamics

• PM 10.1
• Gr 4.4

Let's quickly pause to give a classification of the types of responses that might be expected from materials.

Permanent polarization: here, \({\bf P} \neq 0\) even if \({\bf E} = 0\). These are the electrets.

Nonlinear dielectrics: for these, there are non-negligible corrections to the purely linear behaviour. One can express these nonlinearities through an expansion of the form \[ P_i = \sum_j \alpha_{ij} E_j + \sum_{jk} \beta_{ijkl} E_j E_k E_l + ... \] (even-order terms usually absent, but could be there in principle).

Linear dielectrics: for these, the only significant term is the linear term. The most general form is then \[ P_i = \varepsilon_0 \chi_{ij} E_j \] where \({\bf \chi}\) is the electric susceptibility tensor (it's a tensor of rank 2), which cannot depend on \({\bf E}\) (otherwise we go back to the nonlinear case). This is common in crystals, leading to phenomena such as double refraction.

Linear isotropic dielectrics: at a given point, the electrical properties of the dielectric are independent of the direction of \({\bf E}\) (isotropy). Liquids fall into this category. We here have